EASL – Vol 4 – Issue 4 (2021) – PISRT https://old.pisrt.org Sun, 23 Jan 2022 16:08:45 +0000 en-US hourly 1 https://wordpress.org/?v=6.7 Predicting COVID-19 cases, deaths and recoveries using machine learning methods https://old.pisrt.org/psr-press/journals/easl-vol-4-issue-4-2021/predicting-covid-19-cases-deaths-and-recoveries-using-machine-learning-methods/ Thu, 30 Dec 2021 13:19:12 +0000 https://old.pisrt.org/?p=6148
EASL-Vol. 4 (2021), Issue 4, pp. 43 - 49 Open Access Full-Text PDF
Mohamed Lounis, Farhan Mohammad Khan
Abstract:In the presented work we applied three machine learning techniques to forecast and predict COVID-19 cases, deaths ad recoveries numbers in Algeria for the next six months using data from February 25th, 2020 to April 26th , 2021. These models are represented by the Gaussian process regression (GPR), the support vector machine (SVM) and the decision tree (DT). The plotting results and parameters evaluation pointed out that the Gaussian Process Regression (GPR) has the best performance. Prediction with this model showed that the number of cases, deaths and recoveries will increase in the next months Algeria recording a peak in the month of August and the curve will tend to decrease later.
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Engineering and Applied Science Letter

Predicting COVID-19 cases, deaths and recoveries using machine learning methods

Mohamed Lounis\(^1\), Farhan Mohammad Khan
Department of Agro-veterinary Science, Faculty of Natural and Life Sciences, University of Ziane Achour, BP 3117, Road of Moudjbara, Djelfa 17000, Algeria.; (M.L)
Department of Civil Engineering, BITS Pilani, Pilani Campus, India.; (F.M.K)

\(^{1}\)Corresponding Author: lounisvet@gmail.com

Abstract

In the presented work we applied three machine learning techniques to forecast and predict COVID-19 cases, deaths ad recoveries numbers in Algeria for the next six months using data from February 25th, 2020 to April 26th , 2021. These models are represented by the Gaussian process regression (GPR), the support vector machine (SVM) and the decision tree (DT). The plotting results and parameters evaluation pointed out that the Gaussian Process Regression (GPR) has the best performance. Prediction with this model showed that the number of cases, deaths and recoveries will increase in the next months Algeria recording a peak in the month of August and the curve will tend to decrease later.

Keywords:

COVID-19; Machine learning; Gaussian process regression; Support vector machine; Decision tree.

1. Introduction

Seventeen months after its emergence, the coronavirus disease 2019 (COVID-19) continues its propagation affecting more than 165 million patients leading to more than 3.4 million deaths surpassing all expectations. Algeria has seen its former case emerged on February, 25th 2020. After multiple facets of the epidemiological curve, the number of cases has attained 125,896 subjects. This number seems to be lower than the number of cases reported in the bordered countries like Morocco (515,758 cases) and Tunisia (329,925 cases). With 3,395 deaths and 87,746 recovered persons these numbers determine until now a fatality rate of 2.7 % and a cured rate of 69.7% respectively [1]. To understand the epidemiological traits of this disease and to predict its evolution and its probable end-point multiple approaches have been used in Algeria and around the world. These approaches varied from epidemiological and mathematical/statistical to deep learning/machine learning models [2]. In this way, machine learning models are of great importance [3]. These tools which have proved their role in different complicated problems in different field in the last years including health, agriculture, engineering, sport, climate and robotics [4] have been widely used in the current context of COVID-19 [5,6,7,8].

Among these models we can find auto regressive integrated moving average (ARIMA) models [5], BSTS (Bayesian structural time series) [4], simple RNN (recurrent neural network) [7], artificial neural network (ANN) [8], long-short term memory (LSTM) [9], linear regression [10], adaptive neurofuzzy inference system (ANFIS) [11], least absolute shrinkage and selection operator (LASSO) regression [12], CUBIST (cubist regression) [13], Gaussian process regression (GPR) [14], exponential smoothing (ES) [15], random forest (RF) [8,13,16], ridge regression (RIDGE) [13], support vector machine (SVM) [8,13], Naïve bayes (NB) [8], decision tree (DT) [8], box-jenkins method [17], variational auto encoder (VAE) [7,10], gated recurrent units (GRU) [7,9] and multi-layer perceptron (MLP), models [18].

2. Related works

Multiple researchers have been carried out using machine learning methods in the actual COVID-19 context. Below are cited some conducted researches using Gaussian process regression (GPR), support vector machine (SVM) and decision tree (DT):

After analyzing historical COVID-19 data, Velásquez and Lara [14] forecasted COVID-19 affection with reduced-space Gaussian process regression associated to chaotic dynamical systems using obtained information of the two first months (January 21, to April 12, 2020). Their work demonstrated the usefulness of the Gaussian models in the COVID-19 infection prediction.

In their study, Ribeiro et al., [13] set as objectives the evaluation of the performance of multiple models like autoregressive integrated moving average (ARIMA), cubist regression (CUBIST), random forest (RF), RIDGE regression, support vector regression (SVR), and stacking-ensemble learning in a COVID-19 cases short projection of 1, 3 and 6 days of the ten most affected states in Brazil. The performance evaluation has given the following classification: SVR, stacking-ensemble learning, ARIMA, CUBIST, RIDGE, and RF models.

In a comparison made by Ball [14], the support vector machines (SVM) has demonstrated higher performance than linear regression, multi-layer perceptron, random forest models in predicting COVID-19 trend in USA, Germany and the global. In Mexico logistic regression, decision tree, support vector machine, naive Bayes, and artificial neutral network to study COVID-19 cases by Muhammed et al., [8]. The researchers observed that decision tree, support vector machine and Naïve bayes model have the highest accuracy (94.99%), sensitivity (93.34%) and specificity (94.30%) respectively.

Daniyal et al., [19] in Pakistan, compared the performance of three regression models including linear, logarithmic, and quadratic in modeling of COVID-19 deaths using data of about 5 months. Later, they deduced that the rate of mortality will decrease by the end of October as shown by the quadratic regression model which has shown the best performance.

Prediction of COVID-19 mortality in Korea was the main objective of the study of An et al., [12]. The study begun by testing the least absolute shrinkage and selection operator (LASSO), linear support vector machine (SVM), SVM with radial basis function kernel, random forest (RF), and k-nearest neighbors. As a result, LASSO and linear SVM has shown high sensitivities (90.7% and 92.0%, respectively) and specificities (91.4% and 91.8%, respectively). In the same country, Das et al., [20] predicted mortality in 3,524 COVID-19 patients using five machine learning models (logistic regression, support vector machine, K nearest neighbor, random forest and gradient boosting). The logistic regression model was proposed as an open-source online prediction tool for decision-making due to its high performances.

3. Methodology

In this paper, COVID-19 time series data available till 26th April 2021 in Algeria were used for a projection of daily cases, deaths and recoveries for the next six months using three machine learning techniques that are Gaussian process regression (GPR), support vector machine (SVM) and decision tree (DT). Data regarding the number of cases reported in Algeria, were extracted from Worldometer. The COVID-19 curve evolution is shown in Figure 1.

Figure 1. COVID-19 curve evolution in Algeria. 

4. Results and discussion

In the current wok, three machine learning approaches were applied to predict the number of COVID-19 cases, deaths and cured persons in Algeria. We first evaluated the forecast performance of these models by the estimation of parameters like the root mean square error (RMSE), the mean square error (MSE), the mean absolute error (MAE), and the coefficient of determination (R2) values for COVID-19 daily cases. Results showed that if the three models have shown acceptable performances (Table 1), the GPR model was the most efficient showing an RMSE of 31.126 and an R2 of 0.98. These parameters were calculated by comparing actual/predicted cases after a 10-fold cross-validation. Figures 2, 3 and 4 showed response plots the three models GPR, SVM, and DT respectively. Figures 5, 6 and 7 present the predicted/observed pattern of each model.

Table 1. Performance parameters of the three model.
Model parameters GPR Quadratic SVM DT
RMSE 31.126 42.485 37.93
R-squared 0.98 0.97 0.97
MSE 968.83 1804.9 1438.7
MAE 18.334 26.268 22.996

Figure 2. Plotting response of the GPR model.

Figure 3. Plotting response of the SVM model.

Figure 4. Plotting response of the DT model.

Figure 5. Predicted/Observed cases pattern of the GPR model.

Figure 6. Predicted/Observed cases pattern of the SVM model. 

Figure 7. Predicted/Observed cases pattern of the DT model.

We then, used available data till 26th April 2021, of daily confirmed, recovered, and deceased cases of COVID-19 cases in Algeria and forecasted them using the three models for the next six months. Predicted daily new cases, recovered and dead persons are shown in Figures 8, 9 and 10 respectively. Results showed that confirmed cases will increase in the next months and will start their declining from the first week of October according to the GPR model. The number of recoveries (Figure 9) and deaths (Figure 10) follow generally the same evolutionary curve.

Figure 8. Forecast of Daily Confirmed Cases using machine learning methods.

Figure 9. Forecast of Daily Recovery Cases using machine learning methods. 

Figure 10. Forecast of Daily Deceased Cases using machine learning methods.

It is to mention that these projections were done without considering the effect of preventive measures which are considered to be the same in the next months. Prediction performance could be ameliorated if their effect will be added. The performance of our models has shown a high value for the coefficient of determination of the three models used in this study. As a comparison we can show that our models have better performances in term of R2 than other models like ARIMA (0.95) [21] and ANFIS (0.956) [22]. Other models like MPL-ICA (0.9971) [22], logistic regression (0.996) [23] and lasso regression (1.0) [24] have demonstrated higher performances.

5. Conclusion

As a conclusion, we used in this work three machine learning methods to forecast COVID-19 cases, deaths and recoveries in Algeria. Results showed that all models showed acceptable performances and the exponential GPR network was the most efficient. Prediction with this model showed that Algeria will probably recorded a third wave with a peak in the month of August 2021.

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of Interest

''The authors declare no conflict of interest.''

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Dependence of reflectance on angular deposition and film thickness of ZnS/Ag nanolayers https://old.pisrt.org/psr-press/journals/easl-vol-4-issue-4-2021/dependence-of-reflectance-on-angular-deposition-and-film-thickness-of-zns-ag-nanolayers/ Thu, 30 Dec 2021 13:08:54 +0000 https://old.pisrt.org/?p=6146
EASL-Vol. 4 (2021), Issue 4, pp. 26 - 42 Open Access Full-Text PDF
Edward Bwayo, Willy Okullo, Daniel Mukiibi, Denis Okello, Robert Lugolole, Tumps Winston Ireeta
Abstract:This paper presents the spectral reflectance of thermally evaporated ZnS/Ag nanostructures. The coating of ZnS/Ag nanostructures was performed in two steps while varying the film thickness and deposition angle. Silver metal wire (99.99% purity) was heated under vacuum at a pressure of \(2.5 \times 10^{-5}\) mBars and deposited on glass slide substrates in the diffusion pump microprocessor vacuum coater (Edwards AUTO 306). Pieces of zinc sulphide (99.99% purity) were heated and deposited to the glass slides previously coated with silver to form the ZnS/Ag/glass composite. The optical reflectance of the samples was studied by the UV/Vis/NIR spectrometer (Perkin Elmer Lambda 19) with UV-WinLab software. The reflectance was measured at angles of incidence between \(15^o\) and \(75^o\). Spectrophotometric studies showed that reflectance decreased with decrease in film thickness and decreased with increase in deposition angle of silver nanoparticles. The reflectance of ZnS/Ag nanostructures decreased with increase in deposition angle of zinc sulphide.
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Engineering and Applied Science Letter

Dependence of reflectance on angular deposition and film thickness of ZnS/Ag nanolayers

Edward Bwayo\(^1\), Willy Okullo, Daniel Mukiibi, Denis Okello, Robert Lugolole, Tumps Winston Ireeta
Department of Physics, School of Physical Sciences, College of Natural Sciences, Makerere University, Uganda.; (E.B & W.O & D.M & D.O & R.L & T.W.I)

\(^{1}\)Corresponding Author: bwayoedward@gmail.com

Abstract

This paper presents the spectral reflectance of thermally evaporated ZnS/Ag nanostructures. The coating of ZnS/Ag nanostructures was performed in two steps while varying the film thickness and deposition angle. Silver metal wire (99.99% purity) was heated under vacuum at a pressure of \(2.5 \times 10^{-5}\) mBars and deposited on glass slide substrates in the diffusion pump microprocessor vacuum coater (Edwards AUTO 306). Pieces of zinc sulphide (99.99% purity) were heated and deposited to the glass slides previously coated with silver to form the ZnS/Ag/glass composite. The optical reflectance of the samples was studied by the UV/Vis/NIR spectrometer (Perkin Elmer Lambda 19) with UV-WinLab software. The reflectance was measured at angles of incidence between \(15^o\) and \(75^o\). Spectrophotometric studies showed that reflectance decreased with decrease in film thickness and decreased with increase in deposition angle of silver nanoparticles. The reflectance of ZnS/Ag nanostructures decreased with increase in deposition angle of zinc sulphide.

Keywords:

ZnS/Ag nanostructures; Film thickness; Deposition angle; Reflectance.

1. Introduction

The reflectance properties of surfaces of optical components in several devices in use today require careful tailoring to meet the requirements of technological advances. Reflective mirrors for use in domestic and industrial housing units in hot climates should have the potential to reflect a high percentage of infrared radiation but transmit a desirable amount of visible light. However, these reflective surfaces are susceptible to environmental degradation. This is a fundamental setback in optical applications. It is very vital for the reflective surfaces to be protected and made resistant to weather and chemicals in the environment [1,2]. Most reflective mirrors are coated with metals films such as silver, copper, gold, aluminium to improve there reflective characteristics. These metals at nano-scale have remarkably low optical absorption in the visible region and high reflectance in the infrared region and frequency dependent refractive index [3,4,5]. The additional technical characteristic of silver at nano-scale is that of high chemical and optical activity with the capacity to fuse with dielectric materials to form hybrid semiconductor layers [6,7,8].

Nevertheless, whenever materials such as silver, gold, copper, and aluminium are exposed to the environment, they tend to age out due to environmental degradation. They thus have to be protected by coating them with a suitable material in form of a transparent dielectric coating material. A dielectric material is a transparent film which has strong ionic or directed covalent bonds. Most dielectrics are transparent to visible and/or infrared light. When an electromagnetic radiation falls on a dielectric material, it interacts with valency band electrons. The valency band electrons absorb energy and under go electronic transitions to the conduction band. This changes the the optical parameters of the dielectric films [9,10]. The change in optical properties can be quantified by solving Maxwell equations at the boundary between different optical media subject to boundary conditions [10]. Dielectrics used for this purpose include zinc oxide, magnesium floride, zinc sulphide, indium tin oxide among others [11,12]. The optical characteristics of zinc oxide, magnesium floride, zinc sulphide, indium tin oxide are among other factors governed by deposition conditions which include deposition pressure, temperature, deposition rate and deposition angle [13]. Other factors like film thickness are required when tuning the spectral reflective properties of nanostructured components for optical, electrooptic, telecommunications and architectural applications in energy conservation [14].

The energy efficiency of buildings for residential and industrial storage facilities in most instances is regulated with external cooling mechanisms such as electric powered air conditioners. The cooling systems are very expensive to maintain on the short and long run. The architectural windows and doors are part of the alternative natural cooling options that are cheap to install and maintain. Several buildings are fitted with float glass windows whose spectra properties have not been quantified by any spectrophotometric processes. However, glass panes used on windows should be designed to reflect infrared radiation but at the same time allow a considerable amount of visible light to pass through them. On this note, the optical structural adjustment of the window surface has always not been a simple task to under take [15,16,17]. Many attempts have been made to modify the optical properties of glass by applying transparent polystyrene or coating the glass surface with transparent dielectric or metallic thin films. During thin films deposition processes, it is vital to control the vapour flux coming from evaporating materials. The evaporating material may reach the substrate normally or at an oblique angle. At normal incident, the vapour flux upon deposition tends to form a relatively uniform homogeneous film. Nevertheless, the growth of very thin films is always accompanied by the formation of atomic islands on the surface of the dielectric [18,19].

However, during oblique deposition, the adhesion of initial particles on the substrate may hinder the direct impact of preceding atoms on the substrate. Along the direction of initially deposited atoms, columns are formed that shadow the incoming vapour atoms from reaching directly to the substrate surface [20,21]. This leads to the formation of columner structures which restrict adatom mobility and surface diffusion of deposited particles. Oblique deposition therefore, produces rough surfaces but also changes film micro-structure [22,23]. Atomic islands are also formed for very thin nanostructures (\(< 12\) nm) [24]. The microstructure may have pores or internal voids that affect the density, optical reflectance, transmittance and optical absorbance of the thin films [25,26].

When an electromagnet radiation is incident on a thin film coated on glass surfaces, it is either transmitted with attenuation or upon reflection may surfer interference effects due multiple reflections from different media interfaces. However, the extent of these effects depend on the values of optical constants, angle of incidence, microstructure and surface roughness of the reflective surface. Additionally, reflectance of a surface may be affected by the relative motion between the Earth and the Sun. The intensity of electromagnetic radiation from the Sun changes with time along side other factors like, location, the Sun angle and solar shading. Hence, it is very important to measure and establish the reflectance values for different angles of incident radiation for every reflecting surface. This information is very important in structural design of fenestration especially transparent windows and doors [27]. Although several attempts have been made to control thermal conductivity through walls and the roof of buildings [28,29], thermal radiation transfer from the ambient through the transparent windows and doors into the building envelope remains one of the biggest challenges in the housing industry. Thus in this paper, the authors present the spectral reflectance of ZnS/Ag nanostructures hinged on film thickness, deposition angle and angle of incidence of electromagnetic radiation.

2. Experimental procedures

The deposition of ZnS/Ag thin films was performed in two steps while varying the film thickness and deposition angle. Silver metal wire (99.99% purity) was heated on a refractory tantalum boat under vacuum at a pressure of \(2.5 \times 10^{-5}\) mBars and deposited on glass slide substrates in the diffusion pump microprocessor vacuum coater (Edwards AUTO 306). The heating current was slowly raised to a current of 38 A because silver metal has low melting point of 961\(^o\)C. The evaporation rate of 1 nms\(^{-1}\) was used. Glass slides were fixed on a rotary holder placed 11 cm away from the refractory boat vertically above it. The substrate holder together with the glass slide was rotated at an angle \(\theta\) about the horizontal so that the vapor is incident at an angle \(\theta\) to the substrate normal (the angle between the normal to the substrate and direction of incidence of the evaporated atoms). The setup is shown in Figure 1 below. Three sets of samples of silver films of thickness 4 nm were deposited at different angles of \(\theta= 0^o\), 30\(^o\) and 60\(^o\). This was repeated for each of thickness of 7, 10 and 15 nm onto glass substrate.

Figure 1. Setup for Vacuum vapour deposition.

The pieces of ZnS (99.99% purity) were heated in a molybdenum boat with source cover to reduce the spreading of the vapor. Because ZnS has a high melting point of 1830\(^o\)C, the heating current was increased to 56 A. The ZnS was heated and deposited to film thickness of 4 nm at vapour incidence angle \(\theta = 0, 30\) and \(60^o\) to the glass slides previously coated with silver to form the ZnS/Ag/glass multilayer system shown in Figure 2.

This procedure was repeated for thicknesses 7, 10 and 15 nm. The thickness monitor was calibrated separately for each of the materials in the vacuum chamber. The thickness of the films were measured by a thickness monitor connected to a quartz crystal monitoring system placed inside the diffusion chamber [30]. In vacuum vapour deposition, it is very had to produce films that have perfectly smooth surfaces with high homogeneity and uniform thickness. This effect compromises the accuracy of the spectrophotometric results since the measurement of transmittance and reflectance can hardly be done at exactly the same spot on the specimen [31].

Figure 2. An expanded representation of the ZnS/Ag/glass composite

The optical reflectance of the samples were studied by the UV/Vis/NIR spectrometer (Perkin Elmer Lambda 19) with UV-WinLab software. This is a double beam instrument covering the ultraviolet, visible and near infrared spectral wavelength. Baseline measurement using a clean piece of substrate was done. Reflectance was measured at angles of incidence; 15\(^o\), 30\(^o\), 45\(^o\), 60\(^o\), and \(75^o\) of the incident radiation to the optical system in the wavelength range of 250 to 2500 nm [32]. Since it is very had to measure reflectance at normal incidence, the lowest angle of incidence for reflectance measurements was \(15^{o}\) subject to the limitations of the spectrometer model used [33].

3. Results and discussion

3.1. Effect of thickness and deposition angle on reflectance of ZnS(0)/Ag films

The spectrophotometric measurements on reflectance was analysed on specimens fabricated with different thickness and deposition angles. The reflectance of normally deposited ZnS(0)/Ag(0) nanostructures was relatively low (ranged between 5% - 35%) in the visible range (400-780 nm) of the electromagnetic spectrum followed by a rise in reflectance towards the infrared at about \(\lambda = 800\) nm, Figure 3 (a). The reflectance results showed remarkable interference effects in the infrared region in all the samples. These effects were dependent on the film thickness and wavelength of the electromagnetic radiation. The interference effects ware observed in the infrared region 800-2200 nm and the tended to disappear or diminshed as the wavelength approached the the upper edge of the visible spectrum. This observation was also reported in the obliquely deposited Nb\(_2\)O\(_5\) [34] and ZnO [35] thin films.

Figure 3. Effect of thickness and deposition angle of Ag on reflectance of ZnS/Ag nanostructures. The deposition angle of ZnS films was kept constant at normal incidence, \((0^{o})\).

The interference effects in the infrared region were more intense in the (15 nm)ZnS(0)/Ag(0) nanostructures. These effects were as a result of reflection from different optical interfaces of air-ZnS, ZnS-Ag, and Ag-glass interfaces [36]. The interference effects infer that the prepared samples were relatively smooth and uniform [37,38]. There was a further rise in reflectance to about 54% in the infrared region at about \(\lambda = 1300\) nm. It was also observed that the reflectance decreased with decrease in film thickness of ZnS/Ag multilayer films Figure 3 (a). However, there was a big drop in reflectance with film thickness from (7 nm)ZnS(0)/Ag(0) to (4 nm)ZnS(0)/Ag(0). The decrease in reflectance was partly due to decrease in film thickness and formation of atomic islands during the deposition of very thin metal films and dielectric nanostructures [39,40].

For wavelength below \(\lambda =400\) nm, there were interference minima in reflectance of the samples. These effects according to Wlodarski et al., [41] and Zhou and Liu [42] were as a result of increase in absorption due to interband electronic transitions of the zinc and silver atoms. Thin nanostructures of (4 nm)ZnS(0)/Ag(0), had very low reflectance values i.e. between 2.6% - 10.5% in the visible region and reflectance increased to a maximum value of 22.1% at \(\lambda =2000\) nm in the infrared wavelength as shown in Figure 3 (a).

When the deposition angle of Ag nanoparticles was increased to \(30^o\) in the ZnS/Ag composite i.e. ZnS(0)/Ag(30) Figure 3 (b), the reflectance decreased progressively as the thickness of Zn/Ag films dencreased. The reflectance of (15 nm)ZnS(0)/Ag(30) in the visible region (400-800 nm) was between 13% -31%. In the infrared region, the reflectance had maximum value of about 50.5%. Highest reflectance values were recorded for thin films of (15 nm)ZnS/Ag. This trend was also observed in other thin films of thickness (7 nm)ZnS/Ag and (4 nm)ZnS/Ag but with reduced values of reflectance in the in the visible and infrared wavelengths. A significant drop in reflectance was recorded for (4 nm)ZnS/Ag nanostructures. This was due to increased optical absorption and transmittance of the (4 nm)ZnS/Ag nanostructures. Additionally, the increase the deposition angle increased film discontinuities and roughness which in turn increased optical absorption due to aggregation of silver islands on the glass substrate [43,44]. In a separate study on very thin nanolayers by Hu et al., [45] destructive interference enhances optical absorption and transmittance but suppresses reflectance of the electromagnetic waves. There were interference effects which alternated with decreasing thickness of the ZnS/Ag composite and the deposition angle of silver nanoparticles. This had pronounced effects on reflectance of near infrared radiation. On a whole, reflectance decreased with decrease in film thickness and decreased with increase in deposition angle of Ag.

Further increase in deposition angle of Ag from ZnS(0)/Ag(30) to ZnS(0)/Ag(60) Figure 3 (c), decreased the reflectance. However, the nanostructure response to reflectance was not so sensitive to the small increase in deposition angle of silver metal films. The reduction in reflectance according to Liedtke et al., [43] was attributed to atomic shadowing that created areas with reduced grain size of silver atoms. This created imperfections in the Ag films. These imperfections generate grain boundaries in the thin films that decreased the reflectance of the thin films. As the number of grain boundaries increases with deposition angle, there is reduction in specular reflection for very thin films due to scattering of incident light. The increase in reflectance at wavelengths \(\lambda > 800\) nm was related to the increase of the Drude-like absorption in which the amount of voids in the metal phase increases [46].

3.2. Effect of thickness and deposition angle on reflectance of ZnS(30)/Ag films

The reflectance of obliquely deposited ZnS(30)/Ag(0) films presented in Figure 4 (a), can be described from two perspectives. Thus, the reflectance decreased with decrease in film thickness but also the reflectance increased along with the increase in wavelength of electromagnetic radiation.

Figure 4. Effect of thickness and deposition angle of Ag on reflectance of ZnS/Ag films. The deposition angle of ZnS nanoparticles was maintained at (\(30^o\)) for three deposition angles of Ag nanoparticles.

On the other hand, the deposition angle of ZnS had little impact on reflectance in the visible wavelength. The reflectance values in the visible region at about \(\lambda=800\) nm were 25.2% for (10 nm)ZnS(30)/Ag(0), 18.8% for (7 nm)ZnS(30)/Ag(0) and 6.4% for (4 nm)ZnS(30)/Ag(0) samples. However, the reflectance in the infrared wavelength decreased with increase in deposition angle of zinc sulphide. At about \(\lambda =1800\) nm, the reflectance values were 45.8% for (10 nm)ZnS(30)/Ag(0), 37.0% for(7 nm)ZnS(30)/Ag(0) and 9.4% for (4 nm)ZnS(30)/Ag(0) samples. This implies that increasing the deposition angle of ZnS increased the optical absorption in the infrared spectral region.

Reflectance of the samples in Figure 4 (b) decreased with decrease in film thickness of the ZNS/Ag composite. When the deposition angle of Ag was increased from (ZnS(30)/Ag(0) to (ZnS(30)/Ag(30), the reflectance of the multilayer was further reduced. This decrease was observed both in the visible wavelengths at about \(\lambda=800\) nm was as follows: 24.0% for (10 nm)ZnS(30)/Ag(30), 17.0% for (7 nm)ZnS(30)/Ag(30) and 6.3% for (4 nm)ZnS(30)/Ag(30). The reflectance values obtained in the infrared wavelengths at \(\lambda=1800\) nm were 45.1% for (10 nm)ZnS(30)/Ag(30), 36.5% for(7 nm)ZnS(30)/Ag(30) and 9.5% for (4 nm)ZnS(30)/Ag(30) specimens. The reflectance values in the infrared region were higher than those in the visible region. The reflectance of (4 nm)ZnS(30)/Ag(30) was exceptionally very low. This was largely contributed by the small film thickness, the oblique deposition angle of ZnS and silver thin films and optical absorption by the discontinuous islands of silver formed on glass substrate.

However, as the deposition angle of both ZnS and Ag was increased to ZnS(30)/Ag(60) Figure 4 (c), the reflectance of (10 nm)ZnS(30)/Ag(60), (7 nm)ZnS(30)/Ag(60) and (4 nm)ZnS(30)/Ag(60) further decreases with film thickness. The reflectance of (10 nm)ZnS(30)/Ag(60) and (7 nm)ZnS(30)/Ag(60) samples increased with increase in deposition angle of Ag. The reflectance of (4 nm)ZnS(30)/Ag(60) was extremely low at an average of \(< 3%\) in the visible region and \(< 5%\) in the infrared wavelength.

3.3. Variation of thickness and deposition angle on reflectance of ZnS(60)/Ag films

The refleectance of the specimens increased with increase in film thickness. The reflectance was observed to increase from the visible wavelength to infrared wavelength of the electromagnetic spectrum (Figure 5 (a)). Interference effects were observed and alternated with film thickness. The reflectance of (10 nm)ZnS(60)/Ag(0), (7 nm)ZnS(60)/Ag(0) and (4 nm)ZnS(60)/Ag(0) in the visible region had maximum values of 29.9%, 24.1%, 7% respectively at about \(\lambda=800\) nm. The reflectance values of (10 nm)ZnS(60)/Ag(0), (7 nm)ZnS(60)/Ag(0) and (4 nm)ZnS(60)/Ag(0) in the infrared region at \(\lambda=1800\) nm were 54.9%, 41.9% and 7.4% respectively. This was attributed to low deposition angle of Ag though the deposition angle of ZnS is high. This infers that for these particular samples, the deposition angle of ZnS has a small effect on reflectance of the ZnS/Ag multilayer films. The reflectance of (4 nm)ZnS(60)/Ag(0) was very low \(< 5%\) both in the visible and infrared spectral regions. In this sample, the most influencing factor on reflectance was the film thickness and discontinuities in the nanostructured - silver films.

When the deposition angle of Ag was increased from ZnS(60)/Ag(0) to ZnS(60)/Ag(30), (Figure 5 (b)), The reflectance of the samples increased with increase in film thickness. The reflectance values further decreased in the visible as follows: 18.3% for (10 nm)ZnS(60)/Ag(30), 16.6% for (7 nm)ZnS(60)/Ag(30) and 5.1% for (4 nm)ZnS(60)/Ag(30). In the infrared regions, reflectance reduced as follows: 37.7% for (10 nm)ZnS(60)/Ag(30), 27.4% for (7 nm)ZnS(60)/Ag(30) and 6.0% for (4 nm)ZnS(60)/Ag(30). Interference effects observed along the entire electromagnetic spectrum. This trend is not observed in the (4 nm)ZnS(60)/Ag(30) multilayer film and its reflectance values are below 6% in the visible region. Interference effects were observed to diminish since much of the light was either absorbed or transmitted by the ZnS/Ag multilayer nanostructures.

The reflectance of the obliquely deposited ZnS(60)/Ag(60) samples decreased with decrease in film thickness as shown in Figure 5 (c). The increase in deposition angle of silver further decreased reflectance in the visible region. Interference effects are intense in the (10 nm)ZnS(60)/Ag(60), (7 nm)ZnS(60)/Ag(60) and (4 nm)ZnS(60)/Ag(60) nanostructures. The reflectance values in the visible wavelength for (10 nm)ZnS(60)/Ag(60), (7 nm)ZnS(60)/Ag(60) and (4 nm)ZnS(60)/Ag(60) mulitlayers were 17%, 13.7% and 10.5% respectively. In the infrared region, the reflectance of (10 nm)ZnS(60)/Ag(60) and (7 nm)ZnS(60)/Ag(60) nanolayers increased to 31% and 25.1% respectively.

Figure 5. Effect of thickness and deposition angle of Ag on reflectance of ZnS/Ag nanastructures when ZnS was deposited at (\(60^o\)) to the substrate normal

3.4. Effect of angle of incident radiation and deposition angle on reflectance of (10 nm)ZnS/Ag(0) films

Generally, reflectance increased from the visible towards the infrared wavelengths in the entire electromagnetic spectrum Figure 6 (a). The reflectance of electromagnetic radiation for normally deposited films i.e. ZnS(0)/Ag(0), increased from \(i= 15^o\) to \(i= 30^o\). The reflectance then decreased progressively from \(i=45^o\) to \(i=75^o\). Reflection peaks were observed at 700 nm for different angles of incidence. For \(i= 15^o\), the peak was observed at 36.8%. And for \(i= 30^o, 45^o, 60^o\) and \(75^o\) the peaks were at 52.3%, 22.1%, and 16.6% respectively. At \(\lambda > 800\) nm, the reflectance peaks were obtained in the range of \(\lambda = 1100 - 1300\) nm. In the infrared wavelengths, very low reflectance values (\( < 35% \)) were recorded at \(i=75^o\). The decrease in reflectance for angles of incidence \(i= 45^o\) to \(75^o\) was due to high attenuation and optical absorption of the infrared light by the discontinuous nanolayers. However, interference effects were visible in the entire spectrum. In the event that this material was to be adopted for use as a window pane, then the angle of inclination of the window pane to the direction of incident sun light has to be determined for minimum or enhanced reflectance.

When the deposition angle of ZnS was increased from ZnS(0)/Ag(0) to ZnS(30)/Ag(0) Figure 6 (b), there was a slight decrease in reflectance both in the visible and infrared wavelength. This implies that increasing the deposition angle of ZnS decreased the reflectance of the ZnS/Ag nanofilms. Comparatively, these results were lower than those obtained in Figure 6 (a). At \(\lambda = 720\) nm, the peak reflectance values at \(i=30^o, 45^o, 60^o\) and \(75^o\) were 28.0%, 59.7%, 22.9%, 21.8%, and 17.0% respectively. Reflectance then started to increase at \(\lambda=800\) nm towards the infrared spectral region. Very strong interference effects were observed in the infrared region.

Further increase in the deposition angle of ZnS from ZnS(30)/Ag(0) to ZnS(60)/Ag(0) Figure 6(c), the reflectance values increased from the visble wavelength towards the infrared wavelengths. The reflectance peaks moved toward the short wavelength. At \(\lambda = 660\) nm, the peak reflectance values for \(i= 30^o\), \(45^o\), \(60^o\) and \(75^o\) were 29.2%, 59.7%, 22.9%, 21.8% and 17.0% respectively. After these peak values, there was a drop in reflectance at the upper edge of the visible spectrum. The reflectance values increased from \(i= 15^o\) to \(i= 30^o\). The reflectance then decreased from \(i= 45^o\) to \(i= 75^o\). The reflectance of visible light for which the angle of incidence \(i=30^o\), had a maximum value of about 59.7% at wavelength, \(\lambda=660\) nm. After this point, reflectance then dropped to about 46% at wavelength \(\lambda=883\) nm and it then increased progressively to about 82% at \(\lambda=2400\) nm. The reflectance for \(i=15, 45, 60\) and \(75^o\) in the visible region was quite different from the reflectance whose angle of incidence was \(i=30^o\). However, in the infrared region the the reflectance values did not differ so much as the wavelength increased. The low reflectance values revealed that the specimens ware transparent to visible light but highly reflective to infrared light. The interference effects in the infrared region (\(\lambda > 800\) nm) ware minimal compared to those obtained in Figure 6 (b).

Figure 6. Effect of angle of incident radiation and deposition angle of ZnS on reflectance of (10 nm) ZnS/Ag films. The silver nanoparticles in the ZnS/Ag nanostructure were normally deposited on glass at (\(0^o\)) followed by ZnS at different deposition angles; (a) at (\(0^o\)), (b) at (\(30^o\)) and (c) at (\(60^o\)).

3.5. Effect of angle of incident radiation and deposition angle on reflectance of (10 nm)ZnS/Ag(30) films

When the angle of deposition of ZnS was set to normal angle i.e. ZnS(0)/Ag(30), while the deposition angle of Ag was maintained at \(30^o\), Figure 7 (a). The reflectance value of 30.3% in the visible region was obtained for \(i=30^o\) at \(\lambda= 800\) nm. This was followed by \(i=45^o, 15^o, 60^o\) and \(75^o\). The reflectance for other angles of incidence was \(< 20%\). The reflectance pattern had been altered compared to the values obtained in Figure 7 (a). The reflectance for \(i=30^o\) in the infrared region reached 73% at \(\lambda =2400\) nm. Other spectra attained maximum reflectance values (41% - 54.5%) in the infrared region at \(\lambda = 1978\) nm. There were sharp peaks in the wavelength range 350 - 450 nm. These peaks showed the presence of monodispersed ZnS nanoparticle distribution. The reflectance values obtained on this sample indicates poor reflectance in the visible wavelengths. This was a good modification for use in transparent nanostructures. The high reflectance in the infrared wavelength implies that infrared transmittance was poor for this sample. In terms of thermal infrared control, the specimen should be aligned such that light is incident on it at \(i=15^o\) in the visible region and at \(i=30^o\) in the infrared region.

On increasing the angle of deposition of ZnS from ZnS(0)/Ag(30) to ZnS(30)/Ag(30) (Figure 7 (b), the reflectance values in the visible region was \(< 27%\). The spectra for \(i= 15^o\) had higher values in the visible wavelength followed by \(i= 30^o\). The spectra for \(i= 30^o\) had high reflectance values (62.7% at \(\lambda=2400\) nm) in the infrared wavelength. Other spectra had peak reflectance values in the range between 36.7% - 51.9% at 1979 nm. The reflectance decreased with increase in angle of incident radiation from \(i=15\) to \(45, 65\), and \(75^o\). Low values of reflectance in the visible region meant that this sample has reciprocally high transmittance values to visible light.

When the angle of deposition of ZnS was adjusted to \(60^o\) i.e. ZnS(60)/Ag(30), Figure 7 (c), reflectance values for different angles of incidence in the visible region were 28.0%, 59.5%, 22.4%, 21.3% and 17.2% for \(i=15^o, 30, 45, 65\), and \(75^o\) respectively. While the reflectance in the infrared region, reflectance values ranged between 36.4% - 57.9% for \(i=15^o\), 45.7% - 80.3% for \(i=30^o\), 22.3% -52.7% for \(i=45^o\), 20.5% - 51.2% - 51.2% for \(i=60^o\) and 16.1% - 34.0% for \(i=75^o\). The reflectance for \(i=30^o\) was higher than for \(i=15^o\) in the visible wavelength. This implies that this sample was transparent to both visible light and highly reflective to infrared light. The physical significance of this sample was that it could be used in applications involving antireflectors or in the reflection of thermally energetic infrared radiation with tilted angles of incidence between for for \(i=15^o\) to \(i=30^o\).

Figure 7. Reflectance spectra of (10nm)ZnS/Ag nanostructures. The silver nanoparticles were deposited at \(30^0\) on glass substrate followed by ZnS at different deposition angles; (a) at (\(0^o\)), (b) at (\(30^o\)) and (c) at (\(60^o\))

3.6. Transmittance and reflectance spectra for (10 nm)ZnS/Ag at different deposition angles

Most of the spectrophotometric studies on multilayer structures involve the measurement of reflectance and transmittance to ascertain optical quality of the reflecting surfaces. This approach provides the opportunity to establish a correlation between deposition processes, conditions and optical parameters such as reflectance, transmitance and optical absorbance. When an electromagnetic radiation is incident on the surfaces of nanostructures such as Ag and ZnS, electrical oscillations of conducting electrons takes place on the surface of the metal. These electrical oscillations are called localized surface plasmons [47].

The excitation of surface plasmons by an external electrical field results in charge polarization on the metal surface. At resonance point (point at which frequency of applied field is equal to frequency of waves from electrical excitation), surface plasmon resonance occurs which leads to strong absorption or scattering of incident light. Surface plasmon absorption bands of Ag are in the visible and near infrared spectral regions. This is very useful for technological applications. When Zinc sulphide nanoparticles are subjected to the external electromagnetic field, coherent oscillations (surface Plasmon resonance) of the conduction electrons also occurs [48]. Surface plasmon effects are however, affected by several factors which include frequency of incident radiation, film thickness and formation of atomic islands on the surface of the dielectric substrate. Notebly, film thickness and deposition angle present a profound effect on spectral properties of thin film nanolayers. It is against this simple background that the Authors were prompted to carryout a comparison between reflectance and transmittance measurements on the samples.

The transmittance spectra of (10nm)ZnS/Ag decreased with increase in deposition angle of ZnS in the wavelength range between (480 nm - 1200 nm), Figure 8 (a). In the wavelength range of 380 nm - 520 nm, the transmittance values were between 48% - 56%. The high transmittance of ZnS/Ag nano multilayers was due to high refractive index of ZnS that enhanced antireflection of incident electromagnetic radiation [49] and [50]. In the infrared region, at about \(\lambda = 800\) nm, the transmittance was \(< 20%\) and then the transmittance decreased rapidly as the wavelength was increasing. In the visible region, reflectance values were low ranging between 5% to 32%. The reflectance in the infrared wavelength increased from \(\lambda\) =800 nm to 2200 nm to a reflectance value of 58% at \(\lambda =\) 2000 nm.

The high transmittance in the visible spectrum is due to small thickness and relative homogeneity of the thin films at normal deposition angle of silver [51]. From the graph, there exist points which the researchers have referred to as equivalence points. These are points at which transmittance curves intersect the reflectance curves. These points occur in the wavelength range \(\lambda=570 - 640\) nm. High reflectance in the infrared region signifies poor transmittance to thermal infrared radiation while the same samples have relatively average values values to transmittance of visible light.

Figure 8. Reflectance and transmittance spectra of (10nm)ZnS/Ag nanostructures. (a) The deposition angle of ZnS was increased from \(0^o\) to \(60^o\) when the deposition angle of was fixed at \(0^o\). (b) The deposition angle of ZnS was increased from \(0^o\) to \(60^o\) when the deposition angle of was fixed at \(30^o\). (c) The deposition angle of ZnS was increased from \(0^o\) to \(60^o\) when the deposition angle of was fixed at \(60^o\).

The transmittance of the deposited films decreased with increase in deposition angle of ZnS in the ZnS/Ag multilayer film in the visible region, Figure 8 (b). However, in the infrared region, transmittance increased with increase in deposition angle of ZnS. The reflectance in the infrared wavelength decreased with increase in deposition angle of ZnS films. The equivalence point were shifted and spread in the wavelength between 620 - 830 nm. This behaviour was also been reported in a recent study by [52].

At high deposition angle of Ag i.e. ZnS/Ag(60) Figure 8 (c), the transmittance in the visible region increased considerably. However, transmittance decreased with increase in deposition angle of ZnS in the visible wavelengths. The transmission peaks at the wavelength of about 500 nm decreased with increase in deposition angle of ZnS. Where us in the infrared region, transmittance increased with deposition angle of ZnS. This shifted the equivalence point to about \(\lambda= 780\) nm. The physical shift in equivalence wavelength was due to high deposition angle of both zinc sulphide and silver films. The high deposition angle of Ag decreased the reflectance in the infrared region.

3.7. Transmittance and reflectance spectra for (7 nm)ZnS/Ag at different deposition angles

The composite made by depositing Ag films normally followed by ZnS at different deposition angles Figure 9 (a) was characterised with high transmittance in the visible wavelengths and low transmittance in the infrared region. The optical transmittance increased with increase in deposition angle of ZnS in visible region. The transmission peaks in visible spectrum increased with the deposition angle of ZnS. The transmission peaks decreased progressively towards the the long wavelengths i.e. 74% for (7 nm)ZnS(60/Ag(0), 68.7% for (7 nm)ZnS(30/Ag(0) and 65.9% for (7 nm)ZnS(0/Ag(0). At the wavelength \(\lambda \geq 800\) nm, the transmittance was \(< 20%\). The reflectance of the nanostructures in the visible region decreases with increase in deposition angle of ZnS. It was observed from Figure 9 (a), that the reflectance values in the infrared wavelengths was lower than reflectance values in the visible wavelengths. Though the reflectance values of the samples were above average (\(< 50%\)) in the infrared region, the samples could reflect more infrared light than what they can transmit. On the other hand, visible transmission for the samples was fairly above 65% which is a good property for visibility.

Additionally, reflectance increased from visible wavelengths to near infrared region of electromagnetic spectrum. These results were so close in the infrared region. Though the deposition angle for ZnS was increased from \(0^o\) to \(60^o\), the spectral response to deposition angle of ZnS did not change much. This shows that Ag nanoparticles strongly influence the transmittance and reflectance of the ZnS/Ag multilayer films. The reflectance of the samples in the visible region (\(\lambda =400-800\) nm) was low and ranged between 24% - 31% for the three deposition angles of ZnS. Reflectance in the infrared region for \(\lambda \geq 800\) nm was fairly good (between 55% and 58%). The transmittance in the visible spectrum was above average and ranged between 65% and 75% , Figure 9 (a).

When the angle of deposition of Ag was increased from ZnS/Ag(0) to ZnS/Ag(30) (Figure 9 (b)), the transmittance in the visible region increased with increase in deposition angle of ZnS. The transmission peaks in the visible spectrum were obtained at around \(\lambda = 421\) nm with the following transmittance values: 65% for ZnS(0)/Ag(30), 63% for ZnS(30)/Ag(30) and 60% for ZnS(60)/Ag(30). The transmittance decreased progressively from 34.4% at \(\lambda= 800\) nm to 16.8% at \(\lambda = 2400\) nm. The reflectance in the visible region at \(\lambda =800\) nm was very low (ranged between 6.5% - 22.2%) and decreased with increase in deposition angle of ZnS. The reflectance then increased toward the near infrared spectral region. Interference effects decreased with increase in deposition angle of ZnS. Further analysis of Figure 9 (b), revealed that the transmittance values were very close in the entire electromagnetic spectrum. This meant that the changes in deposition angle of ZnS had less impact on the optical transmittance of the electromagnetic radiation. In the infrared spectral wavelengths, the reflectance values for ZnS(0)/Ag(30) and ZnS(30)/Ag(30) were greater than the transmittance of ZnS(0)/Ag(30), ZnS(30)/Ag(30) and ZnS(60)/Ag(30). The equivalence points were located at around 886 nm and 1054 nm in the infrared spectral wavelength.

When the deposition angle of Ag nanoparticles in the composite was raised to \(60^o\) i.e. from ZnS/Ag(30) to ZnS/Ag(60) Figure 9 (c), there were remarkable spectral characteristics both in the visible and infrared wavelengths. The transmittance in the visible wavelength decreased with increase in the deposition angle of ZnS. The transmission peaks were observed in the wavelength range of 400 nm to 560 nm. The transmission peaks for specimen ZnS(0)/Ag(30), ZnS(30)/Ag(60) and ZnS(60)/Ag(60) were at 62.0%, 53.3% and 45.0% respectively. At about \(\lambda = 734\) nm, the trend in transmittance changed towards the infrared spectral range. The transmittance increased with increase in deposition angle of ZnS. This implies that the samples were more transparent to infrared radiation at high deposition angles of ZnS. Nevertheless, the transmittance of ZnS(60)/Ag(60) in the infrared wavelength was higher than the transmittance in the visible region by 11%. The reflectance spectra in the visible had low values of 18.6%, 15.6% and 13.6 for specimen ZnS(0)/Ag(30), ZnS(30)/Ag(60) and ZnS(60)/Ag(60) respectively at \(\lambda =\)800 nm. The reflectance then increased toward the infrared region but dependent on deposition angle of ZnS. The reflectance decreased with increase in deposition angle of ZnS and reflectance interference effects were observed in the infrared wavelength. High values of reflectance were obtained at \(\lambda = 1843.9\) nm and they were as follows: 43.5%, 34.9% and 25.0% for ZnS(0)/Ag(30), ZnS(30)/Ag(60) and ZnS(60)/Ag(60) respectively.

Figure 9. Reflectance and transmittance spectra of (7nm)ZnS/Ag nanostructures. (a) The deposition angle of ZnS was increased from \(0^o\) to \(60^o\) when the deposition angle of was fixed at \(0^o\). (b) The deposition angle of ZnS was increased from \(0^o\) to \(60^o\) when the deposition angle of was fixed at \(30^o\). (c) The deposition angle of ZnS was increased from \(0^o\) to \(60^o\) when the deposition angle of was fixed at \(60^o\).

3.8. Transmittance and reflectance spectra for (4 nm)ZnS/Ag at different deposition angles

The (4 nm)ZnS/Ag nanostructures displayed quite different spectral characteristics compared to (7 nm)ZnS/Ag and (10 nm)ZnS/Ag nanostructures. In Figure 10 (a), the optical transmittance values were higher than the reflctance values both in the visible and infrared spectral wavelengths. The transmittance peaks in the visible region exited at around 352 - 402 nm but increasing toward the short infrared wavelengths. At about \(\lambda = 1017\) nm, the transmittance of (4 nm)ZnS(0)/Ag(0) continued to decrease however, the transmittance of (4 nm)ZnS(30)/Ag(0) and (4 nm)ZnS(60)/Ag(0) started to increase at about 790 nm towards the infrared wavelength. The reflectance of the specimens was very low \(< 12% \) at 800 nm. The deposition angle of ZnS had little impact on the reflectance in the visible region but the reflectance in the infrared region decreased with increase in deposition angle of ZnS. Highest reflectance of 23% was observed at \(\lambda = 1995\) nm. The decrease in reflectance of ZnS/Ag nanostructures may also be attributed to the formation of ZnS and Ag clusters or discontinuous Ag islands on glass substrate which enhanced transmittance or absorption of the visible light [53,54]. The (4 nm)ZnS/Ag(0) nanostructures had near average optical transparency which qualifies them for potential applications in transparent windows and optoelectronic devices. The high transmittance at low deposition angles in the visible region is due to fairly smooth surface and relative homogeneity of the films [51]. The sharp decrease in transmittance was observed in the ultraviolet region and this was attributed to the light absorption by the dielectric layers.

When the deposition angle of Ag nanoparticles was increased to \(30^o\), Figure 10 (b) the transmittance in the visible region increased with increase in deposition angle of Ag. However, the transmittance decreased with increase deposition angle of ZnS. The transmission peaks progressively moved toward the infrared wavelength of the electromagnetic spectrum. The recorded peaks in transmittance were 62.0%, 66.8%, and 53.5% for ZnS(0)/Ag(30), ZnS(30)/Ag(30) and ZnS(60)/Ag(30) respectively. At around \(\lambda =1072\) nm, the transmittance tends to increase with the deposition angle of ZnS. Nevertheless, the transmittance of ZnS(0)/Ag(30) remained relatively high. The reflectance values in the visible were low ranging between 3% - 9%. A slight increase in reflectance was observed at 803 nm. However, the reflectance decreased with increase in deposition angle of ZnS. Highest reflectance in the infrared region was obtained at \(\lambda = 1985\) nm as follows: 16.3% for ZnS(0)/Ag(30), 11.2% for ZnS(30)/Ag(30) and 6.1% for ZnS(60)/Ag(30). The reflectance values were generally lower than the transmittance values. These samples were transparent to infrared and visible light wavelengths. However, they might not be suitable for thermal infrared control applications.

Figure 10. Reflectance and transmittance spectra of (4 nm)ZnS/Ag nanostructures. (a) The deposition angle of ZnS was increased from \(0^o\) to \(60^o\) when the deposition angle of was fixed at \(0^o\). (b) The deposition angle of ZnS was increased from \(0^o\) to \(60^o\) when the deposition angle of was fixed at \(30^o\). (c) The deposition angle of ZnS was increased from \(0^o\) to \(60^o\) when the deposition angle of was fixed at \(60^o\).

When the deposition of Ag nanoparticles was increased further to \(60^o\) i.e. ZnS(30)/Ag(60) 10 (c), transmittance of the samples in the visible region decreased with increase in deposition angle of ZnS. The transmission peaks in the visible were located between 381 - 580 nm. The peak values for transmittance in this region were 55.0%, 51.5% and 45.3% for specimen ZnS(0)/Ag(30), ZnS(30)/Ag(30) and ZnS(60)/Ag(30) respectively. The transmittance values in infrared wavelength were higher than those in the visible region. The reflectance of the samples was very low and ranged between 3.3% - 10.3% in the wavelength range 383 - 780 nm. At around 800 nm, the reflectance increased towards the infrared wavelengths. Highest reflectance was obtained at wavelength of 1983 nm for different specimens. The reflectance values obtained were as follows: 10.1% for ZnS(0)/Ag(30), 27.3% for ZnS(30)/Ag(30) and 15.5% for ZnS(60)/Ag(30). The transmittance values were generally higher than the reflectance values in the entire electromagnetic spectrum. The significance of these results implies that these specimens are not desirable for use in thermal infrared control and anti-reflection applications.

4. Conclusions

The reflectance results of the multilayer structures of normally and obliquely evaporated ZnS/Ag films on glass substrate show that the reflectance decreased with decrease in film thickness and deposition angle of ZnS. However, the reflectance decreased with increase in deposition angle of silver metal films. By controlling the thickness and deposition angle, it is possible to design windows that are transparent to visible light and the same time opaque to near infrared photons. When the deposition angle of ZnS increased in the ZnS/Ag multilayer, the reflectance decreased in the infrared region. The deposition of silver films at higher deposition angles suppressed reflectance in the visible spectral wavelength which is a desirable property for visible transparent optical structures. Therefore, it is recommended that silver films be deposited at higher deposition angles for high visibility. Considering the (4 nm)ZnS/Ag nanostructures, the films had very low values of reflectance in both the visible and infrared regions. These nanostructures were transparent to the entire optical spectrum. Thermal regulation cannot be achieved with these films. Therefore, this composite was a poor heat mirror in the near infrared wavelength. The reflectance (7 nm)ZnS(0)/Ag(0) and (10 nm)ZnS(0)/Ag(0) at near normal incidence was around 33% in the visible region and over 50% in the infrared wavelength. Therefore, these structures can be considered as a transparent window materials for thermal infrared control. The reflectance of (15 nm)ZnS/Ag was high both in the visible and infrared regions. Thus, these samples can be regarded as opaque to both visible light and infrared radiation. Therefore, they cannot be used as transparent heat mirror. The installation of reflective structures such as reflective mirrors should be done so as to achieve high transmittance (low reflectance) to visible light but high reflectance to infrared radiation energy. Thus for these nanostructures, the suitable orientation of the reflecting surface to the incoming electromagnetic radiation should at angle of incidence between \(i=15^o\) and \(30^o\). For angles of incidence \(i > 30^o\), reflectance values tend to decrease as the angle of incidence increases.

Acknowledgments

The Authors would like to acknowledge the financial and material support from SIDA (The Swedish International Cooperation Agency) through ISP (the International Science Programme, Uppsala University) and Uganda Independent Scholarships Trust Fund Board, Ministry of Education and Sports.

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of Interest

''The authors declare no conflict of interest.''

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Gallery of integrating factors for non-linear first-order differential equations https://old.pisrt.org/psr-press/journals/easl-vol-4-issue-4-2021/gallery-of-integrating-factors-for-non-linear-first-order-differential-equations/ Thu, 30 Dec 2021 12:58:51 +0000 https://old.pisrt.org/?p=6142
EASL-Vol. 4 (2021), Issue 4, pp. 17 - 25 Open Access Full-Text PDF
Albert Adu-Sackey, Gabriel Obed Fosu, Buckman Akuffo
Abstract:This paper discusses a gallery of useful results in connection with integrating factors that are often left as problems for discovery learning and are generally not taught in typical Ordinary Differential Equations courses. Most often than not the approach earlier writers employ is to give a possible form for an integrating factor that may results in an integrating curve without practical prove as far as the subject matter is concerned. In this write-up, an attempt is made by solving the resulting partial differential equation emanating from an underlining general differential equation of a non-exact form, by the use of the ratio theorem to establish various intricate possibilities of integrating factors that are seldom and often relegated to the background, even though they may be equally be applied as a function of a unitary variable or a linear combination of both the dependent and independent variables under certain conditions. Granted an integrating factor is found and such a function applied, the benefit is enormous especially the non-exact differential equation reduces into a known type which may be identified as exact, homogeneous, and or separable that yields a solution.
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Engineering and Applied Science Letter

Gallery of integrating factors for non-linear first-order differential equations

Albert Adu-Sackey, Gabriel Obed Fosu\(^1\), Buckman Akuffo
Department of Applied Mathematics, Koforidua Technical University, Ghana.; (A.A.S & B.A)
Department of Mathematics, Kwame Nkrumah University of Science and Technology, Ghana.; (G.O.F)

\(^{1}\)Corresponding Author: gabriel.obed@presbyuniversity.edu.gh

Abstract

This paper discusses a gallery of useful results in connection with integrating factors that are often left as problems for discovery learning and are generally not taught in typical Ordinary Differential Equations courses. Most often than not the approach earlier writers employ is to give a possible form for an integrating factor that may results in an integrating curve without practical prove as far as the subject matter is concerned. In this write-up, an attempt is made by solving the resulting partial differential equation emanating from an underlining general differential equation of a non-exact form, by the use of the ratio theorem to establish various intricate possibilities of integrating factors that are seldom and often relegated to the background, even though they may be equally be applied as a function of a unitary variable or a linear combination of both the dependent and independent variables under certain conditions. Granted an integrating factor is found and such a function applied, the benefit is enormous especially the non-exact differential equation reduces into a known type which may be identified as exact, homogeneous, and or separable that yields a solution.

Keywords:

Integrating factor, Ratio theorem, ODE, PDE.

1. Introduction and preliminaries

The extreme challenge in finding an integrating factor of a simple or complex form, for a non-exact differential equation is borne out of solving a partial differential equation base on certain conditionality imposed on the ordinary differential equation (ODE) to transform it into an exact type. The snappy introduction of partial differential equation in the study of ordinary differential equations may pose some difficulty [1] of a sort to people who may only be taking the course briefly and may not advance or stray into the study of calculus of several variables, mathematical methods, integral equations, and specialized fields like special functions of mathematical physics, thermodynamics, and quantum mechanics. These advanced areas require a grave deal of useful repertoire of the methods or the techniques that provides a complete procedure for solving mercenaries of differential equations [2,3,4]. For instance, the functions that are usually classified as special functions of mathematical physics are all characterized by second-order differential equations and are implicit in the use of transforming their forms into exactness, leading to the derivation of their orthogonal properties by the application of certain integrating factor.

The use of an integrating factor has the advantage that is susceptible in the handling of most ordinary differential equations. In literature and for practical reasons, simple integrating factors involving a single variable could routinely be found [5]. This makes it possible for the non-exact differential equation to become solvable [6], so that the limitation or difficulty of having to deal with the partial differential equation in order to extract the possible integrating factor(s) from is completely taking care of. This fact makes it unattractive and a quick avenue for the subject matter to be touched on and short-lived in favor of the other aspects of the study of ordinary differential equations. Such intervention and notion may rather put readers at a disadvantage in general for two main reasons.

The first being that, readers may run along with the idea that once a first order non-linear equation is seen not to be exact nor meet the litmus test of the known types with their tailored appropriate methods that go with them, in the determination of their respective solutions, then one can always obtain an integrating factor in a single variable only or for want of a better word, by inspection [5,7], the entire equation is re-arranged into an integral form and the solution deduced without fully giving such method the in-depth attention it fully deserves as is often the case.

Secondly, the passion for a possible extension for the subject matter especially for integrating factors in two or more variables as well as the development of such techniques for higher-order differential equations may be completely incomprehensible or out of reach, even though quiet an extensive work has been done by earlier writers, particularly in the reduction of higher orders of ordinary differential equations under symmetric considerations [8,9]. The interesting aspect of the use of symmetric conditions for systems of ordinary or partial differential equation is that they tell much about the closed-form or analytic solvability of the system without having to routinely solve it in advance [1,10]. In contrast, this parallels the approach for the determination of integrating factors, in the sense that if there exist infinitely many integrating factors to the same non-exact differential equation, then its general solution may be obtained in terms of any two of such functions expressed as a direct proportion without actually solving the given differential equation.

An integrating factor is guaranteed to exist provided the given differential equation is solvable and in fact the general conditions of existence of integrating factors are derived under the theory of Lie group and Lie symmetries [1,11,12]. Once such a function is determined and applied, the exact equation now obtain, play out as a total derivative [13] of some appropriate function referred to as the potential function [14,15]. The resulting general solution from this procedure is implicitly [5] defined and geometrically it represents a family of level curves for that potential function [6,11].

In this paper, we shall primarily dwell on first order differential equations by resorting to a combination of partial differential equation and the ratio theorem to establish various integrating factors, of great value which could remarkably be applied in reducing a given differential equation into exactness and hence separable form for its solution to be determined.

A cursory look at differential calculus reveals that there is no one size fit straight jacket approach or in general an algorithm in the determination of integrating factors [16] for nonlinear first-order differential equations. Hither, we shall attempt to construct a variety of non-zero [6] functions from the necessary and sufficient conditions that require a none exact differential equation to convert into an exact form. We shall extensively apply the theory on ratios and more importantly to identify certain exact forms to develop a number of cases involving the use of integrating factor formulas, some of which are obviously going to be in terms of a single variable whiles others may appear as sums and products or quotients involving linear combinations of two variables of the types \( xy, \ x/y, \ y \pm x \), and \( y^{2} \pm x^{2} \).

2. Forms of integrating factors

Let the differential form of a first-order differential equation assumed to be non-exact be given by
\begin{equation}\label{e1} M(x, y)dx + N(x, y)dy = 0. \end{equation}
(1)
Then the necessary and sufficient condition for Equation (1) to transform into exact is based on the partial differential relation
\begin{equation}\label{e2} [\mu M(x, y)]_{y} = [\mu N(x, y)]_{x}, \end{equation}
(2)
where \(\mu (x, y) \) is an integrating factor, and \( M, \ N \) are arbitrary functions with continuous first derivatives within a closed simply connected region in \( x \) and \( y \). Once this condition criterion is satisfied, Equation (2) expands into
\begin{equation}\label{e3} M \mu_{y} - N \mu_{x} = \mu [N_{x} - M_{y} ]. \end{equation}
(3)
The auxiliary equation associated with the resulting expansion in (3) yields
\begin{equation}\label{e4} \dfrac{dx}{N} = \dfrac{dy}{-M} = \dfrac{d \mu}{\mu [M_{y}- N_{x}]}. \end{equation}
(4)
Equation (4) may be hinged upon as a basis to afford us the opportunity to develop a number of useful cases of integrating factor types that can be deduced to make the non-exact differential equation (ODE) transforms into exact form depending on the nature of the referenced differential equation as given by Equation (1). The cases are expounded below:

Case 1

If we seek an integrating factor purely in \( x \), then by pairing the first and last expressions in Equation (4) we get
\begin{equation}\label{e5} \dfrac{dx}{N} = \dfrac{d \mu}{\mu [M_{y}- N_{x}] } \implies \mu (x) = e^{\int f (x)dx}, \end{equation}
(5)
where \( f(x) = ({M_{y}- N_{x}})/{N} \). This integrating factor (5) would always transform the standard first-order non-exact differential equation into an exact type, provided it is expressed in the same appearance and form of Equation (1), that is \( [ p(x)y - q(x)]dy + dx = 0 \) [4,17]. The result of the transformed equation has the benefit of always yielding a unique solution without any singularity [13] defined in them whenever, the coefficients \( p(x) \) and \( q(x) \) are continuous on a prescribed open interval I [4,17].

Case 2

For an integrating factor in the dependent variable \( y \) alone, one may similarly take from Equation (4) the relation
\begin{equation}\label{e6} \dfrac{dy}{-M} = \dfrac{d \mu}{\mu [M_{y}- N_{x}] } \implies \mu (x) = e^{\int f (y)dy}, \end{equation}
(6)
where \( f(y) = \left( \dfrac{N_{x}-M_{y}}{M} \right) \).

Case 3

Other combinations are possible by applying the ratio theorem and choosing constant multipliers of 1 for the first two terms. By such multipliers, the Equation (4) is modified into the form
\begin{equation}\label{e7} \dfrac{dx}{N} = \dfrac{dy}{-M} = \dfrac{d \mu}{\mu [M_{y}- N_{x}]} = \dfrac{d (x \pm y)}{N \pm M}. \end{equation}
(7)
Now for integrating factor formulas to be churned out in terms of sums and differences, we shall need to pair the third and fourth expressions in Equation (7) to give
\begin{equation}\label{e8} \dfrac{d \mu}{\mu [M_{y}- N_{x}] } = \dfrac{d (x \pm y) }{ N \pm M } \implies \mu (x \pm y) = e^{\int f (x \pm y)d (x \pm y)}, \end{equation}
(8)
where \( f(x \pm y) = \left( \dfrac{M_{y}- N_{x}}{N \pm M} \right) \).

Case 4

For an integrating factor of the combination \( xy \), it is necessary to take up the multipliers \( y \) and \( x \) for the first two terms of Equation (4), and again by pairing the results with the last term of the same equation, one is led to the relation
\begin{equation}\label{e9} \dfrac{d \mu}{\mu [M_{y}- N_{x}]} = \dfrac{y dx + x dy}{y N - xM} \implies \mu (xy) = e^{\int f(xy) d(xy)}, \end{equation}
(9)
where \( f(xy) = \left(\dfrac{M_{y}-N_{x}}{yN-xM} \right) \).

Case 5

For an integrating factor involving sums of squares of the form \( x^{2}+ y^{2} \), it is necessary to take up the multipliers \( 2x \) and \( 2y \) for the first two terms of Equation (4), and thus by pairing the results with the last term of the same equation, then it leads to the relation
\begin{equation}\label{e10} \dfrac{d \mu}{\mu [M_{y}- N_{x}]} = \dfrac{2x dy + 2y dx}{2 [xN - yM]} \implies \mu (x^{2}+ y^{2}) = e^{\int f(x^{2}+ y^{2})d(x^{2}+ y^{2})}, \end{equation}
(10)
where \( f(x^{2}+ y^{2}) = \dfrac{1}{2} \left( \dfrac{My- N_{x}}{xN- yM} \right) \).

Case 6

It can be seen that the differential equation of an innocently looking form \( ydx-xdy = 0 \) is indeed not exact, yet separable. This equation can be shown to transform into exactness by the use of the following non-trivial [6] integrating factors \[ \dfrac{1}{x^{2}}, \, \dfrac{1}{y^{2}}, \, \dfrac{1}{x^{2}-y^{2}}, \, \dfrac{1}{x^{2}+y^{2}} .\] These choices of integrating factors for the differential pair \( ydx - xdy \) [7,18,19], could be exploited to our advantage in introducing five extra useful integrating formulas defined exponentially. A striking coincidence of such a differential pair is arrived at, by noting that when multipliers \( y \) and \(-x \) are imposed on the first two terms of the Equation (4) and the resulting expression paired with the last term of that same equation, one is led to the relation
\begin{equation}\label{e11} \dfrac{d \mu}{\mu [M_{y}- N_{x}]} = \dfrac{y dx- x dy}{ [yN + xM]} \implies \mu (\theta) = e^{\int f(\theta)d\theta}, \end{equation}
(11)
where \( f(\theta) = \tau \left( \dfrac{My- N_{x}}{yN+xM} \right) \).

Now, when \( \tau \) takes on the values \( x^{2}, \ y^{2}, \ xy, \ x^{2}- y^{2}, \ x^{2} + y^{2} \), we obtain five different integrating factors which may be used to convert the non-exact differential Equation (1) into an exact form, where such functions are well-defined as shown below:

  1. \( \mu \left( \dfrac{x}{y} \right) = e^{\int f(x/y)d(x/y)} \), where \( f \left( \dfrac{x}{y} \right) = y^{2} \left( \dfrac{M_{y}- N_{x}}{yN+ xM} \right) \);
  2. \( \mu \left( \dfrac{y}{x} \right) = e^{\int f(y/x)d(y/x)} \), where \( f \left( \dfrac{y}{x} \right) = x^{2} \left( \dfrac{M_{y}- N_{x}}{yN+ xM} \right) \);
  3. \( \mu \left( \ln \dfrac{x}{y} \right) = e^{\int f(\ln (x/y))d(\ln (x/y))} \), where \( f \left(\ln \dfrac{x}{y} \right) = xy \left( \dfrac{M_{y}- N_{x}}{yN+ xM} \right) \);
  4. \( \mu \left( \arctan \dfrac{x}{y} \right) = e^{\int f(\arctan (x/y))d(\arctan (x/y))} \), where \( f \left( \arctan \dfrac{x}{y} \right) = (x^{2}+ y^{2}) \left( \dfrac{M_{y}- N_{x}}{yN+ xM} \right) \);
  5. \( \mu \left(\xi \right) = e^{\int f(\xi)d\xi} \), where \( f \left( \xi \right) = (y^{2}- x^{2}) \left( \dfrac{M_{y}- N_{x}}{yN+ xM} \right) \), and \( \xi = \dfrac{1}{2}\ln \dfrac{x+y}{y-x} \).
Apart from obtaining these various possibilities for the integrating factor formulas expressed in at least one variable, and in terms of the exponential function in connection with Equation (1), it may be possible to further develop some integrating factors under some characterization associated to the non-exact differential equation. Such cases are considered and proved as follows:

Case 7

If it is known that the non-exact differential Equation (1) is homogeneous with respect to the coefficients \( M(x, y) \), and \( N(x, y) \) of the same degree, then the integrating factor takes the form
\begin{equation}\label{e12} \mu (x,y) = \dfrac{1}{xM + yN}. \end{equation}
(12)

Proof. For homogeneous functions, we may express \( M \) and \( N \) as

\begin{equation}\label{e13} M(x,y) = x^{n} \phi_{1} (v), \ N(x,y) = x^{n} \phi_{2} (v), \, \mbox{ with } v = \dfrac{y}{x}. \end{equation}
(13)
Substituting (13) into (1), we obtain \[ x^{n} \phi_{1} (v) d x+ x^{n} \phi_{2} (v) [v dx + x dv] =0 .\] This implies that
\begin{equation}\label{e14} x^{n} [\phi_{1} (v) + v \phi_{2} (v) ] dx + x^{n+1} \phi_{2} (v) dv = 0. \end{equation}
(14)
For Equation (14) to be exact and separable we need to divide through by \( x^{n+1} [\phi_{1} (v) + v \phi_{2} (v) ] \), leading to
\begin{equation}\label{e15} \dfrac{1}{x} dx + \dfrac{\phi_{2} (v) dv}{ [\phi_{1} (v) + v \phi_{2} (v) ]} = 0. \end{equation}
(15)
Clearly the form of Equation (15) is separable and so such a divisor must be an integrating factor. That is
\begin{equation}\label{e16} \mu (x,y) = \dfrac{1}{x^{n+1} [\phi_{1} (v) + v \phi_{2} (v) ] } = \dfrac{1}{xM + yN}. \end{equation}
(16)

Case 8

If it is known that the coefficients of the non-exact differential Equation (1) are functions of products of \( x \) and \( y \) defined by \( M(x, y) = y f_{ 1}(xy) \), and \( N(x, y) = x f_{ 2}(xy) \), then the integrating factor can readily be shown to be \[ \mu (x,y) = \dfrac{1}{xM - yN} .\]

Proof. By substituting the forms of \( M(x, y) \) and \( N(x, y) \) together with the expression \( [y f_{2}(xy) - y f_{ 2}(xy)]dx \), into Equation (1) we get \[ y f_{1} (xy) dx + [yf_{x}(xy)- yf_{2}(xy) ]dx + x f_{2} (xy) dy = 0 .\] Thus \[ y[f_{1}(xy) - f_{2}(xy)]dx + f_{2}(xy)d(xy) = 0 .\] This implies that

\begin{equation}\label{e17} \dfrac{xy}{x}[ f_{1}(xy) - f_{2}(xy)]dx + f_{2}(xy)d(xy) = 0. \end{equation}
(17)
For Equation (17) to be exact and separable, it is expedient to divide through by \( xy [ f_{1}(xy) - f_{2}(xy)] \), leading to
\begin{equation}\label{e18} \dfrac{1}{x} dx + \dfrac{f_{2} (xy)}{xy[ f_{1}(xy) - f_{2}(xy)]} d(xy) = 0. \end{equation}
(18)
Clearly the form of Equation (18) is separable and so such a divisor must be an integrating factor. That is \[ \mu (x,y) = \dfrac{1}{xy[ f_{1}(xy) - f_{2}(xy)]} = \dfrac{1}{xM - yN}, \ \mbox{ provided } xM-yN \ne 0 .\]

In case the powers of each term of the factors of the product are unequal and have the form \( M(x,y)= yf_{1} (x^{m}y^{n}) \) and \( N(x,y) = x f_{2} (x^{m}y^{n}) \), then the integrating factor can readily be shown to be \[ \mu (x,y) = \dfrac{1}{n M_{x} - m N_{y}} .\]

Proof. By substituting the new forms of \( M(x, y) \) and \( N(x, y) \) into Equation (1) we get

\begin{equation}\label{e19} f_{1} (x^{m}y^{n}) y dx + f_{2} (x^{m}y^{n}) x dy =0. \end{equation}
(19)
Now by writing out the total differential of the product \( x^{m}y^{n} \), we see that \[ d (x^{m}y^{n}) = m x^{m-1} y^{n} dx + n x^{m} y^{n-1} dy \] leads to
\begin{equation}\label{e20} x dy = \dfrac{1}{nx ^{m-1} y^{n-1} } [ d (x^{m}y^{n}) - m x^{m-1} y^{n-1} \cdot y dx ]. \end{equation}
(20)
Using the expression in Equation (20) into Equation (19) and grouping the terms for \( dx \) and \( dy \) we obtain \[ \left[ f_{1} (x^{m}y^{n}) - \dfrac{m}{n} f_{2} (x^{m}y^{n}) \right] y dx + \dfrac{1}{nx^{m-1}y^{n-1}} f_{2} (x^{m}y^{n}) d (x^{m}y^{n}) =0 .\] This is expressed as \[ \dfrac{xy}{x} \left[ n f_{1} (x^{m}y^{n}) - m f_{2} (x^{m}y^{n}) \right] dx + \dfrac{1}{x^{m-1}y^{n-1}} f_{2} (x^{m}y^{n}) d (x^{m}y^{n}) =0 .\] This implies that
\begin{equation}\label{e21} \dfrac{1}{x} \left[ n f_{1} (x^{m}y^{n}) - m f_{2} (x^{m}y^{n}) \right] dx + \dfrac{1}{x^{m}y^{n}} f_{2} (x^{m}y^{n}) d (x^{m}y^{n}) =0. \end{equation}
(21)
Finally, to get separable differentials, we need to divide Equation (21) through by the expression \( \left[ n f_{1} (x^{m}y^{n}) - m f_{2} (x^{m}y^{n}) \right] \). This yields
\begin{equation}\label{e22} \dfrac{1}{x} dx + \dfrac{f_{2} (x^{m}y^{n})}{x^{m}y^{n} \left[ n f_{1} (x^{m}y^{n}) - m f_{2} (x^{m}y^{n}) \right]} d (x^{m}y^{n}) =0. \end{equation}
(22)
Thus, the function \[ \mu (x,y) = \dfrac{1}{xy \left[ n f_{1} (x^{m}y^{n}) - m f_{2} (x^{m}y^{n}) \right]} = \dfrac{1}{n M_{x}- m N_{y}} \] must be the integrating factor for Equation (19) provided \( n M_{x} - m N_{y} \ne 0 \).

Case 9

An integrating factor of the form \( \mu (x,y) = x^{p} y^{q} \) may be assumed if the differential equation is given by the general form
\begin{equation}\label{e23} x^{\alpha} y^{\beta}(my dx + nx dx) + x^{\rho} y^{\sigma} (ay dx + bx dy) = 0, \end{equation}
(23)
where \( \alpha, \beta, \rho, \sigma, m, n, a, b \) are constants, with \( p \) and \( q \) being unknown constants to be determined. In order to apply such an integrating factor, the strategy [6] is to ensure that the unknown index constants \(p\) and \(q\) must be selected in such a way that when Equation (23) is multiplied by this integrating factor \( \mu (x,y) \) exactness is achieved. That is
\begin{equation}\label{e24} [mx^{\alpha+p}y^{y^{\beta+q+1 }} + ax^{\rho+p}y^{\sigma +q+1}]_{y} = [mx^{\alpha+p}y^{y^{\beta+q+1 }} + ax^{\rho+p}y^{\sigma +q+1}]_{x}. \end{equation}
(24)
On performing the partial derivatives and equating coefficients of the terms \( x^{\alpha + p}y^{\beta +q} \) and \( x^{\rho + p}y^{\sigma +q} \), we get the following equations
\begin{equation}\label{e25} mq - np = n(\alpha + 1) - m(\beta + 1); \qquad aq -bp = n(\alpha + 1)- m(\beta + 1). \end{equation}
(25)
Now, provided the determinant of the coefficients of \( q \) and \( p \) in the systems of linear equations in two unknown is non-singular, that is \( \begin{vmatrix} m & n \\ a & b \end{vmatrix} \ne 0 \), then can \( p \) and \( q \) be found uniquely from Equation (25) by \[ \begin{split} p &= \dfrac{1}{bm - an}[am (\sigma -\beta)+ an (\alpha + 1) - mb (\rho +1) ], \\ q &= \dfrac{1}{bm - an}[bn (\alpha -\rho)+ an (\sigma + 1) - mb (\beta +1) ]. \end{split} \] If it happens that the coefficients of the products \( x^{m}y^{n} \) and \( x^{a}y^{b} \) of the differential pairs of Equation (23) are arbitrarily defined by functions \( f_{1} (x^{m} y^{n}) \) and \( f_{2} (x^{a}y^{b}) \), such that their index indices coincides with the constants associated with the given differential pairs, then it is may be possible to obtain a much simpler integrating factor to the differential equation
\begin{equation}\label{e26} f_{1} (x^{m} y^{n}) (my dx + nx dy) + f_{2} (x^{a}y^{b}) (ay dx + bx dy) = 0 \end{equation}
(26)
by observing that the total differentials of \( x^{m}y^{n} \) and \( x^{a}y^{b} \) can be written as
\begin{equation}\label{e27} d(x^{m}y^{n}) = x^{m-1}y^{n-1} (my dx + nx dy); \qquad d(x^{a}y^{b}) = x^{a-1}y^{b-1} (ay dx + bx dy). \end{equation}
(27)
Now by substituting Equation (27) in (26) and multiplying through by the quantity \( 1/xy \) we obtain
\begin{equation}\label{e28} \dfrac{1}{x^{m}y^{n}} f_{1}(x^{m}y^{n}) d(x^{m}y^{n}) + \dfrac{1}{x^{a}y^{b}} f_{2} (x^{a}y^{b}) d(x^{a}y^{b}) = 0. \end{equation}
(28)
Clearly Equation (28) is indeed separable, and so an integrating factor for Equation (26) must be of the form \[ \mu (x,y) = \dfrac{1}{xy} .\]

Case 10

Some differential forms may not have associated integrating factors to them, but if such differential equations are characterized as having Homogeneous coefficients, then it suffices to fall on certain useful appropriate substitutions, that transform the differential equations into other standard equations which may have readily available integration factors [20] to make them solvable. A consideration of the differential equation of the form
\begin{equation}\label{e29} [f_{1} (x,y) + f_{3} (x,y) x^{k+1} ] dy = [f_{2} (x,y) + f_{3} (x,y) yx^{k} ]dx, \end{equation}
(29)
where \( f_{1} (x,y) \), \( f_{2} (x,y) \) and \( f_{3} (x,y) \) are homogeneous functions of \( x \) and \( y \), while \( f_{1} (x,y) \) and \( f_{2} (x,y) \) are of the same degree have no direct integrating factor though, but the use of the common substitution \( y = vx \) is priceless when handling homogeneous differential equations. This particular substitution transform Equation (29) into a standard Bernoulli equation which is known to be non-linear and can further be reduced into linear standard equation which of course will still remain non-exact.

Proof. By the hypothesis that the arbitrary functions \( f_{1}(x, y),\ f_{2}(x, y) \) and \( f_{3}(x, y) \) are homogeneous, let \( y = vx \implies dy = vdx + xdv \) and noting from Equation (11) that when Equation (29) is carefully arranged the last term of that equation yields total differential form given by \( d(y/x) = (xdy - ydx)/x^{ 2} = dv \), so that in effect the differential equation for case 10 reduces to the form \[ [ f_{1}(v)(vdx + xdv)- f_{2}(v)dx]+ f_{3}(v)x^{k+2} dv = 0 .\] Implying that

\begin{equation}\label{e30} [vf_{1}(v) -f_{2}(v)]dx + x f_{1}(v)dv + f_{3}(v)x^{k+2}dv = 0. \end{equation}
(30)
Now by assuming that \( x \) is a function of \( v \), Equation (30) is written into the form
\begin{equation}\label{e31} \dfrac{dx}{dv} + \dfrac{-f_{1}(v)}{f_{2}(v) - vf_{1} (v) }x = \dfrac{f_{3} (v)}{f_{2}(v) - vf_{1} (v)} x^{k+2}. \end{equation}
(31)
Equation (31) is clearly the well known Bernoulli non-linear equation and so by setting \( w= x^{-k-1} \), where \( w \) and \( x \) are all functions of \( v \), then Equation (31) turns out as
\begin{equation}\label{e32} \dfrac{dw}{dv} + \dfrac{ (k+1)f_{1}(v)}{f_{2}(v) - vf_{1}(v)}w = \dfrac{ - (k+1)f_{3} (v)}{f_{2}(v) - vf_{1} (v)}. \end{equation}
(32)
This is clearly the standard non-exact differential equation. Now per case 1, if \( f (x) \) is replaced with \[ \dfrac{(k+1)f_{1}(v)}{f_{2}(v)- vf_{1} (v)} = \dfrac{(k+1)xf_{1}(y/x)}{xf_{2}(y/x)- yf_{1} (y/x) } ,\] then the transformed Equation (32) when written out in its differential form will possess the integrating factor given by \[ \mu (x,y) = e^{\int \Lambda d(y/x)} ,\] where \( \Lambda = \dfrac{(k+1)x f_{1} (y/x)}{xf_{2} (y/x) - yf_{1} (y/x)} \).

3. Discussion of results

An erroneous impression usually created in the study of differential equations lies in the result of solving the partial differential equation from which an integrating factor could be extracted as being relatively difficult to handle than the original non-exact equation to which a solution is being sorted. The general rule for solving a differential equation is to identify its type and to map up the method that best solves it. However, many differential equations do not fall under the category of separability, homogeneity, and exactness and so require some integrating factors to reduce them to any of such types that lead to their solutions. In general, there are no known algorithms for finding integrating factors and so it is essential to explore their derivations and not to resign them to only the simple cases involving either the independent or the dependent variable only. An advantage that can be adduced about integrating factors is that it may happen that at least two integrating factors \( \mu_{1}, \mu_{2} \) to the particular non-exact differential equation can be easily found from any of the cases outlined, such that their ratio is not a constant, then it will be superfluous and waste of effort to go through the normal procedure in finding the potential function that solves the differential equation. In such instance, the general solution is best written out as an implicit relation in the form \( \mu_{1} (x,y) = C\mu_{2} (x,y) \). This can be anticipated in the sense that, a partial differential equation may have more than one possible solution to it, and each of the integrating factors \( \mu_{1}, \mu_{2} \) satisfies the established partial differential equations at Equation (2). The constant \( C \) has to do with the associated order of the original differential equation.

We adopted the ratio theorem together with specific multipliers to derive some integrating factors and further observed keenly that certain differential forms were equally exploitable in advancing other important integrating factors. Generally integrating factors are not unique, for the single fundamental reason that the partial differential equation from which these integrating factors are developed from do not have unique solutions and so such intrinsic characterization are exhibited by them as well.

It is important to underscore that not, every integrating factor turns out as an exponential function as may be observed for the cases 7, 8 and 9. These cases have been meticulously introduced to give a firm grip on the subject matter. They are developed from the viewpoint of certain peculiarity of their forms and characterization such as the homogeneity of their coefficients and even much more on the instance that some are easily turned into product of exact forms.

Case 9 was more or less developed on a hunch base on the form of the coefficients of the differential pairs with the condition that the determinant of the resulting linear equations should be non-singular. Another fair form of case nine with arbitrary functions whose total derivatives are coincidental to the equation expressed in differential form was also exploited to give a simpler integrating factor for that particular case.

Finally, the case 10 was introduced to show that not every differential equation in differential form may have direct integrating factor related to it. However, with appropriate substitutions, they transform into named differential equation such as the Bernoulli equation which is known to have some integrating factor linked to it when transformed into the standard first-order differential equation.

4. Conclusion

In this paper, an exploration is made into the derivation of integrating factors that are dependent on certain linear combinations involving two variables. Such useful functions are not unique in general, and the particular ordinary differential equation that calls for its use may have an infinite number of them associated with the very same differential equation. Thus, with some dexterity and ingenuity, any of such integrating factors could be applied to turn the defining non-exact differential equation into solvable type without recourse to the usual forms they are adopted either in terms of the single independent variable \( x \) or that of the dependent variable \( y \). We hope that the research on the gallery of integrating factors will spark a renewed interest and also make the subject matter have its rightful place it fully deserves in the study of differential equation.

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of Interest

''The authors declare no conflict of interest.''

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The relationship between the energy efficiency of buildings and occupants: A review https://old.pisrt.org/psr-press/journals/easl-vol-4-issue-4-2021/the-relationship-between-the-energy-efficiency-of-buildings-and-occupants-a-review/ Thu, 30 Dec 2021 12:49:15 +0000 https://old.pisrt.org/?p=6140
EASL-Vol. 4 (2021), Issue 4, pp. 5 - 16 Open Access Full-Text PDF
Muhammad Usman Farooq, Abdul Ahad, Zeeshan Maqsood, Niranjan Devkota, Syed Naqi Raza
Abstract:Green buildings are supposed to provide a sustainable solution for energy usage, but their low performance raised some questions in the literature. The researchers determine that occupants are the key factor for this energy deficiency. In the last two decades, a stream of research focuses on the greening of occupants, but a synthesis of findings and results are absent in the literature. In this study, we reviewed the literature on green buildings and occupants. Based on the findings we classified four classes. The first class consists of green occupants and green buildings, which is the ideal solution for high-energy efficiency. The second class is of brown occupants and green buildings and is the prime reason behind outperformed green buildings and yields negative-medium level efficiency. The third class comprises green occupants and brown buildings and yields positive-medium level efficiency, which helps to start the journey towards sustainability. The fourth class is the combination of brown buildings and brown occupants and has the lowest efficiency and worst impact on the environment throughout the lifecycle. Further, we link these classes with the energy-saving efficiency of buildings and finally recommended an efficient solution for second and third world countries. The study contributes to green building literature and packed with managerial implications to gain the maximum benefits of green buildings.
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Engineering and Applied Science Letter

The relationship between the energy efficiency of buildings and occupants: A review

Muhammad Usman Farooq, Abdul Ahad, Zeeshan Maqsood, Niranjan Devkota\(^1\), Syed Naqi Raza
Department of Civil Engineering and Architecture, University of Sialkot, Pakistan.; (M.U.F)
Department of Management Sciences, National College of Business Administration \& Economics, Pakistan.; (A.A)
Department of Statistics, University of Sialkot, Pakistan.; (Z.M)
Department of Economics, Quest international University, Nepal.; (N.D)
Department of Electrical Engineering, University of Sialkot, Pakistan.; (S.N.R)

\(^{1}\)Corresponding Author: niranjandevkota@gmail.com

Abstract

Green buildings are supposed to provide a sustainable solution for energy usage, but their low performance raised some questions in the literature. The researchers determine that occupants are the key factor for this energy deficiency. In the last two decades, a stream of research focuses on the greening of occupants, but a synthesis of findings and results are absent in the literature. In this study, we reviewed the literature on green buildings and occupants. Based on the findings we classified four classes. The first class consists of green occupants and green buildings, which is the ideal solution for high-energy efficiency. The second class is of brown occupants and green buildings and is the prime reason behind outperformed green buildings and yields negative-medium level efficiency. The third class comprises green occupants and brown buildings and yields positive-medium level efficiency, which helps to start the journey towards sustainability. The fourth class is the combination of brown buildings and brown occupants and has the lowest efficiency and worst impact on the environment throughout the lifecycle. Further, we link these classes with the energy-saving efficiency of buildings and finally recommended an efficient solution for second and third world countries. The study contributes to green building literature and packed with managerial implications to gain the maximum benefits of green buildings.

Keywords:

Sustainable environment; Green buildings; Energy; Building efficiency; Occupants; Greening.

1. Introduction and preliminaries

The concept of green buildings introduced in late 90's but after the year 2000 due to the establishment of rating systems, they got high attention. Green buildings are water and resource-efficient in addition to energy efficiency and are considered to adopt energy efficiency practices on a priority basis, thus offer sustainable solution (National Stone Institute). Also there is an exponential increase in the energy demands over the globe due to increasing population, automated systems, and climate change. It is estimated that it will reach 50% only in the building sector until 2050 if the experts do not take appropriate action to deal with it [1]. The green building is a head-turning topic nowadays to deal with energy requirements and expected to use fewer resources especially energy as compared to conventional (brown) buildings.

Anyhow, there are contradicting findings present in the literature that the green buildings are often found fail to achieve the required performance. The researchers state that there is a huge difference between the calculated and real-time energy consumption of certified green buildings keeping the function and size constant [2,3,4,5]. This difference varies case to case in the range of 30% to 100% [6,7,8]. Here raises the issue, that why the green buildings are underperforming and what are the factors which could help green buildings meeting their target performance? This issue motivated the researchers to check non-technical drivers of energy consumption in green buildings, with a focus on activities of occupants in buildings [9,10]. There is a consensus of researchers that the low performance of green buildings can be dealt with working on occupants' behavior because energy consumption behavior of occupant significantly affect the energy performance of buildings [11,12]. So we argue that earning green certification for a building is not enough in the journey of a sustainable environment.

The researchers found a positive impact of energy consumption behaviors of occupants on the performance of building in terms of energy usage [11,12]. In [13] Brockman states that occupants are very important to consider as they account for 50 % usage of the energy of a building. The second problem arises about the performance of brown buildings. There are countless brown buildings which will still serve for more than 20 years and consumes a larger portion of energy. There is almost no work in the literature which could help to increase efficiency of these buildings. Mixed mode buildings (building which contains both green and brown counterparts) could be the greening practice in developing countries to move forward in the path of sustainability. Anyhow, still the numbers of brown building are very high and need robust solution.

Therefore as a result of work on this aspect: Incentive programs, education and training regarding green behaviors, trash management, use of vegetation as facades of buildings, green certification of tenants, use of artificial machinery to develop reports of energy usage at individual level, development of management practices, baby steps of government like awareness signs and banners on public places and use of media came as solutions. But each of these solutions consulted their own technique and discussed them separately resulting in fragmented literature. A holistic framework is absent in the literature and a need exists to synthesize all provided results and solutions. These reasons generated a gap for our paper.

Thus our review is determined to clarify four research issues in the literature. First, the underperformance of green buildings having all technical facilities. Second, the role of occupants in reducing the efficiency of green buildings. Third, the role of occupants in increasing the efficiency of brown buildings. Four, the methods to change energy efficiency behavior of occupants.

These problems are very critical to deal specially when the literature is fragmented into different categories and results are inconsistent. In this paper, we establish a systematic research framework to answer the above-mentioned questions. Our review fulfills the following purposes. First, we provided simplified results of past work and their relationship with the efficiency of buildings which will help governments, companies and even individuals to take steps towards sustainable environment using these existing brown buildings. Second, we highlight the four different domains of green building research providing the research direction for under research domain. Third, with the help of systematic literature review of the greening of occupants regarding energy consumption behaviors we have classified four classes of occupants and buildings which have an impact on the energy consumption efficiency of the buildings and illustrate them in a building efficiency matrix. The study provides the synthesized findings of the literature on green buildings energy efficiency and occupants. It will also help the managers to achieve the maximum energy efficiency from the green buildings and can turn the brown buildings more performing by working on the behaviors of occupants.

The remaining of the paper is organized in the following manner. In section two methods are discussed. In section three the previous work is analyzed and section four comprised of results. We discussed the study findings and provided the suggestions in the last section. This section also includes study limitations, future direction, and conclusion.

2. Methodology

We followed a systematic literature gathering process to collect the studies for this review. Anyhow to get a broader understanding of the phenomena we started with a general search to check studies available on the topic of the greening of occupants. We found numerous results however a handful of papers investigated greening of occupants in terms of energy use. After this as the first step of systematic literature gathering process, we selected the terms "green building", "greening", "occupants" and "tenants" in the title of papers, on keyword lists, or in the abstracts.

2.1. Inclusion and exclusion criteria

We included the articles that are published in English language during the time span of 2000 to September 2018. After this basic screening we included the review and research papers and focused on articles which discussed greening and occupants in detail. We excluded the data which just discussed greening and occupants as side topic. There were many topics where we achieved consensus after discussing the matter briefly. It was our basic concern that the whole structure of the paper revolve around greening of occupant.

3. Procedure

We conducted the search from the well-known six databases that includes Web of Science, Science Direct, ProQuest, JStor, EBSCOHost and Google Scholar. For further clearance, we also collected some data from the reference lists of selected articles. Doing all this we found 453 articles fulfilling our criteria out of which 179 were duplicate. After reading the title and abstract of remaining articles, 240 papers were found irrelevant. Remaining thirty-four full length articles were studied carefully and twenty articles were found out of the context. Mostly research have been done on residential buildings and least considered type is office buildings. Finally, this exercise results in a set of thirteen articles as a set of research which are shown in Table 1, while the flowchart diagram [14] is shown in Figure 1. The data was coded by two researchers and the agreement level was achieved after the discussion when a disagreement was found.

Figure 1. PRISMA flow diagram

Table 1. Papers selected for review.
Sr. No. Author & Year Title Journal
1 Koehler [30] Green facades-a view back and some visions Urban Ecosystems
2 Steinberg et al., [18] Developing a focus for green building
occupant training materials
Journal of Green
Building
3 Mirel et al., [20] Certifiable Green Journal of Property
Management
4 Deuble et al., [21] Green occupants for green buildings:
The missing link?
Building and
Environment
5 Milenkovic and Amft [22] Recognizing energy-related activities using
sensors commonly installed in office buildings
4th International
Conference on Ambient
Systems, Networks and
6 Hope and Booth[23] Attitudes and behaviors of private sector landlords
towards the energy efficiency of tenanted homes
Energy Policy
7 Cao et al., [32] Development of an Energy-Aware Intelligent
Facility Management System for Campus Facilities
Defining the Future of
Sustainability and
Resilience in
Design, Engineering,
and Construction
8 Mokhtar et al., [25] Strategies for improving energy saving
behaviour in commercial buildings in Malaysia
Engineering, Construction
and Architectural  Management
9 Aghili et al., [26] Key Practice for Green Building
Management In Malaysia
4th International Building
Control Conference 2016
10 Huebner et al., [27] Understanding electricity consumption:
A comparative contribution of building factors,
socio-demographics, appliances,
behaviours and attitudes
Applied Energy
11 Aulia et al., [28] Identification of Increasing Green Behaviour
in Citraland Bagya City, Medan
1st Annual Applied
Science and Engineering
Conference
12 Azar and Menassa [11] Framework to investigate energy conservation
motivation and actions of building occupants:
The case of a green campus in Abu Dhabi, UAE
Applied Energy
13 Ohueri Chukwuka et al., [29] Energy efficiency practices for Malaysian
green office building occupants
Built Environment Project
and Asset Management

We found no research relevant to our criteria from 2000 to 2006. Description of studies which are included in the review is tabulated below:

Table 2. Quick review of selected papers .
Study type Place Sample size Contribution
Case study U.S 41 initiatives Multi criteria decision analysis
Case study U.S 43000 Certification of occupants
Case study Georgia One building Energy aware intelligent system
Research paper Indonesia 45 respondents Motivation and socializing of occupants is necessary
Research paper Australia 02 building Forgiveness factor
Research paper Netherlands 01 building Recognition of energy-related activities
Research paper\
Case study
Abu Dhabi -
UAE
227  campus users Factors such as respondents’ demographics,
the level of control over building systems,
and motivation drivers (e.g., ?nancial, social,
and environmental) highly affect energy saving
actions and need through consideration for effective
human-focused energy conservation strategies.
Research Paper Malaysia 53 respondents integration of technological strategy; organizational
strategy; and occupants behavioral strategy will
critically reduce energy consumption
Research Paper Malaysia 03 buildings Energy efficient use of electric appliance – Computer
Research Paper United
Kingdom
845 households appliance ownership and usage are the most influential
variables in understanding electricity consumption
Review Multiple Green building
standards
Key management practices
Review Paper Germany 16 articles Green facades
Survey Malaysia 53 landlords Integration of technological strategy, organizational
strategy and occupants behavioral strategy critically
reduce energy consumption
of green office buildings in Malaysia.

4. Findings

After careful construing of literature it is found that from 2004 to 2006 with the efforts of Levitt Northfield Sustainability Tenant Incentive Program NSTIP was conceived. They were also first to enroll in the Xcel Design Assistance Program XDAP. In these programs, occupants were incentivized for doing green activities. They developed 51 points of compliance and compiled a handbook containing how to comply and what are the benefits in terms of cost. This method helped them a lot to increase the efficiency of that retail center. In addition to this, it promoted the culture and generated a ripple effect in the area and supply chain as well [27]. McConnell in [28] worked on the greening of the solid waste management program. This program is a four "R" strategy: Review, Reduce, Reuse and Recycle. This is a very organized and effective program and if followed by occupants will lead towards a sustainable environment. Another step was to provide an incentive to occupants for use of green facades (vegetation on the front of the building). It is widespread in Germany by Koehler [29] and he emphasizes that these living walls must be used all over the globe. Steinberg et al., [16] developed a decision matrix and helped to find relevant and necessary information for the training of occupants. Their work explained briefly that the provision of green policies and mechanisms is not enough. Occupants must be trained, and these actions must be promoted. A manifesto was developed and presented in Passive and Low Energy Architecture PLEA conference. Occupants were termed as active determinants of energy performance of building and conditions and directives were also provided [30].

In the same year, another revolutionary step was taken by Hines (Interests Limited Partnership: Real estate company). They introduced Green Office Tenant certification. In this program, there were a total 100 points from which tenant must score 70 points to earn certification. Tenants were evaluated in seven categories and energy efficiency is one of them. They distributed the tenant guide containing information about taking baby steps to enhance the performance of building in many terms including energy consumption [17]. Here ends the work of a decade regarding the greening of occupants/tenants or also sometimes termed as inhabitants.

In the most recent decade, a progressive pattern is observed in the literature in terms of publications, variety and demographic conditions of research. Further studies on occupant behavior and green buildings showed that users of green buildings offer high forgiveness factor (tolerance to building environment) to the features of their building than any other building [18]. A technical analysis was conducted by Milenkovic and Amft [19] to show that occupant behavior has a significant influence on the energy consumption of a building. They used HMM (Hidden Markov Models)-based recognition of office desk activities and by modifying the HMM transition probability estimated energy consumption in the simulation. A review study by Hope and Booth [20] pointed out some facts that why landlords of the private sector do not focus on the energy efficiency of their buildings. They summarized that fast growth, high upfront cost and ineffective policy measures are the main causes. They also identified the lack of studies available on private landlords regarding energy efficiency behaviors.

Energy-aware intelligent facility management system was designed and tested by Lee et al., [31]. Their focus was on the HVAC system only, but they clearly showed the positive impact. Basically, this system was designed with a concept to help and assist occupants to develop energy consumption behaviors. Observations are the best motivator and they could realize the severity of a cause incomplete sense. In a study on commercial buildings, energy consumption behaviors of occupants of green and brown buildings were compared [22]. The results showed that the occupants have better behaviors in the green buildings. They recommended that there must be the regular distribution of posters and guidelines along emails and the live updates regarding energy use of the particular building must be a regular practice as well. New staff must also be briefed on the company's policy about energy saving. These strategies were successful in encouraging of green behaviors.

At management level, sustainable approaches must be incorporated into daily routine. Aghili et al., [23] developed five key practices after reviewing the data available in the context of management to enhance the efficiency of green buildings. The set of these practices is general and enlighten two very important things: One is time and the other is stakeholders. This study suggests that sustainability can be achieved when these practices are incorporated and prioritized at each phase of the green building starting from procurement and leading to designing, building, operation and demolition. Besides that, it also recommends that every person including designers, buyers, constructors, building users and all other persons related to the building must be aware of sustainable practices.

In a unique study with respect to technicality for residential buildings, variability was measured in electricity consumption and found to be 35% [24]. Four classes of predictors were also explained and the analysis showed that appliance ownership and usage is the only variable with the highest value to explain he use of energy in the residential building [24]. This study suggested that income of occupants and size of households may vary the study results. Aulia et al., [25] suggested that socialization activities are very necessary for the education of occupants for green behaviors. A study in Abu Dhabi, found that only the will of occupants to use less energy is not enough to determine the energy usage behavior. Other factors like demographics, control of occupants over systems of the building, the frequency of energy communication from management and especially the reasons due to which occupants save energy are the real drivers of energy-saving behavior. They contributed a general framework to test their assumptions applicable to any type of building.

In a very well-focused recent study, Chukwuka et al., [26] did a mixed method research on a sample of fifty-three employees in a green building in Malaysia. They suggested that organizational strategy, integration of technological strategy and occupants' behavioral strategy will enhance the energy efficiency of buildings. In the light of their results, they developed a set of energy efficiency practices which is a great contribution.

Table 3 provides the details about the methods and details of the selected studies about the greening of occupants for building energy efficiency. We have also discussed the methods and their usage for a better understanding.

Table 3. Review of literature on occupants & energy saving behaviors .
Methods Details Discussion
Appreciation for
green facades [30]
The plantation was used at the front of buildings as
living facades. Occupants were provided incentives by the
government of Germany for doing so. This work is based on a
review of studies in Germany and other potential
countries regarding this technique. It was recommended
that this technique must be adopted at the global level.
This technique helps to develop habitat for several species, cools
temperature inside buildings, reduce cost as they replace expensive
materials. But studies are limited to Germany and Western countries.
Further research is required especially in countries with harsh weather.
Training of occupants [18] Multi-criteria decision analysis was done to concise the
training material, so that maximum attention could be gained
by occupants. By using this analysis and testing it on a focus
group 40 LEED credits were reduced to two main points which
are energy saving and waste reduction. The focus of research
was to increase the impact of training material.
There was no literature on the effect of occupants on the
performance of green buildings and positive results of
training of occupants on the efficiency of buildings. It is the
best way to cater for this issue as it involves every person.
Certification of
Occupants [20]
A 100-point scale was developed to assess occupants. Points
were termed as leaf and occupant must earn 70 points
to get certified.
This initiative sheds lights upon differentiation of green certification
of building and occupants' role. This clears that occupant has to
work for sustainability too. The only building cannot play
the whole part. Moreover, the certification will further increase
their knowledge about green practices.
Encouragement of
occupants having
energy saving
behavior [25]
Three strategies were developed, and their success was observed
in energy-saving behaviors. First, awareness of occupants about
energy saving behaviors must be raised with the help
of pamphlets, posters, emails, and some appropriate guidelines.
Second, regular live updates of energy use to the users of
building and third briefing to the newly employed staff.
This is very necessary for daily interaction of building
users with their building with respect to energy use.
These strategies change the occupants perspective
and develop their energy-saving behavior.
The energy-aware
intelligent facility
management system [32]
In this system at first knowledge, the database was developed
including basic information on common daily work request
and work instructions. Secondly, the artificial intelligent model
was proposed which could automatically analyze and then
prioritize the future work requests by keeping the focus on
energy consumption impact, safety and occupant satisfaction.
The validity of the system was checked with the help of  case study on campus.
This system helps occupants by prioritizing tasks.
The focus of this paper was on HVAC, further work
should be done on other aspects of energy consumption  as well.
Green Building
management
practices [26]
Five clusters of management practices were identified after
an extensive review of green building practices. These five
clusters included: Sustainable Procurement, Sustainable operation,
Resource management, Repair and Maintenance Management,
and Environmental Health. The clusters cover society, economy
and environmental dimensions.
These are the most general set of practices
and are addressing to achieve sustainability work from the
initial phase of building is necessary. It also emphasizes
that every person from the design phase of the building
to the demolish phase must keep in focus these practices.
Table 4. No. of publications against the type of buildings.
Building Type No. of publications
Academic Institute 3
Office 2
Residential Building 6
Commercial 2
Total 13

In addition to all these explored benefits of greening of occupants, there is lack of diversification in studies such as, certification type (in papers one to three types are discussed at a time), building type (only one building type is concerned), energy use type (if electricity is taken as variable then many common machines like computer and mobile are omitted from the study), research context (studies are case studies, or are for one city, or for one country only). There is a need of more primary work on different building types for the development of the generalized framework. As the literature shows only two studies are conducted on office type settings. Another issue is lesser sample size, in many cases not even fulfilling the criteria of statistical analysis. The possible reason behind this issue could be the lesser number of green buildings in second and third world countries. Trash management and incentive program for occupants are the hot topics which must be explored by researchers. These issues conclude that this area is quite understudied, and this is the right time for researchers to take the initiative for this work.

5. Discussions

The study reviews the literature about the resolution of energy efficiency issues in achieving sustainability with the help of green buildings. An important point of key interest is to understand the forgiveness factor. It is the extent to which the occupant ignores unfavorable conditions and compromise with the building attributes. Green buildings perform efficiently when occupants offer this factor [18]. For example, the famous attribute of the green building is wide openings for the utility of clean air and sunlight. Anyhow, this attribute causes so many problems, such as disturbances due to excess dirt in case of a storm, heat in case of summer, cooling in case of winter and rainwater. If occupants tolerate such problems only then the wide opening could provide a healthy environment in the building and save energy by using sunlight and air. Considering this factor and facts provided in the review section we classified four classes as depicted in Figure 1.

First class is of green occupants living in the green buildings. The occupants offer forgiveness factor and have sound knowledge of green practices (use of automated HVAC systems, water, and energy efficient systems etc.). Green buildings are designed by keeping in view the sustainability, constructed by using recycled materials, and operated with the help of automatic and energy efficient systems which makes it best. While the trained or certified green users are supposed to use the provided systems yielding the highest efficiency. Also according to research this set makes the building the most efficient in the use of resources especially energy and is the best solution for the sustainable environment [18].

The second class consists of brown occupants and green buildings. In this set, occupants offer low to zero forgiveness factor and have negligible knowledge of green practices, which leads to the less efficient building. The main cause is that they could not use the systems as they are not trained or showing non-seriousness. Even in some cases, such green buildings use more energy than conventional buildings of the same size and use. There is also a reason that sometimes occupants think that the green certified building is itself doing sustainable actions. While this is not the case and the matter, in reality, cause drastic effects on the energy efficiency of the building. This is one of the main reasons behind the low performance of green buildings.

The third class includes green occupants and brown buildings, which makes a low impact on the sustainability of the environment. This class is very important, and we recommend it for existing buildings and in developing and underdeveloped countries as well. The literature shows that a number of green buildings are very low in these regions and their implication is also full of barriers [32]. For example, in Pakistan a nuclear country there are only 24 green buildings according to Pakistan Green Building Council. Thus, there is a need for this class in such countries to aware people about greening by generating impact with the help of green occupants. Moreover, this is also the need of time because existing buildings are large in number and will serve for decades. This class will help us to reduce their impact on the sustainability of the environment.

The fourth class comprises of brown occupants and brown building. This is the worst scenario and has a high negative impact on the sustainability of the environment. Brown buildings are not equipped with energy efficient systems, have a conventional design, a material with high-value negative impact on the environment is used during construction. It depicts the real-time situations of many countries. Besides, the occupants are brown with no knowledge of green practices, no training and even not aware of the concepts like sustainability, green buildings, and environmental issues and impact.

Figure 2. Relation of occupants with buildings.

This relationship of occupants and type of buildings could also be better understood by developing the relationship of the efficiency of buildings, occupants, and the type of building in a matrix. Figure 2 shows that we could make four possible combinations. The first combination in this matrix consists of green buildings and green occupants. It is the most effective solution set for the sustainable environment and have the highest efficiency. The second combination is of brown occupants with green buildings it yields high to low efficiency. The third combination is of green occupants and brown buildings which yields low to high efficiency. Here it is worth noting that both second and third combinations result in medium level efficiency. Anyhow, the second class reduces the performance of green building from high to low while the third class increases the performance of brown building from low to high in the positive manner. The fourth combination is of brown buildings and brown occupants which yield the lowest energy efficiency. Figure 3 shows the building efficiency matrix.

We contribute to the literature by recommending a suitable class for high energy efficiency without changing the building type. That class consist of green occupants and brown buildings. Converting the conventional buildings to green one and developing green buildings results in high costs to the investors. Anyhow, greening the occupants is relatively less costly for the organizations. This paper is categorically helpful for the underdeveloped economies as they usually cannot afford high budget strategies to cover the deficiencies.

Our keen observations made us able to deal with some major flaws in the literature. First, the methodology portion of most of the papers is somewhat inappropriate. Second, their applicability is challengeable as they just discussed the techniques. There is no detail about impact of occupants on greening, benefits of the green practice discussed in respective paper. Third, in survey studies the sample size is low and in many cases secondary data was processed. There is high need in the field to introduce primary data studies. It has been observed that some greening techniques of occupants are under practiced and need validation. For example, trash management and incentive for occupants for compliance of green practices which could be a driving factor for occupants to behave green need more research.

Figure 3. Building efficiency matrix. 

Our study is significant in the following manners. The paper provides the complete overview on the topic from 2000 to 2018. No review on this topic has been conducted yet according to our best knowledge. Thus, the study concludes that the green buildings can be more energy efficient by greening the occupants. Second, we developed a relationship among occupants, buildings and efficiency in the form of a matrix. This matrix contributes in green building literature by providing a clear picture of various combinations of buildings and occupants. We discuss four different classes by the pairing of occupants and buildings. It provides synthesized results about the greening of occupants and helps the managers to increase the efficiency of brown buildings by implementing the greening occupants' practices.

6. Study implications

The study is inherited with practical implications for the managers, investors, and governments. First, the findings suggest that the building management must start greening their inhabitants using the suitable technique provided in the literature. The change in the occupants' behavior will help both green and brown building management in long run to save energy, to play their role in the sustainable environment and to save money.

Second, the population of developing and underdeveloped countries is larger than the developed ones. Thus, it is necessary for governments to work on their nation as their impact regarding energy use on the environment will be very high. This is also necessary as due to global linkages they also commute to other parts of the world. The education of occupants' energy saving behavior will develop their positive energy consumption behavior which will be fruitful to the world. After discussing pros and cons of available techniques we selected the most feasible technique for the greening of occupants. This solution lies in educating and training the occupants regarding green practices and to use automated systems in green buildings. The education provides a broader sense of situations, able to modify decision power and develop thinking. Like switching off light, fan or any other electrical appliance when not in use, is a green practice. A prodigious list could be prepared for this purpose. On the other side training generates a sense of competition in occupants, help them to face real time situations, also boost their confidence and as return increase their energy efficiency.

Third, these steps will also help to reduce overall electricity consumption. Nowadays this is very important issues especially in underdeveloped and even in developing countries, to meet the demands of electricity. Another case is also observed that where there is no load shedding per unit rates are quite high. Careful use of electricity by green occupants will also reduce per unit cost of available electricity. Fourth, greening can also be done by working on trash management at the individual level. This will help to reduce pollution and other garbage related costs. Fifth, these approaches will also make healthy ecosystems due to low carbon production, excessive greenery, clean air and the availability of clean water.

7. Limitations and future directions

Our study does not provide the historical perspective of green buildings rather it aims to cover the elements that can help in gaining energy efficiency through their occupants. The study also does not cover the other elements that may help in achieving high green building efficiency. Future research should incorporate these elements for the review.

8. Conclusion

The review shows that the green buildings are not performing up to the mark and the cause could be their occupants. Previous research proposed numerous solutions for greening of occupants, but their synthesis was absent, and our research fulfilled the purpose. After systematic literature, we developed a relationship between occupants and building with respect to energy efficiency, identified four classes, and discussed their relation and effect on energy saving of building. A robust solution for sustainability issues of second and third world countries is the introduction of class three (green occupants \& brown buildings). The best strategy to green the occupants is the early education of occupants regarding green practices. The greening of occupants not only enhances the energy efficiency of buildings but also helps the environment by reducing garbage and rehabilitation of ecosystems. We provide recommendations for future research by enhancing demographic variety, diversification of context and external validity elements

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of Interest

''The authors declare no conflict of interest.''

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Digital high-speed data modulation techniques https://old.pisrt.org/psr-press/journals/easl-vol-4-issue-4-2021/digital-high-speed-data-modulation-techniques/ Thu, 30 Dec 2021 12:40:16 +0000 https://old.pisrt.org/?p=6138
EASL-Vol. 4 (2021), Issue 4, pp. 1 - 4 Open Access Full-Text PDF
Winston Tumps Ireeta, Esther Nabadda, George Isoe
Abstract:Most radio stations use frequency modulation (FM) to broadcast yet amplitude modulation (AM) ensures long distance modulation. The limitations of FM reception are the line of sight and the area of reception. These two parameters are much smaller in FM compared to AM which makes AM modulation have an added advantage over FM modulation. The results presented in this paper include; direct modulation at different bias currents and different transmission fiber lengths and the amplitude modulation using the Mach-Zehnder. The results show the possibility to transmit huge data at high speeds to over 100Gbps.
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Engineering and Applied Science Letter

Digital high-speed data modulation techniques

Winston Tumps Ireeta\(^1\), Esther Nabadda, George Isoe
Department of Physics, Makerere University, Kampala, Uganda.; (W.T.I & E.N)
Department of Physics, Nelson Mandela University, Port Elizabeth, South Africa.; (G.I)

\(^{1}\)Corresponding Author: winston.ireeta@mak.ac.ug

Abstract

Most radio stations use frequency modulation (FM) to broadcast yet amplitude modulation (AM) ensures long distance modulation. The limitations of FM reception are the line of sight and the area of reception. These two parameters are much smaller in FM compared to AM which makes AM modulation have an added advantage over FM modulation. The results presented in this paper include; direct modulation at different bias currents and different transmission fiber lengths and the amplitude modulation using the Mach-Zehnder. The results show the possibility to transmit huge data at high speeds to over 100Gbps.

Keywords:

Amplitude modulation; Eye diagram; Data transmission; Optical fiber.

1. Introduction

Direct modulation is very useful in high speed data transmissions though this method has several problems. The maximum bandwidth that can be modulated is just a few GHz and the maximum quantum efficiency \((\eta)\) is 100%, which places an upper limit on the slope efficiency and therefore the gain [1,2] shown in equation (1).

\begin{equation} \label{e1} S_{c}=\eta \frac{hc}{q\lambda}. \end{equation}
(1)
The use of external modulation is more advantageous compared to direct modulation. In direct modulation where a laser diode is used for example, it suffers from chirp which then introduces large dispersion penalty in the signal being transmitted. The distorted signal will have its slope efficiency limited by fundamental quantum efficiency (100% max) and its capacity limited to 30 GHz at most unless optical injection locking is applied. It is only in the use of external modulation that at least 100 GHz transmission capacities have been reported [3,4].

2. Research design

The experimental setup for the characterization of the Mach-Zehnder for different bias voltages is shown in Figure 1. When the bias voltage is varied, the power is varied so that the most suitable voltage at which transmission should be carried out is obtained. This is known as Mach Zehnder optimization.

Figure 1. Set up for Mach-Zender characterisation (performance at different bias voltages)

Figure 2. Amplitude/ Intensity modulation. A fiber optic Mach-Zehnder interferometer based on lithium niobate components is used to modulate the signal at 10GBps and 8.5GBps.

Figure 2 shows the experimental set up that was used for modulation of signals at 10GBps and 8.5GBps for a fiber optic Mach-Zehnder interferometer based on Lithium Niobate.

In amplitude modulation also known as Amplitude-shift keying (ASK), digital data is represented as variations in the amplitude of the carrier wave. For an ASK system, the binary symbol 1 is represented by transmitting a fixed-amplitude carrier wave and a fixed frequency for a bit duration of T seconds. When the signal value is 1 it means that the carrier signal will be transmitted; otherwise, a signal value of 0 will be transmitted. This such carrier signal is shown in Equation (2).

\begin{equation} \label{e2} X_{c}t=A_{c}cos2\pi f_{c}t+\varphi. \end{equation}
(2)
In equation, the amplitude, \(A_{c}\) , of the signal is what has been modulated and hence amplitude modulation.

3. Result

For high speed data transmission over long distances, direct modulation may be inefficient and therefore the use of external modulation techniques for example the amplitude/ intensity and phase modulations using a Mach-Zehnder modulator.

Figure 3. shows the Mach-Zehnder bias voltage characteristic curve. From the two graphs it shows that the power increases with increase in the voltages both to the negative and positive biases.

Figure 3 shows the Mach-Zehnder bias voltage characteristic curve. From the two graphs it shows that the power increases with increase in the voltages both to the negative and positive biases.

3.1. Intensity/Amplitude modulation at 8.5GBps and 10Gbps

A back to back (with no fiber) transmission was performed at different rates and the sensitivity determined. The results showed better sensitivity for an 8.5Gbps compared to the 10Gbps transmission as shown in Table 1. This means that the increase of the data transmission rates exposes the signal to greater errors (penalty) and hence the reduction in sensitivity.

Table 1. Sensitivities of a Mach-Zehnder modulator at different modulation bit rates .
Rate 8.5Gbps 10Gbps
Sensitivity(dBm) -19.7972 -18.1763

The BER was also measured for the two data transmission speeds and the results are shown in Figure 4. The increase in data transmission rates increased the power penalty.

Figure 4. B2B BER measurements at different transmission rates i.e. 8.5Gbps and 10Gbps

Figure 5. Eye Diagrams for 8.5Gbps and 10Gbps data transmission

Figure 5 shows the Eye diagrams for data transmissions at 8.5GBps and 10GBps. There is a closure in the eye (10Gbps) compared to the eye (8.5Gbps) which signifies increased attenuation as the rate increases. For the case of 8.5Gbps the eye is clear implying it's easier to distinguish between the zeros and ones by the receiver.

3.2. Transmission through a fiber of length of 74.91km

In this study, an optical fiber (74.91km) was inserted and transmission at 8.5Gbps done. The results showed a very small power penalty, implying, external modulation (amplitude modulation) enables transmission at longer distances with minimal dispersion as shown in Figure 6. This is consistent with the results of the study by Peter Winzer in 2012 [5].

4. Conclusion

Very small power penalty was achieved at data transmission at a rate of 8.5GBps in a 74.91km single mode fibre which means that even with an increase in the rates to up to 100GBps, transmission at longer distances with minimal dispersion is possible.

Figure 6. BER measurements of B2B and 74.91km distance of an amplitude modulated data transmitted at 8.5Gbps

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of Interest

''The authors declare no conflict of interest.''

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