1. Introduction

Since the discovery of CNT by Iijima [1], a considerable number of studies on carbon nanotubes with multi branched arrangements have been analyzed [2, 3, 4]. Subsequently, it has been established that such structures have merits when applied to the functionability of transistors and diodes. As a result, logical investigations and analysis of carbon nanotube have been a subject of interest such as the vibrations of a micro-resonator that is excited by electrostatic and piezoelectric actuations. Studies have been performed on beams, CNT, nano-wires, nano-rods and nano-beam so as to specifically understand and achieve their area of best fit. In achieving this, the well known beam models were employed and dynamic ranges were obtained in the scope of the structures [5, 6, 7, 8]. Different studies have been performed to determine the resonance point and to enhance the mechanical properties of single and multi-walled nanotubes, using initial stress and compressive axial loading, to study the non-coaxial resonance [9, 10, 11, 12, 13, 14]. Elishakoff et al., [15] performed an investigation on the fundamental natural frequencies of double-walled carbon nanotubes and ascertained some stability criteria using some controlling parameters. Mass detection with nonlinear nano-mechanical resonator as well as dynamic range of nanotube- and nanowire-based electromechanical systems have also been studied [16, 17].

In order to be able to predict the dynamic behaviour of embedded carbon nanotubes, Fu et al.[18] and Xu et al. [19] performed some analysis of nonlinear vibration for embedded carbon nanotubes. In application to electrical and electronics, the static and dynamic analysis of carbon nanotube-based switches, micro-resonator under piezoelectric and electrostatic actuations have been presented [20, 21, 22]. As a continuation of the tremendous work, Abdel-Rahman and Nayfeh [23], Hawwa and Al-Qahatani [24] and, Hajnayeb and Khadem [25] performed vibration and instability studies of double wall carbon natotube (DWCNT) using a nonlinear model and considering electrostatic actuation as external excitation agent. In their investigation, DWCNT was situated and conditioned to direct and alternating voltages and different behaviors of the nanotubes were recorded as the exciting agent was varied. They went further to determine the bifurcation point of the DWCNT and concluded that both walls had the same frequency of vibration under the two resonant conditions considered. Belhadj et al. [26] presented their work on a pinned-pinned supported SWCNT employing nonlocal theory of elasticity and obtained natural frequency up to the third mode. They also explained how the high frequency obtained in their work might be harnessed to be of high merit in optical applications. Lei et al. [27] studied the dynamic behaviour of DWCNT by employing the well-known Timoshenko beam theory. The nonlinear governing equations obtained by Sharabiani and Yazdi [28] were applied to nanobeams that were graded and had surface roughness. Wang [29] obtained a closed form model for the aforementioned surface roughness effect for an unforced fluid conveying nanotube and beams based on nonlocal theory of elasticity and ascertained the significance of stability analysis for reasonably small thickness of the tube considered.

Interesting foundation studies have been considered after modelling of CNTs as structures resting on or embedded in elastic foundations such as Winkler, Pasternak and Visco-Pasternak media [30, 31, 32, 33, 34, 35]. Other interesting works through modelling and experiment have also been presented to justify the widespread application of SWCNTs [36, 37, 38, 39, 40, 41]. To the best of the authors' knowledge, no work has been done to dynamically determine the shearing force and bending moment diagram of a tensioned and pressurized SWCNT. Motivated by these considerations, this work aims to dynamically determine the response, shearing force and bending moment of a tensioned SWCNT under an external uniform pressure by employing Integral transform techniques.

2. Problem description and governing equation

Consider a SWCNT with a uniformly distributed surface pressure as shown in Figure 1. The assumptions used for the analysis include:

1. the SWCNT is modeled according to Euler-Bernoulli beam
2. the exciting agent is a uniform pressure distribution at the SWCNT surface governed by:
   \begin{align}\label{equ1} P(x) = \mu A^{CNT} \frac{d}{dx}\left( P_{0} \left( 1 + \frac{\delta}{L^{CNT}}x \right) \right) \end{align}

   (1)
3. the SWCNT is simply supported at both ends.
   \begin{align}\label{equ2} \chi(0, t) = \chi^{"}(0, t) = 0 \\ \notag \chi (L^{CNT}, t) = \chi^{"} (L^{CNT}, t ) = 0 \end{align}

   (2)
4. CNT is homogenous and have constant properties.
5. CNT have constant cross-section, hence constant moment of Inertia.

By incorporating the above assumptions into the classic Euler-Bernoulli beam model, the vibration of Figure 1 can be described by the model: Substituting Equation (1) into Equation (3) and simplifying, the model becomes;

3. Methods of solution using integral transforms

Since the resulting model in Equation (4) contains both spatial and temporal part, a superb approach of applying Laplace transform to the temporal and Fourier transform to the spatial terms is adopted. In this present study, the linear transient model in Equation (3) will be solved exactly using Integral transform.

3.1. Basic principle of the integral transforms

Fourier transforms can be used to transform second and higher-order spatial derivatives. Although, the inversion of Fourier transform is usually easier than the inversion of Laplace transform, there are some functions which do not have Fourier transform as they have in Laplace transform. Fourier transform could be used to solve problems in.

1. Finite domains \((0 \leq x \leq L)\)
2. Semi-infinite domain, and \(( 0 \leq x \leq \infty, -\infty \leq x \leq 0)\)
3. Infinite domains unlike Laplace transform \((-\infty \leq x \leq \infty )\)

The transform of a finite fourier cosine function is: \begin{align*} F_{c} = \bar{f}(n) = \int_{0}^{l} cos \left( \frac{n \pi x}{L} \right) f(x) dx \ \ \ 0 \leq x \leq L, \ \ where \ n = index \ counter \ 1, 2,3, \dots. \end{align*} The operational properties of the finite fourier cosine function are: Generally, \begin{align*} F_{c} \left[ \frac{d^{2N}f }{dx^{2N}} \right] = (-1)^n \beta^{2n} \bar{f_{c}} - \sum_{m =0}^{n -1} (-1)^m \beta^{2m} \frac{\partial^{2n - 2m -1} f(0^{+})}{\partial x^{2n -2m -1}} \ for \ n \geq 1 \ \textrm{provided }\ \textrm{that} \ |\frac{d^mf}{dx^m}|\rightarrow 0 \;\textrm{as}\; x \rightarrow \infty \end{align*} \begin{align*} \left[ \frac{d^{2n+1}f(x)}{dx^{2n+1}} \right] = (-1)^{n} \beta^{2n+ 1} \bar{f_{s}} - \sum_{m = 1}^{n}(-1)^m \beta^{2m} \frac{\partial^{2n - 2m } f(0^{+})}{\partial x^{2n -2m}} \ n \geq 1. \end{align*} The finite Fourier cosine inverse transform is: The finite Fourier sine transform of a function is defined as: \begin{align*} F_{s} [f(n)] = \bar{f(x)} = \int_{0}^{l} f(x) sin \left( \frac{n \pi x}{L} \right) \ \ 0 \leq x \leq L. \ \ n = 1, 2, 3, \dots. \end{align*} The operational properties of the sine transform of a function are: \begin{align*} F_{s} \left[ \frac{df}{dx} \right] = - \frac{n \pi }{L} \bar{f_{c}} (n) \end{align*} \begin{align*} F_{s} \left[ \frac{d^2f}{dx^2} \right] = - \frac{n \pi }{L} \left[ f(0) - (-1)^n f(L) \right] - \frac{n^2 \pi^2}{L^2} \bar{f_{c}}(n) \end{align*} Generally, we have Thus, if the boundary value problem of a 2nd-order derivative extends over a finite domain and has 1st-type boundary condition at both ends \([f(0)=f_{0} \; at \; x=0, \ f(L)=f_{L} \; at \; x=L]\),the finite Fourier sine transform can be used to transform the 2nd- order derivatives. The finite Fourier sine inverse transform is:

Method of solution: Laplace and Fourier transform

Recall that the linear transient governing equation as shown in Equation (4) may be expressed as subject to the initial and pinned-pinned conditions: \begin{align*} \chi (x, 0 ) = \dot{\chi}(x, 0 ) = 0, \\ \notag \chi (0, t) = \chi^" (0, t) = 0, \\ \notag \chi (L^{CNT}, t) = \chi^" (L^{CNT}, t) = 0. \end{align*} Applying Laplace transform on the temporal term gives Using the initial condition and applying Fourier transform to the spatial terms, we have \begin{eqnarray}\label{equ12} &&\mu A^{CNT} P_{0} \frac{\delta }{sL^{CNT}} \left( \frac{L^{CNT}}{i \pi } \right) \left[ 1 - (-1)^i \right]= Ms^2 \widetilde{\bar{\chi}} + EI^{CNT} \left( \left( \frac{i \pi}{L^{CNT}} \right)^4 \widetilde{\bar{\chi}} - \frac{i \pi}{L^{CNT}} \right)^3 \left[ \chi(0)\right. \nonumber\\ &&\left.- (-1)^{i} \chi (L^{CNT} ) \right] + \frac{i \pi}{L^{CNT}} \left[ \chi^" (0) - (-1)^i \chi^" (L^{CNT}) \right] + \left[\frac{EA}{(1- 2V^*)}\alpha^* \theta -T\right] \left[ - \left( \frac{i \pi}{L^{CNT}} \right)^2 \widetilde{\bar{\chi}}\right. \nonumber\\ &&\left.+ \frac{i \pi}{L^{CNT}} \right] \left[ \chi(0) - (-1)^i \chi (L^{CNT} ) \right] \nonumber + K\widetilde{\bar{\chi}}. \end{eqnarray} Applying the boundary conditions and grouping like terms

Determination of the natural frequency

From Equation (13), the denominator may be arranged to take the form: The natural frequencies can be determined from the poles of Equation (13) as: The displacement of the nanotube will be realized by finding inverse Laplace transform of Equation (13). That is and the Fourier inverse of Equation (16) becomes: Equation (17) is the desired exact solution that represents deflection of the SWCNT.

3.4. Determination of the SWCNT Bending moment and Shear force

The bending moment is related to the SWCNT deflection by [39]: Similarly, the shear force is related to the SWCNT deflection by: Substituting Equation (16) into Equations. (17, 18), we have \begin{align*} B (x, t) = -EI^{CNT} \frac{\partial^2 }{\partial x^2} \left[ \frac{2}{L^{CNT}} \sum_{i =1}^{\infty} \frac{\mu A^{CNT} P_{0} \delta (L^{CNT})^4 }{ i \pi \left( EI^{CNT} (i \pi)^4 - \left( \frac{EA} {1 -2v^*} \alpha^* \theta - T \right) ( i \pi L^{CNT} )^2 + K (L^{CNT})^4 \right) } sin \frac{n \pi x}{L^{CNT}} \right] \nonumber\\ \times \frac{\left[ 1 + (-1)^{ i + 1} + cosh \left\{ \frac{ \sqrt{(L^{CNT} )^4 M \left( -EI^{CNT} (i \pi )^4 + \left( \frac{EA} {1 -2v^*} \alpha^* \theta - T \right) ( i \pi L^{CNT} )^2 - K (L^{CNT})^4 \right) } t }{(L^{CNT})^4 M} \right\} 1 + (-1)^i \right]}{i \pi \left( EI^{CNT} (i \pi)^4 - \left( \frac{EA} {1 -2v^*} \alpha^* \theta - T \right) ( i \pi L^{CNT} )^2 + K (L^{CNT})^4 \right)} \end{align*} The bending moment using the series at \(i = 3\) which is the converged point becomes; And the shear force becomes:

4. Results and Discussion

Figure 2 depicts the convergence criteria based on the number of iteration \((i)\) in the close form solution for the deflection of the simply supported SWCNT. The computational time associated with each iteration is shown in Table 1. It is clear that at \(i = 3\), the solution has already converged, hence, extending the iteration above three will only increase the computational time and cost with negligible effect on the improvement of the established solution.

Table 1. Convergence criteria based on the number of iteration \((i)\) in the close form solution for the deflection of the SWCNT.

Iteration			Maximum deflection		Computational time (secs)
\(i\)	\(chi (pm)\)	\(n =1\)	\(n = 2\)	\(n =3\)	\(n = 4\)
1	1.600	0.256784	0.323885	0.435583	0.436961
2	1.601	0.296788	0.323900	0.436781	0.446962
3	3.700	0.456733	0.466385	0.468551	0.486001
10	3.700	1.236734	1.523877	1.835101	1.999921
15	3.700	3.456780	3.723335	3.935513	4.001610

4.1. Effect of the modal number on the deflection of the SWCNT

Figure 3 displays the simply supported SWCNT deflection along its length for the first five mode shapes. Critical visualization shows that as the modal number associated with the kernel that defines the boundary condition increases, the stability of the SWCNT under study decreases as a result of an increase in the cycles covered by the SWCNT for the same length. These occur because the kernel depends on the modal number.

4.2. Dynamic response of the SWCNT

Figure 4, 5, 6, 7 depict the three dimensional dynamic response associated with the SWCNT for the first four modes. As the mode number increases, the number of cycles completed as shown by the dynamic behaviour of the system increases. A critical assessment shows that it is possible to track the behaviour of the CNT at any instance. The dynamic analysis is important as it helps in the quick monitory and adjustment of the CNT during application.

The model derived in the present study was reduced to the model of Coskun et al. [40] beam model and a very good agreement was obtained as shown in Table 2.

Table 2. Comparison of present study with Coskun et al. [1]40} exact method for pinned - pinned condition.

	Mode shape
Mode	Coskun et al. [1]	Present study
1	3.14159265	3.14159265
2	6.28318531	6.28318531
3	9.42477796	9.42477796
4	-	12.56637061
5	-	15.70796327

\end{table}

4.3. Effect of modal number and length on the frequency of the SWCNT

Figure 8 depicts the influence of modal number and length on the frequency of the SWCNT. A careful study helps visualizes the effect of these two important parameters on stability of the SWCNT. The frequency of SWCNT which is a vital parameter in the study of the SWCNT stability reaches some THz and continues to increase as the modal number increases. This astonishing property enables SWCNT to offer exceptional optical and mechanical properties although there is always need to dampen the frequency to an application limit. These two parameters as a result of their tremendous effects on frequency may be used to annul the effect on each other when one of them is desired based on the requirement of the engineering design and applications.

The dimensional frequency model obtained in the present study was also reduced to the level of Belhadj et al. [26] SWCNT model and a very good agreement was obtained for the first three modes as shown in Table 3.

Table 3. Comparison of the present study with exact solution of Belhadj et al. [26] for pinned - pinned condition for the frequency of the SWCNT (THz).

Length (nm)			Belhadj et al.[26]			Present study
	Mode 1	Mode 2	Mode 3	Mode 1	Mode 2	Mode 3
1	0.3	1.0	2.5	0.3	1.0	2.5
2	0.2	0.6	1.6	0.2	0.6	1.6
3	0.2	0.4	0.7	0.2	0.4	0.7
4	0.2	0.3	0.5	0.2	0.3	0.5
5	0.2	0.2	0.2	0.2	0.2	0.2

4.4. The Shear force and bending moment of the SWCNT

Figure 9, 10, 11, 12 depict the three dimensional Shear force and bending moment diagram of the SWCNT for the first two modes. A critical assessment shows that it is possible to track the positions of maximum shear and maximum moment for proper design of the CNT device. The dynamic analysis is important as it helps in the quick monitory and adjustment of the CNT during application.

5. Conclusion

In this paper, analytical investigations of dynamic response of a SWCNT subjected to an external uniform pressure have been carried out using Integral transform. The exact solution as presented in the present study was reduced to the model of Coskun et al. [40] beam model and a very good agreement was obtained. The natural frequency as obtained in this study was also reduced to the model of Belhadj et al. [26] and an excellent agreement was reached. It was established that the Integral transform gave a good result and was efficient for the problem investigated. Also, some necessary parametric studies were performed to fully understand the dynamic behaviour of SWCNTs and to justify the widespread application of SWCNTs subjected to a uniformly distributed pressure acting externally. Furthermore, the dynamic study reveals that a point of maximum shear force on the CNT produces the minimum bending moment at any mode and for any parameter value considered. It is envisaged that this work will enhance the use of SWCNT under the influence of uniform surface pressure to structural, electrical and mechanical applications.

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.

1. Introduction

Numerous areas of medicine necessitate observation and evaluation of relatively large zones of interest and with high optical magnification. These observations are generally performed with microscopes that enable to observe a very small part of the area of interest at a time.

Within the last 150 centuries, photography showed a huge development leading to very high quality pictures with relatively simple and light equipment. On the other hand, rise of digital within the last 30 years has considerably changed ways of living and working. Above all, smartphones became more than entertainment objects: they can be used as a reliable and relatively cheap professional tool in a large set of areas [1].

With the development of informatics rose the interest to digitize, annotate, share and enhance pictures obtained with glass slides on light microscopes [2]. In order to facilitate teaching class but also research work with small budget (microscope without camera), smartphone started to be investigated. Morrison and Gardner [3] proposed a simple method to take pictures with a light microscope and a smartphone, without any adapter. Despite the low cost and the easy way of implementation of this method, an operator needs some practice before managing to take clear pictures, and both hands are busy during the process. Several other authors therefore proposed hand-made and cheap adapters [4, 5]. Using an adapter to avoid blurry pictures is essential, since the outlets of this method are numerous. As an example, some authors already proposed smartphone applications in order to perform medical diagnostics [6]. Long-lasting, affordable, efficient and multi-purpose adapters are therefore needed.

As a consequence, companies started to commercialize products that are dedicated to one or several cell phones. Roy et al. [7] performed a comparison between 6 of them (and testing 3 with different smartphones), a home-made one and the direct use of the smartphone, enabling them to provide their own recommendation. On the other hand, some authors started to propose a direct printing of the adapter thanks to additive manufacturing with plastics [8]. This method has the advantage of providing the right tool with a cost-effective manufacturing process, since 3D-printers are becoming more and more affordable and worldwide.

New technologies of light microscope associated with special software tools allow for picture stitching and therefore enable creation of high-resolution pictures of the whole areas of interest, called virtual slides [9]. Thanks to multiple microscope focuses, it is even possible to obtain a 3D view of a slide (whole slide imaging) [10], which can be very useful for particular medicine areas (cytology, hematology, microbiology) [11]. However, these techniques necessitate a substantial investment for the optical tool and the software. These investments could be prohibitive for small budget institution or teaching purpose.

The objective of this work is therefore to find a way to perform 2D planar stitching of non-structured (meaning no particular organization) photographs taken by non-professional camera, with little pre-processing, no post-processing work, and little to no investment needed, in a fast way, to achieve a virtual slide with high-definition. Similar work has already been performed by Lu et al. [12] that proposed a home-made microscope and self-developed software to perform the stitching. Their method is relatively cheap (less than \(400\) and automatic however it necessitates an initial assembly step that is time-consuming and it is more adapted to remote areas that need a diagnostic system than for student classes with the need of dozens of systems.

The methodology that is presented answers that question by using a simple and cheap light microscope, a relatively recent smartphone, an adapter between microscope and smartphone that has been printed on purpose and a free image-stitching software.

2. Experimental setup description

The objective of this work is to propose a cost-effective and simple experimental set-up and method in order to create high-resolution microscope stitched images. Both the equipment and the software products can therefore be exchanged by equivalent.

2.1. Light microscope, Smartphone and smartphone adapter

The used light microscope is a simple model from Zeiss (Primo Star). It has a fixed and 20 unique objective (\(\times 10\)) and four different eyepieces (\(\times\)4, \(\times\)10, \(\times\)40 and \(\times\)10.), leading to four possible magnifications: "\(\times\)40", "\(\times\)100", "\(\times\)400" and "\(\times\)1000".

The smartphone adapter has been created with a 3D printer (printer brand: Dagoma, printermodel: Neva, material: polylactic acid (PLA), material brand: ICE). The original model can be found on the website thingiverse.com [13]. It has been modified to fit with the used smartphone (Brand: Huawei, model: P20 Pro) thanks to the free platform tinkercad.com [14]. Experimental setup: smartphone, support and light microscope is shown in Figure 1.

2.2. Experimental protocol

In order to perform clear, clean and reproducible high resolution pictures, the following 3 protocol has been used:

• Stabilize the smartphone camera onto the microscope thanks to the dedicated adapter.
• Adjust the camera (manual focus to the infinite, ISO at minimum, Auto White Balance (AWB) to artificial light, shutter speed (S factor) adjustment is function of luminosity).
• Adjust the microscope (maximum luminosity, microscope magnification: \(\times\)40).
• Taking the pictures with a N-shape itinerary, with 20-40% overlap, in a high-quality raw format (the P20 Pro allows the .dng format).
• Convert pictures with a software in a format that takes into account transparency (in the present work, XnView software and .tif format were preferred).
• Erase the black background thanks to an editing image software (in the present work, Paint.NET).
• Automatically stitch the pictures thanks to a dedicated software. In the present work, the free software Image Composite Editor (protocol in Annex 1) has been used because other tested software products were not providing satisfying results (see Appendix 2).
• Compress the obtained jpeg files to reduce the total weight using a batch processing (in the present work, XnView software to reduce the given quality to 50%).

3. Results and discussion

70 slides were tested with the proposed method. 61 experiments were successful. The latter will be discussed in the following section while the 9 failures will be discussed in Section 0.

3.1. Obtained pictures

Figure 2 shows one picture of sagittal cross-section of a permanent maxillary central incisor, taken with the experimental set-up that has been described in Section 0 (microscope magnification: \(\times\)40). Picture is round because of the microscope lens. Quality of the picture is acceptable, but an overall view of the tooth cross-section is not possible.

However, this picture is one piece of the created high-resolution complete section of the tooth that is obtained with the method proposed in Section 0, as illustrated in Figure 3.

Other examples of obtained pictures are available in Appendix 3 and online on a Prodibi database [15].

3.2. Needed time and virtual slide weight

One of the advantages of the proposed method is its relatively fast execution for large groups of glass slides. As detailed in Table 1, a total duration of 375 min has been necessary to create the stitched pictures (for instance the example given in Section 0): it results in an average of 5 min per virtual slide.

Table 1. Details about the procedure duration.

Action	Duration (min)
Taking pictures of the glass slides	120
Copy/Paste and conversion to .tif	30
Erasing black background	100
Stitching	100
Jpeg compression	10
Upload	15
Total	375

Weights of the obtained files depend on the number of pictures that were taken in order to obtain the stitched figures. In the present case, the latter have between 20 Mpx and 250 Mpx, for a total weight of 2-10 Mo (compressed jpeg), 10-60 Mo (uncompressed jpeg) or 90-340 Mo (tif).

Given the actual glass slides, reducing the quality of the jpeg files that is given by the software to 50% allowed a 75% loss of weight with a satisfactory resolution. Choice of the compression factor is a trade-off between quality and weight. It also depends on the initial quality of the pictures and their application (for instance, cytology requires more quality, thus preventing the use of an important compression factor).

3.3. Increasing resolution/microscope magnification

The same method was applied to one sample with a higher microscope magnification (\(\times\)100). Results are compared to the lowest magnification in Figure 4.

The higher magnification did not show significant enhancement of the obtained pictures, for a relatively important increase of weight (150 Mo for an uncompressed .jpeg format and 750 Mo for a .tif format, against 60 Mo and 340 Mo, respectively). However, this result should not be generalized since the potential image resolution is inherent in the optical system [16].

Even higher magnifications (\(\times\)400 and \(\times\)1000) were available with the microscope; however, the lack of multiple focus planes reduces the depth of sharpness, making such magnifications unusable for virtual slides without z-axis.

3.4. Probable causes of failure

Despite its easy way of use, the proposed method might fail to create the virtual slide. In the present case, 9 stitchings out of 70 failed. The 9 failed stitching consist of three different kinds:

• The stitching process is successful, but necessitates an intermediate step.
• The whole stitching process is not successful, but two or three blocks are obtained.
• The stitching process is completely failed, some pictures being not possible to stitch to the others.

In the present work, the main cause of failure seems to be the monotony of the image. Indeed, all failed stitching concern areas close to the tooth's root; no failure was found with coronal structures. Example of a very monotonous fragment of a slide (part of the root) is shown in Figure 5.

Out of the monotony issue, two possible causes of failure were found, yet they can be fixed by slightly modifying the experimental procedure, as illustrated in Table 2.

Table 2. Probable causes of failure and proposed solutions.

Failure	Solution
Not enough overlap between pictures	Take more pictures (higher overlap)
Monotonous subject (not enough forms andcolour variations within the studied slide)	Choosing better slides
	Lowering light to increase variations
	Adding a non-monotonous element on thepicture by displacing the microscope focus
Not enough contrast	Take new pictures with better adjustment
	Modify obtained image in post processing

4. Conclusions and perspectives

4.1. Conclusions

This work is in line with a global investigation about enhancement of microscope sample pictures with high quality, easy way of use and low-cost. Thanks to the progress in informatics, photography and additive manufacturing, it is feasible to meet all these requirements and create satisfying high-resolution stitched pictures of microscope glass slides with limited resources.

While previous works focused on the way of performing pictures of the microscope viewpoint with smartphone, the present method goes beyond and create large pictures that stitch them into a single high-resolution image representing the objects of interest (in the present case, teeth). Up to now, image stitching of microscope pictures was reserved to high-quality high cost microscopes and/or proprietary software products.

As a conclusion, the present works gathered all the separate knowledges that were presented to provide a simple and efficient method that can be used by researchers, teachers or students to create virtual slides that can be studied as a whole and also feed online database.

4.2. Perspectives/Alternatives

In order to enhance the proposed method, one may:

• Use professional software (additional cost).
• Use a dedicated support for the smartphone (additional cost).
• Use a higher-resolution microscope (additional cost).
• Use a microscope with an integrated picture function (additional cost).
• Perform a thorough pre-processing (time consuming).
• Manually assemble the pictures with an image editor (time consuming).
• Use better microscope glass slides (additional cost).

A very interesting method that must be investigated is the use of the panorama mode of the smartphone, the latter performing itself the image stitching.

5. Appendix 1: Image Composite Editor's stitching protocol

In order to reproduce the protocol that has been executed in this work, one may follow the procedure hereinbelow:

• Open the software ICE.
• Add images by using "New Panorama" or by doing drag-and-drop.
• In "Import", stay in "Simple Panorama" and put "Camera Motion" in "Auto-Detect" or "Planar Motion" (preferred here).
• Click on "Next" or "Stitch".
• In "Stitch", modify picture orientation if necessary.
• Click on "Next" or "Crop".
• In "Crop", cut the image if necessary.
• Click on "Next" or "Export"
• Save the picture on the hard drive (preferred format: "JPEG image" with quality "Superb".

6. Appendix 2: Tested software products 1

Different software products were tested for the proposed protocol. They were discriminated because of the reasons that are summarized in Table 3.

Table 3. Discriminated software products.

Software	Reason
Photoshop (with tracing paper fusion)	Stitching fails: resulting image is out of shape
DeepSkyTracker	Necessitates numbered images and following a logical path (i.e. forbidding the use of a non-assisted by microscope protocol)
Fiji (ImageJ) with "stitching" plug-in	Necessitates numbered images and following a  logical path (i.e. forbidding the use of a non-assisted  by microscope protocol)
PhotoStitcher (free version)	Limited functions on free version
Histostitcher	Does not handle numerous images and with high overlapping
Image stitching	Stitching fails: resulting image is out of shape
Autostitch	Stitching fails: resulting image is out of shape
MIST	Necessitates numbered images and following a logical path

7. Appendix 3: Other examples of obtained stitched pictures

In this appendix, some examples of obtained pictures are given ( Figures 6, 7, 8). A more important set of pictures is provided in [15].

8. Appendix 4: Contextualization of magnification comparison and another example

Here we give Contextualization of magnification comparison (see Figures 9, 10).

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.

1. Introduction

Fins are extended surfaces used for heat enhancement in a thermal system. It finds numerous application in a wide variety of industries such as electronics, automobile, aerospace, thermal plant, processing plant etc [1]. It is a known fact that the activities of some industries such as chemical, nuclear and power plant quite often involves excessive generation of large amount of heat, which requires a careful method of heat dissipation. In contrast, a close look at some other industrial applications reveals that heat enhancement into the thermal system may be required. Typical example of this is found in specialized boilers and heat exchanger equipment. Consequently, increasing demand for heat enhancement devices in the industries have provoked a lot of research interest on the different types of fin design, in order to suit ever increasing industrial need. Notably among the existing fin designs are longitudinal or straight fins, radial fins, porous fins, perforated fins, composite and laminated fins to mention but few. A survey of existing literatures reveal an extensive work on the analysis of different types of fin designs. Ghasemi et al. [2], investigated a solid and porous fin with temperature dependent heat generation using differential transform method (DTM) and found out that temperature distribution is strongly dependent on the Darcy and Rayleigh number for the porous fins and the result obtained is in good agreement with the numerical result. Kosarev [3] researched on steady state thermal analysis of rectangular fins with variable thermal conductivity using grapho-analytical method and shows the possibility of solving heat transfer problems with allowance for variation of thermal conductivity of the materials with temperature. Laor and Kalman [4] worked on the performance and optimum dimension of different cooling fins with temperature dependent heat transfer coefficient using theoretical numerical analysis methods and discovered that fin efficiency of spine is greater than that longitudinal fin because shape factor has less effect on fin efficiency of longitudinal fins as compared to that of spine and the optimum dimension of any fin is attained at a point where the maximum value of dissipation per mass for any fin volume and condition is attained. Taler and Taler [5], examined the steady state and transient heat transfer through fins of complex geometry using finite element finite volume method and arrived at the conclusion that the method gave a very satisfactory result, as such can be used to determine the transient response as well as steady state efficiency of fins attached to oval and circular tubes. In the work of Sun et al. [6], collocation spectral method is used to solve the non-linear heat transfer problem in a convective-radiative rectangular fins and the result obtained when verified, shows a good agreement with the analytical solutions obtained in the literature. Ganji and Dogonchi [7], carried out analytical investigation of convective heat transfer of a longitudinal fin with temperature-dependent thermal conductivity, heat transfer coefficient and heat generation using the differential transform method (DTM), the result obtained indicates that the fin tip temperature increases with an increase in heat generation gradient and also observed that the result obtained by (DTM) is in good agreement with existing numerical data. Jasim [8] examined the transient thermal response of two dimensional composite fin using the finite element technique and observed that the larger the conductivity ratio, the faster the time to reach the steady state and further demonstrated that the efficiency of the fin increase with increase in thermal conductivity ratio. Muzychka and Yovanovich [9], developed an analytical model to compute transient thermal stresses in a biomaterial system using Laplace-numerical methods and discovered that the maximum peel stress under steady state conditions shows smaller or negative stress while the transient solution shows a very large maximum stress during initial stage of the heating process of the biomaterial component. Hatami and Ganji [10] worked on the thermal analysis of a convective-radiative fin with temperature dependent thermal properties using the collocation spectral methods. The result of the study reveals an excellent agreement with the analytical result from the literature and as such, could be used as an effective technique in handling non-linear problems in science and engineering.

Based on the existing knowledge that heat conduction rate decrease from the fin base to the tip, recent fin designed shapes are varied along the direction of conduction for optimum material utilization, which further underscore the effect of fin geometry in heat enhancement process as seen in the work of Vahabzadeh et al. [11], where analytical study of porous pin fins with variable section in fully wet conditions was conducted. The result obtained shows a concave profile having the highest efficiency while the rectangular profile having the least. Hatami and Ganji [12], establishes on similar trend, on 'optimisation of longitudinal fins with different for increasing heat transfer'. Chen and Wang [13], investigated trapezoidal fins for assessment of heat sink performance and material savings. The study reveals 0.88% difference in heat transfer ratio between trapezoidal and rectangular profiles. Mosayebidorcheh \emph{et al.} [14], carried out the transient analysis of longitudinal fins with internal heat generation, considering the effects of thermal conductivity and different profiles. A comparative study of longitudinal fins of rectangular, trapezoidal and concave parabolic with multiple non linearity was presented in the work of Torabi et al. [15], which further demonstrated the effect of geometry in heat enhancement. Torabi and Aziz [16], evaluates the thermal response and efficiency of convective-radiative T-shaped fins with temperature dependent thermal, heat transfer coefficient and surface emissivity and proved that T-shaped fins is significant in engineering applications. In a related work of Hatami and Ganji [17], thermal behaviour of longitudinal porous fins of different profiles of ceramic materials are analysed using the least square method, the result shows that fin of exponential shape with material composition(Si3 N4 ) has the highest heat transfer potentials. The effectiveness of spectral element method (SEM) is demonstrated in the study carried out by Jing et al. [18] on thermal behaviour of longitudinal porous moving fins of different profiles, a comparative analysis of the result obtained with the convex profile with the one obtained by Hatami and Ganji [17] reveals that spectral element method predicts better than the least square method. Meanwhile, there is ever increasing interest in the used of most effective method for predicting thermal responses of fins under different conditions. Hence, in this work, an attempt is made to demonstrate the effectiveness of differential transform method (DTM) in solving a thermal model with high non-linearity, by making a direct comparison with the work of Hatami and Ganji [17] and Ma et al. [18]. To the best of the Arthur's knowledge, no research work up to date has used the differential transform method (DTM) to solve heat transfer problems in a moving porous fin with irregular profiles. Hence DTM will be used to solve for the conductive, convective and radiative heat transfer in moving porous fins of three different geometries. The profiles under consideration are trapezoidal, concave parabolic and convex parabolic shapes. The physical model for the work will be presented and discussed. Also, parametric studies will be conducted to examine the effect of thermo-physical properties such as the Peclet number, porosity, surface emissivity coefficient, non-dimensional heat generation at ambient temperature, heat generation parameter, non-dimensional ambient temperature, power index of heat transfer coefficient, convective-conductive and radiative-conductive parameter and irregular shape on non-dimensional temperature distribution. Fin efficiency will also be examined and discussed.

2. Physical model and the governing equations

This study is focused on porous fins of trapezoidal (Figure 1), convex parabolic (Figure 2) and concave parabolic (Figure 3) profiles moving horizontally with constant velocity. The fins are exposed to convective and radiative environment. Therefore, the thermal analysis is based on the assumptions that the thermal processing of the porous fins is in a steady state and the porous medium is isotropic, homogenous and saturated with single phase fluid. The interaction between the fluid and the porous medium is governed by Darcy law, with the fin size thickness being smaller than in other dimensions, also the transverse Biot is very small, thus making temperature varying significantly only along the longitudinal direction. The surface of the porous fin is assumed grey and diffuse, with the solid and fluid being in local thermal equilibrium. The porous medium is assumed as dispersion medium, however, the heat transfer between internal pores and fluid is omitted in this work. The fin base is maintained at uniform temperature with no contact resistance, and the fin tip is insulated (adiabatic). Lastly, parameters such as heat transfer coefficient, surface emissivity, heat generation vary with temperature by the following equations: and Based on these assumptions, the energy balance on the element of porous fin is given according to energy conservation law. Energy in the left + heat generated in the element = energy out right face + energy lost by immersed fluid as shown below Thickness of the element local fin is expressed as Where \(n = 1, \frac{1}{2}, 2\) for the trapezoidal, convex parabolic and convex parabolic respectively.

The mass flow rate equation is expressed below as: where \(v_{w}\) is the fluid velocity where \(k =\) permeability of porous fin, \(g = \) gravitational acceleration, \(\beta = \) coefficient of thermal expansion and \(v_{f} = \) kinematic viscosity of the fluid. The resulting equation below is obtained when Equations (1), (2), (3), (5), (6) and (7) are substituted in Equation (4) where, this relationship holds for effective thermal conductivity, fluid thermal conductivity and solid conductivity. Similarly as referenced in [13], the following non dimensional parameters are adopted, Substituting the non-dimensional quantities in Equation (8), the resulting non-dimensional energy equation is obtained. The boundary condition for Equation (11) is shown below:

3. Concept and Application of differential transformation method

The differential transform method (DTM) was first used by Zhou [14]. The concept described \(f(x)\) to be an analytical function in the domain \(D\), while \(x = x_{j}\) represent any point in the domain. The Taylor's series expansion of \(f(x)\) which has its center point as \(x_{j}\) can be expressed as follows: A case where \(x_{j} = 0\), Equation (14) reduces to McLaurin series expansion of \(f(x)\) which is expressed as In the work of Zhou [14], the differential transformation of the function \(f(x)\) is expressed as: where \(f(x)\) represents the original analytical function and \(F(k)\) is the transformed function. The differential spectrum of \(F(k)\) is confined within an interval \( x \in [0, Y]\) , where \(Y\) is a constant value. Therefore, the differential inverse transform of \(F(k)\) is given as The more the number of terms of the series expansion generated, the higher the accuracy of result. Hence, we approximate \(f(x)\) by a finite series, Equation (17) becomes where \(\infty = n\) in the case of a finite series. More so, the values of function \(F(k)\) and the values of argument \(k\) are referred to as discrete. Some of the useful mathematical function and their differential transformation are given in the Table 1 [1].

Table 1. Transformations of some functions.

	Original function	Transformed function
1.	\(f(x) = u(x) \pm v(x)\)	\(F(x) = U(x) \pm V(x)\)
2.	\(f(x) = \alpha u(x) \)	\(F(x) = \alpha U(x)\)
3.	\(f(x) = \frac{dy(x)}{dx}\)	\(F(x) = (k+ 1) F(k+ 1)\)
4.	\(f(x) = \frac{d^2y(x)}{sx^2}\)	\(F(x) = (k + 1)(k + 2)F(k + 2) \)
5.	\(f(x) = \frac{d^n y(x)}{dx^n}\)	\(F(x) = (k + 1)(k+ 2) \dots (k + n)F(k + n)\)
6.	\(f(x) = u(x)v(x)\)	\(F(k) = \sum_{l = 0}^{k} U(l)V(k - l)\)
7.	\(f(x) = a (constant)\)	\(F(k) = \alpha \delta(k)\)     where  \( \delta(k) = \begin{cases} 1 \ if \ k = 0 \\ 0 \  if \ k \neq 0 \end{cases} \)
8.	\(f(x) = x\)	\(F(k) = \delta (k - 1)\)
9.	\( f(x) = x^n\)	\(F(k) = \delta (k - n)\)
10.	\(f(x) = exp (\lambda x)\)	\(F(k )= \frac{\lambda^k}{k!}\)
11.	\(f(x) = (1 + x)^n \)	\(F(k) = \frac{n(n -1 ) \dots (n- k + 1)}{k!}\)
12.	\(f(x) = sin(\omega x + \alpha )\)	\(F(k) = \frac{\omega^k}{k!}sin\big( \frac{\pi K}{2!} + \alpha\big)\)
13.	\(f(x) = cos(\omega x + \alpha )\)	\(F(k) = \frac{\omega^k}{k!}cos\big( \frac{\pi K}{2!} + \alpha\big)\)

Applying the DTM recursive relation to the governing equation, we have Solving Equations (19), (20), (21) with the transformed boundary conditions, the required series solution is represented as;

4. Verification

Table 2 depicted the comparative result obtained from silicon nitride of convex cross-section using the four different methods. The agreement of DTM employed in the present study with numerical as well as already published works [18], verifies the scheme.

Table 2. Verification of DTM.

X	\(\theta\)
	LSM	SEN	DTM	Numerical Solution
1	0.885112812	0.885112628	0.848340571	0.84965339
0.9	0.886749804	0.886749648	0.8505218	0.852305597
0.8	0.891018373	0.891018069	0.85666425	0.85927805
0.7	0.897194237	0.897193917	0.866233388	0.869518003
0.6	0.904872207	0.904871474	0.878778309	0.882358962
0.5	0.913966193	0.913965953	0.893924218	0.897381433
0.4	0.924709201	0.924708212	0.911365183	0.914332606
0.3	0.937653337	0.937653293	0.93085726	0.933078485
0.2	0.9536698	0.953669404	0.952212099	0.953575046
0.1	0.97394889	0.973948376	0.975291078	0.975851246
0.0	1.0000000000	1.0000000000	1.0000000000	1.0000000000

4.1. The model efficiency determination

The following expression is used to define the efficiency (i.e., the ratio of actual heat transfer rate to the ideal heat transfer if entire fin area is maintained at the base temperature)

5. Analysis of result and discussion

Based on the numerical research performed by Jing Ma et al. [18], the values of these physical parameters are adopted \begin{align*} \psi = 0.1, \ \ \varphi = 0.8, \ \ \alpha = 0.6, \ \ m = 2, \ \ P_{e} = 1.0, \ \ N_{cc} = 1.0, \end{align*} \begin{align*} N_{rc } = 1.0, \ \ Da = 10^{-5}, \ \ \theta_{a} = 0.3, \ \ Q_{a} = 0.1, \ \ c_{1} = c_{2} = c_{3} = 0.2. \end{align*} Thermal physical properties of the selected materials used are the same as in [18].

Figure 4 depicts the effect of different geometry on non-temperature distribution of Aluminium porous fins, with convex parabolic having the highest temperature distribution, while the trapezoidal having the least, this may be due to their respective convective and radiative surface areas, such that the more the surface area available for heat dissipation the lower the lower the non-dimensional heat distribution in the porous fin. This shows that the trapezoidal fins has the widest surface area while the convex parabolic has the least. In Figures 5, 6 and 7, the effect of temperature distribution on different materials of fin with trapezoidal, concave and convex geometry is shown.

It is observed that in all three geometries considered, the trend is the same, that is, copper porous fin has the highest temperature while stainless steel has the least. The is due to higher thermal conductivity of copper material as compared to that of Aluminium, silicon nitrides and stainless steel material used. The lowest temperature recorded for the stainless steel is on account of its lowest thermal conductivity and highest thermal resistance.

The effect of different values of surface emissivity constant on the temperature distribution of trapezoidal silicon nitride porous fin is depicted in Figure 8. Since surface emissivity show a direct relationship between the rates of variation in the surface emissivity with temperature. As it increases, radiative heat dissipation increases from the porous surface of the fin and the non-dimensional temperature distribution increases accordingly.

The effect of different power index of heat coefficient in a concave aluminium porous fin is displayed in Figure 9. The trend shows that the higher the value of power index of heat coefficient, the higher the non-dimensional heat distribution in the Aluminium porous fin. The curve marked \(m=0\) represents temperature distribution with forced convection model, while the curve marked with \(m=0.25\) represent temp distribution with laminar natural convection model and the curve marked with \(m=2\) represent non dimensional temperature distribution with nucleated boiling model which lead to heat loss reduction.

Figure 10 shows a rapid increase in temperature distribution with increase in peclet number for the convex stainless steel porous fins. The trend shows that as the peclet number increases, the non-dimensional temperature becomes higher. This is due to the fact that the porous fin move faster as the peclet number increases and the heat transfer time between the fin surface and the ambient fluid get shorter, while leads to rise in temperature distribution. This phenomenon is established in the work of Sivaraj et al. [19], where the increase in Peclet number is associated with decrease in shear stress. In a similar study, carried out by Makinde and Animashaun [20], it was established that Peclet number is a decreasing property of diffusion gradient of mobile microorganism along the wall.

The effect of internal heat generation Qa and the effect of heat generation parameter (c1, c2 and c3) on the convex copper porous fin is examined in Figure 11 and Figure 12 respectively. The trend shows that Increase in internal heat generation (Qa) and heat generation parameter lead to increase in non-dimensional temp distribution. This is due to the fact that the more the heat generation the higher the non-dimension temp distribution, because fin must dissipate larger heat to the ambient environment.

Non-dimensional temperature increase with increase in porosity as depicted in Figure 13. As porosity increases the heat transfer rate decreases and the effective thermal conductivity of the porous fin decreases as well due to void increment. However, the more the porosity the more the convective and radiative heat enhancement capacity of the fin due to large surface area.

In Figure 14, the effect of convective conductive parameter (Ncc) and in Figure 15, radiative conductive parameter Nrc is examined. Convective conductive parameter (Ncc) connotes ratio of convective heat dissipation to that of conductive heat transfer. As convective conductive parameter (Ncc) decrease from 10-1, convective heat loss from the surface of the porous fin decreases. In contrast, As radiative-conductive parameter Nrc decreases, the non- dimensional temperature increases, this is due to the fact that radiative heat loss is directly proportional to the difference in the fourth power of fin temperature and ambient temperature.

Figure 17 shows the trend of variation in efficiency with Peclect number for Aluminium porous fins. The trend shows an increase in efficiency with increase in Peclet number, with the convex Aluminium has the highest efficiency, followed by the concave profile and the least efficiency is recorded for the trapezoidal profile. However, this result does not take into consideration the difference in material consumption of different irregular profile, as such, may not provide good basis for comparison. For the same thickness and length concave parabolic porous fin consume less materials than the concave parabolic porous fin and the trapezoidal parabolic porous fin . Hence a volume adjusted efficiency is used instead.

Figure 18 depicts the Variation in volume adjusted efficiency with peclect number for Aluminium porous fins. The trend shows an increase in volume adjusted efficiency with peclet number, with concave parabolic porous fin having the highest volume adjusted efficiency, followed by the convex parabolic porous fin and the least volume adjusted efficiency ids recorded in trapezoidal porous fin

In Figure 19 copper porous fins displayed the same trend in volume adjusted efficiency as compared with Aluminium porous fin in Figure 18.

In Figure 20, stainless steel porous fins displayed the same trend in volume adjusted efficiency as compared with both Aluminium porous fin in Figure 18. Copper porous fin in Figure 19. Thus, it can be safely concluded that volume adjusted efficiency which takes into consideration equal volume of different geometry in determining the efficiency of their heat enhancement capacity is a good basis for comparing the efficiency of fins of different geometry.

5. Conclusion

The comparative analysis carried out on the effect of materials on non-dimensional temperature distribution revealed that copper fins has the highest temperature while the stainless steel gets the lowest. More so, the fins with concave geometry gives the highest volume adjusted efficiency with increase in Peclet number while that with convex profile has the least. It is envisage that the obtained results would be useful in the future design of fins with optimum size reduction and high efficiency.

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.

1. Introduction

The practical application of viscoelastic fluids is to polymer industry, food processing, biological structure, chemical engineering etc. In general, a viscoelastic fluid can influence the transport properties of mass and momentum by varying the stability of laminar motion associated with diffusing particles [1,2]. One major difficulty dealt with such fluids is fourth-order derivative included in the Navier-Stokes momentum equation which cannot be treated using perturbation solutions. Therefore, in any given geometry, it is required to distinguish between the inflow and outflow boundaries for deformation of a fluid [3]. It is worth noting that there exist only a few works in the literature for detailed flow investigation of viscoelastic fluids within a given volume due to external fields. In this way, Rajagopal et al. [4] formulated the idea of introducing a viscoelastic fluid past a stretching wall based on Beard and Walter's theorem [5].

They could give rigorous proof of their Runge-Kutta method (RKM) by estimating the norm of series expansion and assuming, for simplicity, that the viscoelastic parameter is small but nonzero. Under this assumption, Nandeppanavar et al. [6] developed those of Subhas and Veena [17] to study phenomena like heat transfer through a porous medium with cooling processes. Furthermore, in a similar manner, Seth et al. [8], Nadeem et al. [9], Abdullah et al. [10], Chang et al. [11], Sivaraj and Kumar [12], Tariq et al. [13] and Prakash et al. [14] presented some thermodynamic review of viscoelastic fluids with both laminar flow velocity and temperature distributions.

Unlike the numerous examples of HBA, especially for cases subjected to external fields [15, 16, 17, 18, 19, 20, 21, 22, 23, 24], yet there is a lack of convergence in the choice of auxiliary parameter for analyzing the viscoelastic fluid past a stretching wall. This paper is intended only as a brief communication to represent conclusively that the HBA is useful tool for achieving much faster convergence. The rest of the paper is organized as follows.

In Section 2, a summary of governing equations based on the viscoelastic Walter's B fluid theory is reviewed. Section 3 provides the HBA and its further correspondence. Section 4 is exclusively devoted to results and discussion. The concluding remarks are summarized in Section 5.

2. Problem formulation

According to basic hypothesis of the viscoelastic Walter's B fluid theory, the governing PDEs and associated boundary conditions can be expressed as [4] where \(v\) is the kinematic viscosity, \(E\) is the elastic parameter and \(\alpha\) is the stretching rate. Introducing the variables \(\eta = \sqrt{\frac{\alpha}{v}}y,\) \(u = \alpha x \varphi_{,\eta}\) and \(v = -\sqrt{\alpha v \varphi},\) the non-dimensional form of governing differential equation is given by with the boundary conditions where \(H = \frac{\lambda E}{v}\) is the viscoelastic parameter. Here, the shear stress at the wall is defined as [4]

3. Solution methodology

Let us suppose that the general nonlinear problem takes the form where \(\mathcal{N}\) is a nonlinear operator. Using \(q \in [0,1]\) as an embedding parameter, the homotopy function \(\mathcal{H}\) is constructed as [25] where \(\bar{\varphi}\) is an unknown function of \(\eta\) and \(q\), \(\mathcal{L} \neq 0\) is an auxiliary linear operator, \(\varphi_{o}\) is an initial guess of \(\varphi\), and \(\hslash \neq 0\) is an auxiliary parameter which provides the desired convergence of series expansion. It is to be noted here that, as \(q\) is increased from \(0\) to \(1\), \(\bar{\varphi}(\eta;q)\) varies from the initial guess to the exact solution. Thus, in view of Equations (7) and (8), \(\bar{\varphi}(\eta;0) = \varphi_{o}(\eta)\) and \(\bar{\varphi}(\eta;1) = \varphi(\eta)\) are the solution of \(\mathcal{H}[\bar{\varphi}(\eta;q);q] |_{q= 0} = 0 \) and \(\mathcal{H}[\bar{\varphi}(\eta;q);q] |_{q= 1} = 0 \), respectively. \(\bar{\varphi}(\eta;q)\) can be expanded in a Taylor's series with respect to \(q\) as where \(\varphi_{j}\) is the jth-order deformation derivative. Equating the homotopy function and \(q\) to zero, the zeroth-order deformation equation is constructed as [25] Also, differentiating \(\mathcal{H}[\bar{\varphi}(\eta;q);q] = 0\), \(j\) times with respect to \(q\) setting \(q = 0\) and dividing it by \(j!\), the \(jth-order\) deformation equation is obtained as where To apply a similar procedure on the governing differential Equation (3) and associated boundary conditions given in Equation (4), the initial guess is selected as The auxiliary linear operator in this case is assumed to be with the property where \(C_{1} - C_{3}\) are the integration constants. The next step is to expand \(\varphi(\eta;q)\) in a Taylor's series as The nonlinear operator for Equation (3) can be defined as The zeroth-order deformation equation can be rewritten in the equivalent form with the boundary conditions Here, the \(jth-order\) deformation equation is given by which goes to zero boundary conditions. Therefore, the \(jth-order\) approximate solution of Equation (19) takes the form where \(\varphi^{*}_{j}(\eta)\) is a particular solution. After solving the \(jth-order\) deformation Equation (19) and then rearranging, the \(jth-order\) approximate solution is obtained as

4. Results and discussion

To illustrate the computational efficiency and validity of the present HBA outlined in Section 3, the variation of shear stress at the wall versus given values of the viscoelastic parameter is compared with the semi-analytical findings analyzed by Rajagopal et al. [4]. It is to be noted here that, due to Hayat et al. [26], the auxiliary parameter in this case is selected as \(\hslash=-0.125\). Based on the results presented in Table 1, as the viscoelastic parameter is increased, the shear stress at the wall enhances and thereby reduces its flow resistance. Furthermore, it is seen that the 9th-order HBA agrees remarkably well with those reported by Rajagopal et al. [4]; because it only suffers from an error of at most 0.096% for all cases listed in Table 1. Therefore, it can be concluded that more accurate results are provided by the 9th-order HBA than those of 5th- and 7th-order ones.

Table 1. Verification of the shear stress at the wall.

H	Present (\(\hslash\)=-0.125)			Rajagopal et al. [4]
	\(k\)=5	\(k\)=7	\(k\)=9
0.005	-0.9915	-0.9936	-0.9965	-0.9975
0.01	-0.9886	0.9919	-0.9939	-0.9949
0.03	-0.9795	-0.9815	-0.9837	-0.9846
0.05	-0.9699	-0.9712	-0.9729	-0.9738

It is worth noting that a series expansion with a faster convergence rate can be expected if the value of auxiliary parameter is optimized. According to the QRL [27], the averaged square residual error is calculated as Solving equation \(\triangle_{k, \hslash} = 0\) in terms of \(\hslash\) and using the fact that \(-2.1 \leq \hslash \leq 0.15\) [27], minimizes its averaged square residual error at the any order of approximation [25,28,29,30]. In this way, the variation of averaged square residual error versus different values of \(\hslash\) is depicted in Figure 1.

As it is seen form Figure 1, the averaged square residual error takes its minimum possible value when the auxiliary parameter is taken to be -0.122. Hence, this important finding can be considered as a tool to accelerate convergence of the present HBA.

Table 2 investigates uniqueness of the present HBA theoretically by making a correspondence between the averaged square residual error and order of approximation. According to this Table 2, it is seen that increasing the order of approximation reduces monotonically the value of averaged square residual error when it is subject to the viscoelastic parameter \(Η=0.2\). Furthermore, one can say that accounting for the minimization of averaged square residual error is so essential to reduce the CPU time without any loss of accuracy. This fact is clearly shown in Figure 2.

Table 2. onvergence of the series expansion.

\(\eta\)	\(k\) = 5		\(k\) = 7		\(k\) = 9
	\(\hslash\)	\(\triangle_{k}\)	\(\hslash\)	\(\triangle_{k}\)	\(\hslash\)	\(\triangle_{k}\)
\(0\)	\(-0.115\)	\(9.89 \times 10^{10}\)	\(-0.119\)	\(7.40 \times 10^{-10}\)	\(-0.122\)	\(4.96 \times 10^{-11}\)
\(0.2\)	\(-0.115\)	\(1.16 \times 10^{9}\)	\(-0.119\)	\(7.69\times 10^{-10}\)	\(-0.122\)	\(5.54\times 10^{-11}\)
\(0.4\)	\( -0.115 \)	\(2.44\times 10^{9}\)	\( -0.119 \)	\(7.91\times 10^{-10}\)	\( -0.122 \)	\(6.23\times 10^{-11}\)
\(0.6\)	\( -0.115 \)	\(3.50\times 10^{9}\)	\( -0.119  \)	\(8.16\times 10^{-10} \)	\( -0.122 \)	\(6.98\times 10^{-11}\)
\(0.8\)	\(-0.115 \)	\(4.73\times 10^{9}\)	\(-0.119  \)	\(8.38\times 10^{-10} \)	\( -0.122 \)	\(7.75\times 10^{-11}\)
\(1\)	\( -0.115 \)	\(6.02\times 10^{9} \)	\( -0.119 \)	\(8.64\times 10^{-10} \)	\( -0.122 \)	\(8.39\times 10^{-11}\)

In view of such configuration seen in Figure 2, there exists an excellent consistency between the 9th- and 12th-order HBA; that is, the local velocity distribution converges certainly for \( \hslash=-0.122\). Furthermore, it is to be noted here that this observation is in agreement with the analytical results reported by Mahabaleshwar et al. [31] for analyzing viscoelastic Walter's B fluid past a stretching wall through a porous medium.

5. Concluding remarks

This paper focused on the problem involving the momentum of viscoelastic Walter's B fluid past a stretching wall using HBA. The QRL was employed to introduce a criterion for minimizing the averaged square residual error at each step. The present findings are compared and verified by those available results in the open literature. Here, the main conclusions are summarized as

1. The 9th-order HBA represents a high accuracy approximation than 5th- and 7th-order ones.
2. Using the auxiliary parameter \(\hslash=-0.122\) yields an absolutely convergent series expansion.
3. The CPU time will be decreased when the averaged square residual error is minimized.
4. Because of the difficulty in evaluating the fourth-order derivative of Equation (3), the 9th-order HBA is applicable only when the initial guess is proportional to the boundary conditions given in Equation (4).

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.

1. Introduction

Over the decades, the fractional calculus has been building a great history and consolidating itself in several scientific areas such as: mathematics, physics and engineering, among others. The emergence of new fractional integrals and derivatives, makes the wide number of definitions becomes increasingly larger and clears its numerous applications. Recently, the existence of solutions of initial and boundary value problems for differential equations involving Hilfer fractional derivative has a considerable attention[1,2,3,4,5,6,7,8].

Very recently, Almeida[9] introduced a new fractional derivative named by \(\psi\)-fractional derivative with respect to another function, which extended the classical fractional derivative and also studied some properties like semigroup law, Taylor's Theorem and so on. Thereafter, Sousa and Oliveira[10,11] initially studied a Cauchy problem for fractional ordinary differential equation with \(\psi\)-Hilfer operator with respect to another function, in order to unify the wide number of fractional derivatives in a single fractional operator and consequently, open a window for new applications and established a new Gronwall inequality to derive a prior bound of a solution. The authors studied the Leibniz type rule: \(\psi\)-Hilfer fractional operator in[12].

The oscillation theory as a part of the qualitative theory of differential equations has been developed rapidly in the last decades and there has been a great deal of work on the oscillatory behavior of integer order differential equations. However, there are only very few papers dealing with the oscillation of FDEs; see[13, 14, 15]. The study of oscillation and other qualitative properties of fractional dynamical systems such as stability, existence, and uniqueness of solutions is necessary to analyze the systems under consideration[16, 17].

Motivated by[18] and the aforementioned papers, we study the oscillatory theory for \(\psi\)-Hilfer fractional type FDEs of the form

where \(D^{\alpha,\beta;\psi}_{a^{+}}\) denotes the \(\psi\)-Hilfer fractional derivative of order \(0< \alpha< 1\) type \(0\leq \beta \leq 1\), \(I^{1-\gamma;\psi}_{a^{+}}\) is the \(\psi\)-Riemann-Liouville fractional integral with \(\gamma=\alpha+\beta(1-\alpha)\) and \(b_{1}>0\).

We assume in this paper that the functions \(f_{1}, f_{2}\) and \(w\) are continuous. The solution representation of (1)-(2) can be written as We only take those solutions which are continuous and continuable to \((a,\infty)\), and are not identically zero on any half-line \((b,\infty)\) for some \(b\geq a\). The term "solution"' henceforth applies to such solutions of equations (1) or (3). A solution is said to be oscillatory if it has arbitrarily large zeros on \((0,\infty)\); otherwise, it is called nonoscillatory.

2. Main results

We will make use of the conditions: and where \(p_{1},p_{2}\in C\left([a,\infty], \mathbb{R}^{+}\right)\) and \(u,v >0\) are real numbers. We will use the following lemma [19, Lemma 1]

Lemma 1.\label{lem2.1} For \(\mathscr{X}\geq 0\) and \(\mathscr{Y}> 0\), we have

and where equality holds if and only if \(\mathscr{X}=\mathscr{Y}\).

Now we may give our first theorem when \(f_{2}=0\).

Theorem 2. Let \(f_{2}=0\) and condition \eqref{e2.1} hold. If

and

Proof. Let \(x(t)\) be a non-oscillatory solution of equations (1)-(2) with \(f_{2}=0\). Suppose that \(T>a\) is large enough so that \(x(t)>0\) for \(t\leq T\). \par Let \(F(t)=w(t)+f_{2}(t,x(t))-f_{1}(t,x(t))\), then we see from (3) that

and hence where, Note that the improper integral on the right is convergent. Applying the limit inferior of both sides of inequality (11) as \(t\) \(\rightarrow \infty\), we obtain a contradiction to condition (8). In case \(x(t)\) is eventually negative, a similar argument leads to a contradiction with (9).

Next we have the following results.

Theorem 3. Let conditions (1)-(2) and (2) hold with \(v>1\) and \(u=1\). If

and where $$ \mathscr{H}_{v}(s)=(v-1)v^{\frac{v}{(1-v)}}p_{1}^{\frac{1}{(1-v)}}(s)p_{2}^{\frac{1}{(v-1)}}(s), $$ then every solution of equation (1) is oscillatory.

Proof. Let \(x(t)\) be a nonoscillatory solution of equations (3), say, \(x(t)>0\) for \(r\geq T>a\). Using (5) in equation (3) with \(u=1\) and \(v>1\) and \(t\geq T\), we find

We apply (6) in Lemma (1) with $$ \lambda=v, \mathscr{X}=p_{1}^{\frac{1}{v}}x \quad \mbox{and} \quad \mathscr{Y}=\left(p_{2}p_{1}^{\frac{-1}{v}}/v\right)^{\frac{1}{(v-1)}} $$ to obtain Using (16) in (15), we have \begin{align*} \Gamma(\gamma)(\psi(t))^{1-\gamma}x(t)\leq c(T)+(\psi(t))^{1-\gamma}\int_{T}^{t}\psi^{'}(t)\left[\psi(t)-\psi(s)\right]^{\alpha-1}\left[w(s)+\mathscr{H}_{v}(s)\right]ds, \quad t\geq T. \end{align*}

The rest of the proof is the similar as in that of Theorem 2.

Theorem 4. Let condition (4) and (5) hold with \(v=1\) and \(u< 1\). If

and where, $$ \mathscr{H}_{u}(s)=(1-u)u^{\frac{u}{(u-1)}}p_{1}^{\frac{u}{(u-1)}}(s)p_{2}^{\frac{1}{(1-u)}}(s), $$ then every solution of equations of (1)-(2) is oscillatory.

Proof. Let \(x(t)\) be a nonoscillatory solution of equations (3), say \(x(t)>0\) for \(t\geq a >1\). Using condition (5) in (3), with \(v=1\) and \(u< 1\), we obtain

Now we use (7) in Lemma 1 with $$ \lambda=u, \ \mathscr{X}=p_{2}^{\frac{1}{u}}x\quad \mbox{and} \quad \mathscr{Y}=\left(p_{1}p_{2}^{\frac{-1}{u}}/u\right)^{\frac{1}{(u-1)}} $$ to get Using (20) in (19) then yields \begin{align*} \Gamma(\alpha)(\psi(t))^{1-\gamma}x(t)\leq c(T)+(\psi(t))^{1-\gamma}\int_{t}^{T}\psi^{'}(s)\left(\psi(t)-\psi(s)\right)^{\alpha-1}\left[w(s)+\mathscr{H}_{u}(s)\right]ds, \quad t\geq T. \end{align*} The rest of the proof is the similar as in that of Theorem 2.

Theorem 5. Let condition (4) and (5) hold with \(v>1\) and \(u< 1\). If

and where \begin{align*} \mathscr{H}_{v,u}(s)&=(v-1)v^{\frac{v}{(1-v)}}\epsilon^{\frac{v}{(v-1)}}(s)p_{1}^{\frac{1}{(1-v)}}(s)+(1-u)u^{\frac{u}{(1-u)}}\epsilon^{\frac{u}{(u-1)}}(s)p_{2}^{\frac{1}{(1-u)}}(s) \end{align*} with \(\epsilon\in C\left([a,\infty], \mathbb{R}^{+}\right)\), then every solution of equation (1)-(2) is oscillatory.

Proof. Let \(x(t)\) be a nonoscillatory solution of (1)-(3), say \(x(t)>0\) for \(t\geq T>a\). Using (5) in equation (3) one can easily write that

We may bound the term \(\left(\epsilon x-p_{1}x^{v}\right)\) and \(\left(p_{2}x^{u}-\epsilon x\right)\) by using inequalities (16) (with \(p_{2}=\epsilon\)) respectively; to get \begin{align*} \Gamma(\alpha)(\psi(t))^{1-\gamma}x(t)\leq c(T)+(\psi(t))^{1-\alpha}\int_{T}^{t} \psi^{'}(s)\left(\psi(t)-\psi(s)\right)^{\alpha-1}\left[w(s)+\mathscr{H}_{v,u}(s)\right]ds, \quad t\geq T. \end{align*} The rest of the proof is the similar as in that of Theorem 2.

Remark 1. The result obtained from (1) are with different nonlinearities and one can observe that the forcing term \(w\) is unbounded, and its oscillatory character is inherited by the solutions.

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.