1. Introduction

The effective conversion of the electrical energy into mechanical energy and vice versa has led the piezoelectric materials to important applications in many engineering structures such as sensors, actuators, intelligent structures, etc. Thermal effects, such as temperature induced deformation and the pyroelectric effect, are especially important for many smart ceramic materials. Thus, a coupling of thermo-electro-mechanical fields is needed to be taken into account if a temperature load is considered in a piezoelectric solid. Models taking into account thermal and piezoelectric effects can be found in [1,2,3].

Thermal effects in contact processes affect the composition and stiffness of the contacting surfaces, and cause thermal stresses in the contacting bodies. When extending contact problems to thermomechanics, additional thermal effects need to be accounted for: heat conduction appears through the contact interface and frictional work is converted to heat. Recent models of frictional contact problems involving thermo-piezoelectric materials can be found in [4,5,6] and the references therein. There, besides the rigorous construction of various mathematical models of contact for thermo-piezoelectric materials, the unique weak solvability of these models was proved, by using arguments of variational and hemivariational inequalities. Numerical analysis of the problems, including numerical simulations, can be found in [7,8]. In [7] the process was assumed to be static, the contact was described with Signorini condition and Tresca's law of dry friction, and a regularized electrical and thermal conductivity conditions and, in [8], the process was assumed to be quasistatic, the contact was assumed to be bilateral, in which contact is always maintained; and associated to the Tresca friction law, and to the heat exchange condition. There, discrete schemes to approximate the problems were considered and implemented in a numerical code, and numerical simulations were provided.

The present paper represents a continuation of [8] and it deals with a mathematical model which describes the frictional contact between a thermo-piezoelectric body and a thermally conductive foundation. We use both the thermo-electro-elastic constitutive law used in [8] but unlike [8], we assume here that the process is dynamic and the foundation is completely rigid and we model the contact with the Signorini condition with Coulomb's law of dry friction. This condition is other physical setting (see, e.g., [9,10]). Also, note that, unlike [8]; here the model includes frictional heat generation and the heat exchange condition, in which the heat exchange coefficient is not a constant but a function of the contact pressure (see [11,12]). The other trait of novelty of the present paper consists in the fact that here we deal with the numerical approach of the problem and provide numerical simulations. The corresponding numerical scheme is based on the spatial and temporal discretization. Furthermore, the spatial discretization is based on the finite element method, while the temporal discretization is based on the Euler scheme. Then, the scheme was utilized as a basis of a numerical code for the problem, in which we develop a specific contact element. We need this element in order to take into account the coupling of the mechanical and thermal unknows on the contact boundary condition. By using the code, simulation results on numerical example are presented.

The rest of paper is structured as follows. In Section 2, we describe our model. Section 3 introduces the variational formulation of the problem, and a fully discrete variational approximation by considering a hybrid formulation. The numerical algorithm used for solving the discrete problem is described in Section 4, where some numerical simulations are also presented to highlight the performance of the method and the effects of the conductivity of the foundation, as well. Finally, in Section 5, we present some conclusions and perspectives.

2. The model

The physical setting is the following: A piezoelectric body occupies a regular domain \(\Omega\subset{\mathbb{R}}^d\,\, (d=2,3) \) with a smooth boundary \(\partial \Omega=\Gamma\). The body is submitted to the action of body forces of density \(\mathbf{f}_{0}\), a volume electric charges of density \(\phi_0\) and a heat source of constant strength \(\vartheta_0 \). It is also submitted to mechanical, electric and thermal constraints on the boundary. To describe them, we consider a partition of \(\Gamma\) into three measurable parts \(\Gamma_D\), \(\Gamma_N\), \(\Gamma_C\), on one hand, and a partition of \(\Gamma_D \cup \Gamma_N\) into two open parts \(\Gamma_a\) and \(\Gamma_b\), on the other hand. We assume that \(meas\,\Gamma_D>0\) and \(meas\,\Gamma_a>0\). The body is clamped on \(\Gamma_D\), therefore, the displacement field vanishes there. Moreover, we assume that a density of traction forces, denoted by \(\mathbf{f}_N\), acts on the boundary part \(\Gamma_N\). We also assume that the electrical potential vanishes on \(\Gamma_a\) and a surface electric charge of density \(\phi_b\) is prescribed on \(\Gamma_b\). We suppose that the temperature vanishes in \(\Gamma_D\cup\Gamma_N\). In the reference configuration, the body is in contact over \(\Gamma_C\) with a thermally conductive foundation. The contact is modelled with a Signorini's conditions and a version of Coulomb's law dry friction. Here, we study the evolution of the state of the system on a finite time interval \([0,T]\), with \(T > 0\). Here and everywhere in this paper \(i, j, k, l\) run from 1 to \(d\), summation over repeated indices is implied and the index that follows a comma represents the partial derivative with respect to the corresponding component of the spatial variable, i.e., \(f_{,i}=\frac{\partial f}{\partial x_i}\).

We denote by \(\mathbf{u} = (u_i) \in\mathbb{R}^d\) the displacement vector, by \(\sigma=(\sigma_{ij})\in \mathbb{S}^d\) the stress tensor, by \(\mathbf{\varepsilon}(\mathbf{u})=(\varepsilon_{ij}(\mathbf{u}))\in \mathbb{S}^d\) the linearized strain tensor, i.e., \(\varepsilon_{ij}(\mathbf{u})=(u_{i,j} + u_{j,i})/2\), by \(\mathbf{E}(\varphi)=-\nabla\varphi= - (\varphi_{,i})\in\mathbb{R}^d\) the electric vector field, where \(\varphi\in\mathbb{R}\) is the electric potential, and by \(\theta\in\mathbb{R}\) the temperature. The notation \(\mathbb{S}^d\) stands for the space of second order symmetric tensors on \(\mathbb{R}^d\). We also use the dot to denote the time derivative, so \(\dot{\mathbf{u}} = (\dot{u}_i)\) represents the velocity vector. The functions \(\mathbf{u}:\Omega\times[0,T]\to\mathbb{R}^d\), \(\sigma:\Omega\times[0,T]\to\mathbb{S}^d\), \(\varphi:\Omega\times[0,T]\to\mathbb{R}\) and \(\theta:\Omega\times[0,T]\to\mathbb{R}\) are the unknowns of the problem, and, for simplicity, we do not indicate the dependence the functions on the variables \(\mathbf{x}\in \Omega\) and \(t\in [0,T]\).

The body is assumed to be thermo-electro-elastic and, therefore, we use the constitutive law

\[\begin{array}{l} \sigma = {\mathcal{F}} \mathbf{\varepsilon}(\mathbf{u}) -{\mathcal{E}}^T \mathbf{E}(\varphi) - \theta\mathcal{M} \quad \hbox{in} \quad \Omega\times(0,T),\\ \mathcal{D}=\mathcal{E}\mathbf{\varepsilon}(\mathbf{u})+\beta \mathbf{E}(\varphi) - \theta\mathcal{P} \qquad \hbox{in} \quad \Omega\times(0,T), \end{array}\] and the heat flux vector \(\mathbf{q} = (q_i) \in\mathbb{R}^d\) is given by the Fourier law of heat conduction \[\mathbf{q} =-\mathcal{K}\nabla\theta \quad \hbox{in} \quad \Omega\times(0,T).\] Here \(\mathcal{D}=(D_i)\) is the electric displacement field and \({\mathcal{F}}=(f_{ijkl})\), \({\mathcal{E}}=(e_{ijk})\), \({{\beta}}=(\beta_{ij})\), \({\mathcal{M}}=(m_{ij})\), \(\mathcal{P}=(p_{i})\) and \(\mathcal{K}=(k_{ij})\) are respectively, the elasticity, piezoelectric, electric permittivity, thermal expansion, pyroelectric and thermal conductivity tensors. \(\mathcal{E}^T\) denotes the transpose of \(\mathcal{E}\); also, the tensors \(\mathcal{E}\) and \(\mathcal{E}^T\) satisfy the equality \({\cal E}\sigma\cdot\mathbf{v}=\sigma\cdot{\cal E}^T\mathbf{v}\;\;\forall\sigma\in\mathbb{S}^d,\, \mathbf{v}\in\mathbb{R}^d\), and the components of the tensor \({\cal E}^T\) are given by \(e_{ijk}^T=e_{kij}\).

Since the process is assumed dynamic, then the equation of stress equilibrium, the equation of the quasistationary electric field, and the heat conduction equation are

\begin{align*} &\mathcal{D}iv \sigma+ \mathbf{f}_0=\rho\ddot{\mathbf{u}} & \hbox{in} \quad \Omega\times(0,T), \\ &{ \div} \mathcal{D} =\phi_0 & \hbox{in} \quad \Omega\times(0,T), \\ & \theta_{ref}\big(\alpha\dot{\theta} + \mathcal{M}\mathbf{\varepsilon}(\dot{\mathbf{u}}) + \mathcal{P}\mathbf{E}(\dot{\varphi})\big) + {\div} \mathbf{q} =\vartheta_0 & \hbox{in} \quad \Omega\times(0,T). \end{align*} Here \(\alpha\) is given as \(\alpha = \rho c_\nu/\theta_{ref}\), where \(\rho\) is the mass density, \(c_\nu\) is the specific heat and \(\theta_{ref}\) is the reference uniform temperature of the body. Moreover, \(Div\) and \({\div}\) represent the divergence operators for tensor and vector functions, i.e., Div\(\sigma=(\sigma_{ij,j}),\, {\div}\mathcal{Y}=(Y_{i,i})\).

We turn to describe the boundary conditions, so we denote by \(\nu = (\nu_i)\) the unit outward normal on \(\Gamma\). Then, on the \(\Gamma_D\cup\Gamma_N\) portion of the boundary, we impose the following conditions

\[\begin{array}{l} \mathbf{u}=\mathbf{0}\qquad \hbox{on} \quad \Gamma_D\times(0,T),\\ \sigma \nu=\mathbf{f}_N \quad \hbox{on}\quad \Gamma_N\times(0,T). \end{array}\] The boundary conditions for the electric potential can be defined in the following forms: \[\begin{array}{l} \varphi=0 \quad\qquad \hbox{on} \quad \Gamma_a\times(0,T),\\ \mathcal{D}\cdot \nu=\phi_b \quad \hbox{on} \quad \Gamma_b\times(0,T). \end{array}\] Next, on \(\Gamma_D\cup \Gamma_N\) we prescribe a Dirichlet condition for the temperature, say, \[ \theta=0 \quad \hbox{on} \quad \Gamma_D\cup\Gamma_N\times(0,T). \] We now describe the thermo-mechanical boundary conditions on the potential contact surface \(\Gamma_C\). We assume that the normal displacement \(u_\nu=\mathbf{u}\cdot \nu\) and the normal stress \(\sigma_\nu=\sigma\nu\cdot \nu\) satisfy the Signorini's contact conditions \[ u_\nu\leq 0,\quad \sigma_\nu\leq 0,\quad \sigma_\nu u_\nu=0\quad \hbox{on} \quad \Gamma_C\times(0,T). \] The corresponding Coulomb law of dry friction may be stated as follows: \[ \left\{ \begin{array}{l} \mbox{if }\;\dot{\mathbf{u}}_{\tau} = \mathbf{0} \quad \mbox{ then }\quad\|\sigma_{\tau}\| \leq \mu|\sigma_\nu|, \\ \mbox{if }\;\dot{\mathbf{u}}_{\tau} \neq \mathbf{0} \quad \mbox{ then }\quad \sigma_{\tau} = - \mu|\sigma_\nu| \displaystyle\frac{\dot{\mathbf{u}}_{\tau}}{\| {\dot{\mathbf{u}}_{\tau}}\|}, \end{array} \right. \quad \mbox{on}\quad\Gamma_C\times(0,T). \] Here \(\dot{\mathbf{u}}_{\tau} = \dot{\mathbf{u}} - \dot{u}_\nu\nu\) is the tangential velocity, \(\sigma_\tau=\sigma\nu -\sigma_\nu\nu\) represents the tangential force on the contact boundary and \(\mu\geq 0\) is the coefficient of friction.

Next, we describe the boundary condition for the temperature on \(\Gamma_C\). We assume that there is heat exchange between the surface and the foundation, which is at temperature \(\theta_f\). Moreover, since the flux of heat generated by the friction traction on the contact surface is proportional to the tangential shear \(\sigma_\tau\) and to the tangential velocity \(\dot{\mathbf{u}}_\tau\) of the surface, we assume a boundary condition of the following form

\[ \mathbf{q}\cdot \nu = k_c(\sigma_\nu)(\theta - \theta_f) - \mu|\sigma_\nu|\|\dot{\mathbf{u}}_\tau\| \quad \mbox{on} \quad\Gamma_C\times(0,T), \] where \(k_c(\cdot)\) is the normal pressure dependent heat exchange coefficient and has to satisfy \(k_c(0) = 0\). This condition guarantees that there is no heat flux between the body and the foundation if they are not in contact. For \(k_c(\cdot)\), we employ a linear model \(k_c(\sigma_\nu) = \bar{k}_{c}|\sigma_\nu|\), where \(\bar{k}_{c}\geq 0\) is model constant, see [11].

We collect the above equations and conditions to obtain the following mathematical problem.

Problem \(P.\) Find a displacement field \(\mathbf{u}:\Omega\times[0,T]\to\mathbb{R}^d\), a stress field \(\sigma:\Omega\times[0,T]\to\mathbb{S}^d\), an electric potential field \(\varphi:\Omega\times[0,T]\to\mathbb{R}\), an electric displacement field \(\mathbf{D}:\Omega\times[0,T]\to\mathbb{R}^d\), a temperature field \(\theta:\Omega\times[0,T]\rightarrow \mathbb{R}\), and a heat flux \(\mathbf{q}:\Omega\times[0,T]\to\mathbb{R}^d\) such that Here, \(\mathbf{u}_0\), \(\mathbf{v}_0\), \(\varphi_0\) and \(\theta_0\) are the prescribed initial displacement, velocity, electric potential and temperature, respectively.

3. Variational formulation and its approximation

3.1. A hybrid variational formulation

We now turn to the variational formulation of Problem \(P\) which is the starting point for the numerical modelling based on the finite element discretization. We denote in the sequel by \("\cdot"\) and \(\|\cdot\|\) the inner product and the Euclidean norm on the spaces \(\mathbb{R}^d\) and \(\mathbb{S}^d\). We introduce the spaces and we use the notation \(H=[L^2(\Omega)]^d\), and we introduce the spaces \begin{align*} V&=\{\mathbf{w} \in [H^1(\Omega)]^d\, ; \ \mathbf{w}=\mathbf{0} \; \hbox{on} \; \Gamma_D\},\\ W&=\{\xi\in H^1(\Omega)\, ;\, \xi=0 \; \hbox{on} \; \Gamma_a\}, \\ Q&=\{\eta\in H^1(\Omega)\, ;\, \eta=0 \; \hbox{on} \; \Gamma_D\cup\Gamma_N\}, \\ {\cal H}&=\{\tau=(\tau_{ij}) \,; \, \tau_{ij}=\tau_{ji}\in L^2(\Omega) \}. \end{align*} The spaces \(H,\ V,\ W,\ Q\) and \({\cal H}\) are real Hilbert spaces endowed with the canonical inner products given by \begin{align*} &\displaystyle (\mathbf{u},\mathbf{w})_H=\int_\Omega \mathbf{u}\cdot \mathbf{w} \, d\mathbf{x}, \\ &\displaystyle (\mathbf{u},\mathbf{w})_V= \int_\Omega \mathbf{\varepsilon}(\mathbf{u})\cdot \mathbf{\varepsilon}(\mathbf{w}) \, d\mathbf{x}, \\ &\displaystyle (\varphi,\xi)_W=\int_\Omega \nabla\varphi\cdot\nabla\xi\, d\mathbf{x}, \\ &\displaystyle (\theta,\eta)_Q=\int_\Omega \nabla\theta\cdot\nabla\eta\, d\mathbf{x}, \\ &\displaystyle (\sigma,\tau)_{\cal H}=\int_\Omega \sigma\cdot\tau\, d\mathbf{x}. \end{align*} We consider the trace spaces \(X_{\nu}=\{{w_{\nu}}_{|_{\Gamma_C}}\; : \; \mathbf{w}\in V\}\) and \(X_{\tau}=\{{\mathbf{w}_{\tau}}_{|_{\Gamma_C}}\; : \; \mathbf{w}\in V\}\) equipped with their usual norms. Denote respectively by \(X_{\nu}^{*}\) and \(X_{\tau}^{*}\) the dual of the spaces \(X_{\nu}\) and \(X_{\tau}\).

We consider the three mappings \(\mathbf{f} : [0,T]\longrightarrow V\), \(\phi : [0,T]\longrightarrow W\) and \(\vartheta : [0,T]\longrightarrow Q\) defined by

\begin{alignat*}{2} &\displaystyle (\mathbf{f}(t),\mathbf{w})_V=\int_\Omega \mathbf{f}_0(t)\cdot \mathbf{w} \, d\mathbf{x}+ \int_{\Gamma_N} \mathbf{f}_N(t)\cdot \mathbf{w}\, da,\;\;\;\; && \forall \mathbf{w}\in V,\nonumber \\ &\displaystyle (\phi(t),\xi)_W=\int_\Omega \phi_0(t) \xi \, d\mathbf{x}- \int_{\Gamma_b} \phi_b(t) \xi\, da, && \forall \xi\in W,\nonumber\\ & \displaystyle (\vartheta(t),\eta)_Q=\int_\Omega \vartheta_0(t) \eta \, d\mathbf{x}, && \forall \eta\in Q.\nonumber \end{alignat*} Then, the hybrid variational formulation of the contact problem \(P\) obtained by multiplying the equations with the relevant test functions and performing integration by part, is as follows. Problem \(P_V\). Find a displacement field \(\mathbf{u} : [0,T]\longrightarrow V\), a normal stress \(\lambda_\nu : [0,T]\longrightarrow X_{\nu}^{*}\), a tangential stress \(\lambda_{\tau} : [0,T]\longrightarrow X_{\tau}^{*}\), an electric potential field \(\varphi : [0,T]\longrightarrow W\) and a temperature field \(\theta : [0,T]\longrightarrow Q\) such that for a.e., \(t\in(0,T)\) \begin{align*} (\rho\ddot{\mathbf{u}}(t),\mathbf{w})_H + ({\mathcal{F}}\mathbf{\varepsilon}(\mathbf{u}(t)),\mathbf{\varepsilon}(\mathbf{w}))_{\cal H} + ({\cal E}^T\nabla\varphi(t),\mathbf{\varepsilon}(\mathbf{w}))_{\cal H} -({\cal M}\theta(t),\mathbf{\varepsilon}(\mathbf{w}))_{\cal H} \\ = (\mathbf{f}(t), \mathbf{w})_V + \displaystyle\int_{\Gamma_C}\lambda_\nu(t)w_\nu\,da + \displaystyle\int_{\Gamma_C}\lambda_\tau(t)\cdot\mathbf{w}_\tau\,da \quad\forall \,\mathbf{w}\in V, \nonumber\\ (\beta\nabla \varphi(t),\nabla\xi)_{H}-({\cal E}\mathbf{\varepsilon}(\mathbf{u}(t)), \nabla \xi)_{H} - ({\cal P}\theta(t), \nabla \xi)_{H} = (\phi(t),\xi)_W\;\;\forall\,\xi \in W, \end{align*} \begin{align*} \Big(\theta_{ref}\big(\alpha\dot{\theta}(t) + \mathcal{M}\mathbf{\varepsilon}(\dot{\mathbf{u}}(t)) - \mathcal{P}\nabla\dot{\varphi}(t)\big) ,\eta\Big)_{L^2(\Omega)} + (\mathcal{K}\nabla \theta(t),\nabla\eta)_{H} + \displaystyle\int_{\Gamma_C} k_c(\lambda_\nu(t))\big(\theta(t)- \theta_f\big)\eta\,da \\ - \displaystyle\int_{\Gamma_C} \mu|\lambda_\nu(t)|\|\dot{\mathbf{u}}_\tau(t)\|\eta\,da = (\vartheta(t),\eta)_Q\;\;\forall\,\eta \in Q,\end{align*} \begin{alignat}{2} & \mathbf{u}(0)=\mathbf{u}_0, \ \ \ \mathbf{v}(0)=\mathbf{v}_0, \ \ \ \varphi(0)=\varphi_0,\ \ \ \theta(0)=\theta_0.&&\nonumber \end{alignat} The inclusion in Equation (16) represents the Signorini contact condition (9). Here, \(\partial\) denotes the subdifferential operator in the sense of convex analysis and \(I_{\mathbb{R}_{-}}\) denotes the indicator function of the negative half-line. Recall also that, the inclusion in Equation in (17) represents the subdifferential form of Coulomb's law of dry friction (10), see [9] for details.

3.2. Numerical approximation

This section is devoted to the numerical discretization of the of problem \({P}_{V}\). First, we consider tree finite dimensional spaces \(V^h\subset V\), \(W^h\subset W\) and \(Q^h\subset Q\) approximating the spaces \(V\), \(W\) and \(Q\), respectively, in which \(h > 0\) denotes the spatial discretization parameter, and let \(U^h=U\cap V^h\). In the numerical simulations presented in the next section, \(V^h\), \(W^h\) and \(Q^h\) consist of continuous and piecewise affine functions, that is, \begin{eqnarray*} V^h=\{\mathbf{w}^h\in [C(\overline{\Omega})]^d \; ;\ \mathbf{w}^h_{|_{Tr}}\in [P_1(Tr)]^d \, \,\,\, Tr\in {\mathcal{T}}^h, \quad \mathbf{w}^h=\mathbf{0} \,\,\, \hbox{on}\,\,\, \Gamma_D\},\\[2pt] W^h =\{\xi^h\in C(\overline{\Omega}) \; ; \; \xi^h_{|_{Tr}}\in P_1(Tr) \,\, \, \, Tr\in {\mathcal{T}}^h,\quad \xi^h=0 \quad \hbox{on} \quad \Gamma_a\}, \\[2pt] Q^h =\{\eta^h\in C(\overline{\Omega}) \; ; \; \eta^h_{|_{Tr}}\in P_1(Tr) \,\, \, \, Tr\in {\mathcal{T}}^h,\quad \eta^h=0 \quad \hbox{on} \quad \Gamma_D\cup\Gamma_N\}, \end{eqnarray*} where \(\Omega\) is assumed to be a polygonal domain, \({\mathcal{T}}^h\) denotes a finite element triangulation of \(\overline{\Omega}\), and \(P_1(Tr)\) represents the space of polynomials of global degree less or equal to one in \({Tr}\).

To discretize the time derivatives, we use a uniform partition of \([0,T]\), denoted by \(0=t_0< t_1< \ldots< t_N=T\). Let \(k\) be the time step size, \(k=T/N\), and for a continuous function \(f(t)\) let \(f_n=f(t_n)\). Finally, for a sequence \(\{w_n\}_{n=0}^N\) we denote by \(\delta w_n=(w_n-w_{n-1})/k\) the divided differences.

We now consider the spaces \(X_{\nu}^h=\{{w_{\nu}^h}_{|_{\Gamma_C}}\, : \, \mathbf{w}^h\in V^h\}\) and \(X_{\tau}^h=\{{\mathbf{w}_{\tau}^h}_{|_{\Gamma_C}}\, : \, \mathbf{w}^h\in V^h\}\) equipped with its usual norm. We also consider the discrete space of piecewise constants \({X^{*}_{\nu}}^{h}\subset L^2(\Gamma_C)\) and \({X^{*}_{\tau}}^{h}\subset L^2(\Gamma_C)^d\) related to the discretization of the normal and the tangential stress, respectively.

The fully discrete approximation of Problem \(P_V\), based on the Euler scheme, is the following:

Problem \(P_V^{hk}\). Find a discrete displacement \(\mathbf{u}^{hk}=\{\mathbf{u}_n^{hk}\}_{n=0}^N\subset V^h\), a discrete velocity \(\mathbf{v}^{hk}=\{\mathbf{v}_n^{hk}\}_{n=0}^N\subset V^h\), a discrete normal stress \(\lambda_{{\nu}_{n}}^{hk}=\{\lambda_{{\nu}_{n}}^{hk}\}_{n=0}^N\subset {X^{*}_{\nu}}^{h}\), a discrete tangential stress \(\lambda_{{\tau}_{n}}^{hk}=\{\lambda_{{\tau}_{n}}^{hk}\}_{n=0}^N\subset {X^{*}_{\tau}}^{h}\), a discrete electric potential \(\varphi^{hk}=\{\varphi_n^{hk}\}_{n=0}^N\subset W^h\) and a discrete temperature \(\theta^{hk}=\{\theta_n^{hk}\}_{n=0}^N\subset Q^h\) such that, for all \(n=1,\ldots,N\), \begin{align*} (\rho\delta\mathbf{v}^{hk}_{n},\mathbf{w}^h)_{H} + ({\mathcal{F}}\mathbf{\varepsilon}(\mathbf{u}^{hk}_{n}),\mathbf{\varepsilon}(\mathbf{w}^h))_{\cal H} + ({\cal E}^T\nabla\varphi^{hk}_{n},\mathbf{\varepsilon}(\mathbf{w}^h))_{\cal H} -({\cal M}\theta^{hk}_{n},\mathbf{\varepsilon}(\mathbf{w}^h))_{\cal H}\\ = (\mathbf{f}_n, \mathbf{w}^h)_V + \displaystyle\int_{\Gamma_C}\lambda_{{\nu}_n}^{hk}w_\nu^h\,da + \displaystyle\int_{\Gamma_C}\lambda_{{\tau}_n}^{hk}\cdot\mathbf{w}_\tau^h\,da \quad\forall \, \mathbf{w}^h\in V^h, \\ (\beta\nabla \varphi^{hk}_{n},\nabla\xi^h)_{H}-({\cal E}\mathbf{\varepsilon}(\mathbf{u}^{hk}_{n}), \nabla \xi^h)_{H} - ({\cal P}\theta^{hk}_{n}, \nabla \xi^h)_{H} = (\phi_n,\xi^h)_W\;\;\forall\,\xi^h \in W^h, \\ \Big(\theta_{ref}\big(\alpha\delta{\theta}^{hk}_{n} + \mathcal{M}\mathbf{\varepsilon}(\delta{\mathbf{u}}^{hk}_{n}) - \mathcal{P}\nabla\delta{\varphi}^{hk}_{n}\big) ,\eta^h\Big)_{L^2(\Omega)} + (\mathcal{K}\nabla \theta^{hk}_{n},\nabla\eta^h)_{H} + \displaystyle\int_{\Gamma_C} k_c(\lambda_{{\nu}_{n}}^{hk})\big(\theta^{hk}_{n} - \theta_f^h\big)\eta^h\,da\\ - \displaystyle\int_{\Gamma_C} \mu|\lambda_{{\nu}_{n}}^{hk}|\|\delta{\mathbf{u}}_{\tau_{n}}^{hk}\|\eta^h\,da = (\vartheta_n,\eta^h)_Q\;\;\forall\,\eta^h \in Q^h, \end{align*} \begin{alignat}{2} &\mathbf{u}_n^{hk} = \mathbf{u}_0^{hk} + \displaystyle\sum_{j=1}^{n} k\mathbf{v}_j^{hk}.&& \end{alignat} Here, \(\mathbf{u}_0^{hk}\), \(\mathbf{v}_0^{hk}\), \(\varphi_0^{hk}\) and \(\theta_0^{hk}\) are appropriate approximation of the initial condition \(\mathbf{u}_0\), \(\mathbf{v}_0\), \(\varphi_0\), \(\theta_0\) and \(\lambda_0^{hk}=\mathbf{0}\).

4. Numerical example

4.1. A solution algorithm

We now describe the numerical solution of the hybrid variational Problem \({{P}_{V}^{hk}}\). The numerical treatment of the conditions (18) and (19) is based on the augmented Lagrangian approach (see [10,13] for more details). To this end we introduce the notation \(\lambda = \lambda_\nu\nu + \lambda_\tau\), where \(\lambda_\nu=\lambda.\nu\) and \(\lambda_\tau=\lambda - \lambda_\nu\nu\). Let \({N^h_{tot}}\) be the total number of nodes and denote by \(\alpha^i,\, \beta^i,\, \gamma^i\) the basis functions of the spaces \(V^h\), \(W^h\) and \(Q^h\), respectively, for \(i=1,\ldots,{N^h_{tot}}\). Then, the expression of functions \(\mathbf{w}^h\in V^h,\, \xi^h\in W^h\) and \(\eta^h\in Q^h\) is given by \begin{eqnarray*} &&\mathbf{w}^h = \sum_{i=1}^{N^h_{tot}}\mathbf{w}^i\alpha^i, \quad \xi^h = \sum_{i=1}^{N^h_{tot}}\xi^i\beta^i, \quad \eta^h = \sum_{i=1}^{N^h_{tot}}\eta^i\gamma^i, \end{eqnarray*} where \(\mathbf{w}^i\), \(\xi^i\) and \(\eta^i\) represent the values of the corresponding functions \(\mathbf{w}^h\), \(\xi^h\) and \(\eta^h\) at the \(i^{th}\) node of \({\mathcal{T}}^h\).

It can be shown that the numerical approach of Problem \(\large{ {P}_{V}^{hk}}\) is governed at each time step \(n\) by a system of non-linear equations of the form

where the functions \(\tilde{\mathbf{M}}\), \(\tilde{\mathbf{A}}\), \( \tilde{\mathbf{G}}\) and \( \tilde{\mathbf{F}}\) are defined below. Here, the vectors \(\delta\mathbf{v}_{n}\in\mathbb{R}^{d\times{N_{tot}^h}}\), \(\delta\varphi_{n}\in\mathbb{R}^{ {N_{tot}^h}}\), \(\delta\theta_{n}\in\mathbb{R}^{ {N_{tot}^h}}\), \(\mathbf{v}_n\in\mathbb{R}^{d\times{N_{tot}^h}}\), \(\mathbf{u}_n\in\mathbb{R}^{d\times{N_{tot}^h}}\), \(\varphi_n\in\mathbb{R}^{{N_{tot}^h}}\), \(\theta_n\in\mathbb{R}^{{N_{tot}^h}}\) and \(\lambda_n\in \mathbb{R}^{d\times {N^h_{\Gamma_C}}}\) are defined by \( \delta\mathbf{v}_n = \{\delta\mathbf{v}_n^i\}_{i=1}^{N_{tot}^h}, \, \delta\varphi_n = \{\delta\varphi_n^i\}_{i=1}^{N_{tot}^h}, \, \delta\theta_n = \{\delta\theta_n^i\}_{i=1}^{N_{tot}^h}, \, \mathbf{v}_n = \{\mathbf{v}_n^i\}_{i=1}^{N_{tot}^h}, \, \mathbf{u}_n = \{\mathbf{u}_n^i\}_{i=1}^{N_{tot}^h}, \, \varphi_n = \{\varphi_n^i\}_{i=1}^{N_{tot}^h}, \, \theta_n = \{\theta_n^i\}_{i=1}^{N_{tot}^h}, \, \lambda_n = \{\lambda_n^i\}_{i=1}^{N^h_{\Gamma_C}}, \) where \(\delta\mathbf{v}_n^i = \frac{\mathbf{v}_n^i - \mathbf{v}_{n - 1}^{i}}{k}\), \(\delta\varphi_n^i = \frac{\varphi_n^i - \varphi_{n-1}^i}{k}\), \(\delta\theta^i = \frac{\theta_{n}^i - \theta_{n-1}^i}{k}\), \(\mathbf{v}_n^i = \frac{\mathbf{u}_n^i - \mathbf{u}_{n - 1}^{i}}{k}\), \(\mathbf{u}_n^i\), \(\varphi_n^i\), and \(\theta_n^i\) represent the value of the function \(\delta\mathbf{v}_n^{hk}\), \(\delta\varphi_n^{hk}\), \(\delta\theta_n^{hk}\) \(\mathbf{v}_n^{hk}\), \(\mathbf{u}_n^{hk}\), \(\varphi_n^{hk}\) and \(\theta_n^{hk}\) at the \(i^{th}\) nodes of \({\mathcal T}^h\). \(\lambda_n^i\) denotes the value of \(\lambda_n^{hk}\) at the \(i^{th}\) node of the discretized contact interface, where \({N^h_{\Gamma_C}}\) denotes the total number of nodes of \(i^{th}\) lying on \(\Gamma_C\).

Next, the generalized acceleration term \( \tilde{\mathbf{M}}(\mathbf{a})\in\mathbb{R}^{d\times N^h_{tot}}\times\mathbb{R}^{N^h_{tot}}\times\mathbb{R}^{N^h_{tot}}\times\mathbb{R}^{d\times N^h_{\Gamma_C}}\), the generalized damping term \( \tilde{\mathbf{A}}(\mathbf{v},\Phi,\Theta)\in\mathbb{R}^{d\times N^h_{tot}}\times\mathbb{R}^{N^h_{tot}}\times\mathbb{R}^{N^h_{tot}}\times\mathbb{R}^{d\times N^h_{\Gamma_C}}\) and the generalized thermo-electro-elastic term \( \tilde{\mathbf{G}}(\mathbf{u},\varphi,\theta)\in\mathbb{R}^{d\times N^h_{tot}}\times\mathbb{R}^{N^h_{tot}}\times\mathbb{R}^{N^h_{tot}}\times\mathbb{R}^{d\times N^h_{\Gamma_C}}\) are defined by \( \tilde{\mathbf{M}}(\mathbf{a}) = \big({\mathbf{M}}(\mathbf{a}),\mathbf{0}_{N^h_{tot}},\mathbf{0}_{N^h_{tot}},\mathbf{0}_{d\times N^h_{\Gamma_C}}\big)\), \( \tilde{\mathbf{{A}}}(\mathbf{v},\Phi,\Theta) = \big({\mathbf{A}}(\mathbf{v},\Phi,\Theta),\mathbf{0}_{d\times N^h_{\Gamma_C}}\big)\) and \( \tilde{\mathbf{ G}}(\mathbf{u},\varphi,\theta) = \big({\mathbf{G}}(\mathbf{u},\varphi,\theta),\mathbf{0}_{d\times N^h_{\Gamma_C}}\big)\). Here \(\mathbf{0}_{N^h_{tot}}\) is the zero element of \(\mathbb{R}^{N^h_{tot}}\) and \(\mathbf{0}_{d\times N^h_{\Gamma_C}}\) is the zero element of \(\mathbb{R}^{d\times N^h_{\Gamma_C}}\); also, \({\mathbf{M}}(\mathbf{a})\in \mathbb{R}^{d\times N^h_{tot}}\), \({\mathbf{A}}(\mathbf{v},\Phi,\Theta)\in\mathbb{R}^{d\times N^h_{tot}}\times\mathbb{R}^{N^h_{tot}}\times\mathbb{R}^{N^h_{tot}}\) and \( {\mathbf{ G}}(\mathbf{u},\varphi,\theta)\in\mathbb{R}^{d\times N^h_{tot}}\times\mathbb{R}^{N^h_{tot}}\times\mathbb{R}^{N^h_{tot}}\) denotes the acceleration term, the damping term and the thermo-electro-elastic term, given by

\begin{align*} & \big({\mathbf{M}}(\mathbf{a}) \cdot \mathbf{w}\big)_{\mathbb{R}^{d\times N^h_{tot}}} = (\rho\mathbf{a}^{h},\mathbf{w}^h)_{L^2(\Omega)^d},\\ &\big({\mathbf{A}}(\mathbf{v}, \, \Phi,\, \Theta) \cdot (\mathbf{w},\xi,\eta)\big)_{\mathbb{R}^{d\times N^h_{tot}}\times\mathbb{R}^{N_{tot}}\times\mathbb{R}^{N_{tot}}} = \big(\theta_{ref}(\alpha\Theta^h + {\cal M}\mathbf{\varepsilon}(\mathbf{v}^{h}) - {\cal P}\nabla\Phi^{h}),\eta^h\big)_{L^2(\Omega)}, \\ & \big({\mathbf{G}}(\mathbf{u}, \, \varphi,\, \theta) \cdot (\mathbf{w},\xi,\eta)\big)_{\mathbb{R}^{d\times N^h_{tot}}\times\mathbb{R}^{N_{tot}}\times\mathbb{R}^{N_{tot}}} = (\mathcal{F}\mathbf{\varepsilon}(\mathbf{u}^{h}) - {\cal M}\theta^{h} , \mathbf{\varepsilon}(\mathbf{w}^h))_{\cal H} + (\mathcal{E}\mathbf{\varepsilon}(\mathbf{w}^h), \nabla\varphi^{h})_H \\ & \qquad \qquad - (\mathcal{E}\mathbf{\varepsilon}(\mathbf{u}^{h}) - \beta\nabla\varphi^{h} + {\cal P}\theta^{h}, \nabla \xi^h)_{H} + (\mathcal{K}\nabla \theta^{h},\nabla\eta^h)_{H} - (\mathbf{f}_n,\mathbf{w}^h)_V - (\phi_n,\xi^h)_W - (\vartheta_n,\eta^h)_Q, \nonumber \end{align*} \( \forall \mathbf{w} \in \mathbb{R}^{d\times N^h_{tot}},\, \xi \in \mathbb{R}^{N^h_{tot}},\, \eta \in \mathbb{R}^{N^h_{tot}},\; \mathbf{w}^h \in V^h,\, \xi^h \in W^h,\) and \( \eta^h \in Q^h. \) Above, \(\mathbf{w},\, \xi\) and \(\eta\) represent the generalized vectors of components \(\mathbf{w}^i,\, \xi^i\) and \(\eta^i\), for \(i = 1,\cdots, N_{tot}^h\), respectively. Finally, the generalized contact operator \(\tilde{\mathbf{F}}(\mathbf{u}, \theta, \lambda)\in\mathbb{R}^{d\times N^h_{tot}}\times\mathbb{R}^{N^h_{tot}}\times\mathbb{R}^{N^h_{tot}}\times\mathbb{R}^{d\times N^h_{\Gamma_C}}\) is defined by \(\tilde{\mathbf{F}}(\mathbf{u}, \theta, \lambda) = \big(\nabla_{\small \mathbf{u}}L^r, \mathbf{0}_{N^h_{tot}}, \bar{k}_c \big(\lambda_\nu - r u_{\nu}\big)_{-}(\theta - \theta_f ) - P_{\small C[\mu(\lambda_\nu - r u_{\nu} )_{-}]}(\lambda_{\tau} - r \delta\mathbf{u}_{\tau} )\cdot\delta\mathbf{u}_\tau ,\nabla_{\small\lambda}L^r \big)\), where \(\nabla_{\mathbf{x}}\) represents the gradient operator with respect the variable \(\mathbf{x}\). Also \(L^{r}\) denote the augmented Lagrangian functional for the contact and friction terms, \begin{eqnarray*} L^{r}(\mathbf{u}^{h},\lambda^{h}) & = & u_{\nu}^{h}\lambda_{\nu}^{h} + \delta\mathbf{u}_{\tau}^{h}\cdot \lambda_{\tau}^{h} + \displaystyle\frac{r}{2}(u_{\nu}^{h})^{2} + \displaystyle\frac{r}{2}|\delta\mathbf{u}_{\tau}^{h}|^{2} -\displaystyle\frac{1}{2r}\Big(\lambda_\nu^h - r u_{\nu}^{h} + \big(\lambda_\nu^h - r u_{\nu}^{h}\big)_{-}\Big)^2\\ && - \displaystyle\frac{1}{2r}\big|\lambda_{\tau}^{h} - r\delta\mathbf{u}_{\tau}^{h} - P_{C[\mu(\lambda_\nu^h - r u_{\nu}^{h})_{-}]}(\lambda_{\tau}^{h} - r\delta\mathbf{u}_{\tau}^{h})\big|^{2}, \label{ltau} \end{eqnarray*} where \(r > 0\) is an augmentation parameter, \(P_{C[\rho]}\) is the orthogonal projection on the Coulomb convex disk of constant radius \(\rho\), and \((\cdot)_{-}\) is the negative part of \(x\in\mathbb{R}\), i.e., \((x)_{-} = \max(-x,0)\).

Let \({\cal F}(\mathbf{u}, \theta, \lambda)\in\mathbb{R}^{d\times N^h_{tot}}\times\mathbb{R}^{N^h_{tot}}\times\mathbb{R}^{d\times N^h_{\Gamma_C}}\) the thermo-mechanical frictional contact operator defined through the relation

\begin{align*} &\big({\cal F}(\mathbf{u}, \theta, \lambda)\cdot(\mathbf{w},\eta,\gamma)\big)_{{\mathbb{R}^{d\times N_{tot}^{h}}}\times{\mathbb{R}^{N_{tot}^{h}}}\times{\mathbb{R}^{d\times N_{\Gamma_{C}}^{h}}}} = \displaystyle\int_{\Gamma_{C}}\nabla_{\mathbf{u}} L^{r}(\mathbf{u}^{h},\lambda^{h}).\mathbf{w}^{h}\ da + \displaystyle\int_{\Gamma_{C}}\nabla_{\small \lambda} L^{r}(\mathbf{u}^{h},\lambda^{h}).\gamma^{h} \ da \\ & \qquad\qquad + \displaystyle\int_{\Gamma_C} \bar{k}_c \big(\lambda_\nu^h - r u_{\nu}^h \big)_{-} (\theta^h - \theta_f^h ) \eta^h \ da - \displaystyle\int_{\Gamma_C} \big(P_{\small C[\mu(\lambda_\nu^h - r u_{\nu}^h )_{-}]}(\lambda_{\tau}^h - r \delta\mathbf{u}_{\tau}^h )\cdot\delta\mathbf{u}_\tau^h\big) \eta^h\ da,\end{align*} \(\forall\, \mathbf{w} \in \mathbb{R}^{d\times N^{h}_{tot}},\,\, \eta\in \mathbb{R}^{ N^{h}_{tot}} ,\,\, \gamma\in \mathbb{R}^{d\times N^{h}_{\Gamma_C}},\ \mathbf{w}^{h} \in V^{h}, \;\, \eta^h\in Q^h,\) and \(\gamma^{h}\in X_{\nu}^{*h}\times X_{\tau}^{*h}.\) The solution of the non-linear system (20) is based on a linear iterative method similar to that used in the Newton method, which permits to treat simultaneously the four unknowns \(\mathbf{u}_n\), \(\varphi_n\), \(\theta_n\) and \(\lambda_n\) and, for this reason, we use in what follows the notation \(\mathbf{x}_n = (\mathbf{u}_n, \varphi_n, \theta_n, \lambda_n)\). This Newton algorithm can be summarized by the following iteration process \begin{align*} & \mathbf{x}^{i+1}_{n+1} = \mathbf{x}^{i}_{n+1}-\left(\displaystyle\frac {\mathbf{P}^{i}_{n+1}} {k^2} + \displaystyle\frac {\mathbf{Q}^{i}_{n+1}} {k} + {\mathbf{K}}^{i}_{n+1} + \mathbf{T}^{i}_{n+1}\right)^{-1}\\ & \qquad \quad \times \mathbf{R}\left(\frac{\mathbf{v}_{n+1}^i - \mathbf{v}_{n}}{k}, \frac{\varphi_{n+1}^i - \varphi_{n}}{k}, \frac{\theta_{n+1}^i - \theta_{n}}{k}, \mathbf{v}_{n+1}^i, \mathbf{u}_{n+1}^i, \varphi_{n+1}^i , \theta_{n+1}^i , \lambda_{n+1}^i\right), \end{align*} where \(\mathbf{x}^{i+1}_{n+1}\) denotes the quadruple \((\mathbf{u}^{i+1}_{n+1}, \varphi^{i+1}_{n+1}, \theta^{i+1}_{n+1}, \lambda^{i+1}_{n+1})\); \(i\) and \(n\) represent respectively the Newton iteration index and the time index; \(\mathbf{P}^{i}_{n+1} = D_{\small\mathbf{u}}{\mathbf{M}}({\delta\mathbf{v}}^{i}_{n+1})\) denotes the mass matrix, \(\mathbf{Q}^{i}_{n+1} = D_{\small\mathbf{u}, \varphi, \theta}{\mathbf{A}}({\delta\mathbf{u}}^{i}_{n+1}, {\delta\varphi}^{i}_{n+1}, {\delta\theta}^{i}_{n+1})\) denotes the damping matrix, \(\mathbf{K}^{i}_{n+1} = D_{\small\mathbf{u}, \varphi, \theta}\mathbf{G}(\mathbf{u}^{i}_{n+1}, \varphi^{i}_{n+1}, \theta^{i}_{n+1})\) represents the elastic matrix and \(\mathbf{T}^{i}_{n+1} = D_{\small\mathbf{u},\theta,\lambda}{{\cal F}}(\mathbf{u}^{i}_{n+1},\theta^{i}_{n+1},\lambda^{i}_{n+1})\) is the contact tangent matrix; also, \(D_{\small\mathbf{u}}\mathbf{M}\), \(D_{\small\mathbf{u}, \varphi, \theta}\mathbf{A}\), \(D_{\small \mathbf{u},\varphi,\theta}\mathbf{G}\) and \(D_{\small \mathbf{u},\theta,\lambda}{{\cal F}}\) denote the differentials of the functions \(\mathbf{M}\), \(\mathbf{A}\), \(\mathbf{G}\) and \({\cal F}\) with respect to the variables \(\mathbf{u}\), \(\varphi\), \(\theta\) and \(\lambda\). This leads us to solve the resulting linear system \begin{align*}\label{eq-l} \left(\displaystyle\frac {\mathbf{P}^{i}_{n+1}} {k^2} + \displaystyle\frac {\mathbf{Q}^{i}_{n+1}} {k}+ \mathbf{K}^{i}_{n+1} + \mathbf{T}^{i}_{n+1}\right)\Delta \mathbf{x}^{i}= - \mathbf{R}\left(\frac{\mathbf{v}_{n+1}^i - \mathbf{v}_{n}}{k}, \frac{\varphi_{n+1}^i - \varphi_{n}}{k}, \frac{\theta_{n+1}^i - \theta_{n}}{k}, \mathbf{v}_{n+1}^i, \mathbf{u}_{n+1}^i, \varphi_{n+1}^i , \theta_{n+1}^i , \lambda_{n+1}^i\right), \end{align*} where \(\Delta\mathbf{x}^{i} = (\Delta\mathbf{u}^{i},\,\Delta\varphi^{i},\,\Delta\theta^{i},\,\Delta\lambda^{i})\) with \(\Delta \mathbf{u}^{i} = \mathbf{u}^{i+1}_{n+1}-\mathbf{u}^{i}_{n+1}\), \(\Delta\varphi^{i} = \varphi^{i+1}_{n+1}-\varphi^{i}_{n+1}\), \(\Delta\theta^{i} = \theta^{i+1}_{n+1}-\theta^{i}_{n+1}\) and \(\Delta \lambda^{i} = \lambda^{i+1}_{n+1}-\lambda^{i}_{n+1}\).

Note that formulation (20) has been implemented in the open-source finite element library GetFEM++ (see [14]).

4.2. Numerical results

For the numerical simulations we consider the physical setting depicted in Figure 1. In this case the body \(\Omega=(0,4)\times(0,1)\subset \mathbb{R}^2\) is clamped on \(\Gamma_{D}= [0,1]\times \{0\}\) and the electric potential is free there (we choose \(\Gamma_{D}= \Gamma_{a}\)). Let \(\Gamma_N = (\{4\}\times[0, 1]) \cup ([0, 4]\times\{1\}) = \Gamma_b\). Vertical tractions act on the part \([0, 4]\times\{1\}\) of the boundary, i.e., \(\mathbf{f}_N(x_1,x_2,t) = (0,-5\, x_1\, t) \,N/m\) and the part \(\{4\}\times[0, 1]\) is traction free. The body is in contact with a conductive foundation on its lower boundary \(\Gamma_{C}= [0,4]\times\{0\}\). We suppose that the temperature vanishes in \(\Gamma_D\cup\Gamma_N\). The body is subjected to action of a volume force of density \(\mathbf{f}_0=(x_1,x_2,t) = (0,-10)\, N/m^2\). No electric charges and no volume heat source are supposed to act in the body, i.e., \(q_0=0\, C/m^2\), \(q_b=0\, C/m\) and \(\vartheta_0 = 0\, W/m^2\).

Here, we use as material the thermo-piezoelectric body whose constants are taken as [2]. The following data have been used in the numerical simulations:

\begin{eqnarray*} && r=10^{7}\, N /m^2 ,\;\; \mu = 0.2,\;\; \bar{k}_c= 1,\;\; \theta_f = 393 \,K,\;\; \theta_{ref} = 293 \,K.\\ && T= 10\, s,\;\; \mathbf{u}_0= \mathbf{0} \, m,\;\; \mathbf{v}_0= \mathbf{0} \, m/s,\;\; \varphi_0= 0 \, V,\;\; \theta_0= 0 \, K. \end{eqnarray*}

Our interest in this example is to study the influence of the thermal conductivity of the foundation on the contact process and, to this end, we consider the problem both in the case when the foundation is insulated there are no heat flux on \(\Gamma_C\) (i.e. \( \mathbf{q}\cdot \nu = 0\mbox{ on } \Gamma_C\)) and in the case when it is thermally conductive.

Figure 2 presents the deformed configurations for the two previously mentioned cases, at final time. We can easily note that considering a thermally conductive foundation reduce the deformations. In order to highlight the influence of the foundation temperature on the electric potential, we plot the electric potential for the two previously mentioned cases (see Figure 3). The first case illustrates the direct piezoelectric effect: the electric potential is generated because of the deformation. However, in the second case, we can easily note that considering a thermally conductive foundation increases the electric potential.

In Figure 4, the Von Mises stress norm is plotted on the deformed configuration. Clearly, effects due to the influence of foundation temperature, can be observed. Both temperature is plotted in Figure 5 at final time for the value \(\theta_f = 393\, K\).

5. Conclusion

In this paper thermo-piezoelectric contact including frictional heat generation and interfacial heat transfer is numerically studied. The novelties arise in the fact that the process is dynamic, the material behavior is described by a thermo-electro-elastic constitutive law and the foundation is thermally conductive. A fully discrete scheme was used to approach the problem and a numerical algorithm which combine the augmented Lagrangian approach with the Newton method was implemented. Moreover, numerical simulations for a representative two-dimensional example were provided. These simulations describe the thermal effect, i.e. the appearance of strain and voltage in the body, due to the action of the temperature field. Also, they underline the effects of the thermal contact, i.e. heat transfer and frictional heating, on the process. Performing these simulations, we found that the numerical solution worked well and the convergence was rapid. This work opens the way to study further models of frictional contact with a coefficient of friction depending on the slip rate or the temperature.

Conflicts of Interest

''The author declares no conflict of interest.''

1. Introduction

At the end of 2019, a new class of SARS (Severe acute respiratory syndrome) virus disease caused by a new type of the corona viruses was discovered in Wuhan, China which named later SARS-CoV-2 (Severe acute respiratory syndrome-Corona Virus-2) or the new coronavirus disease and this disease spread over the world and caused more than three millions deaths over the world up to the writing of this article. Lots of studies discussed the spreading and forecasting of SARS-CoV-2 disease and influence of the disease on different locations. For instance, Lounis and Bagal [1] found the parameters of the SIR model for Algeria. Neto et al., [2] discussed the modelling of spreading of the disease for São Paulo in Brazil. Ebohon et al., [3] discussed the influence of SARS-CoV-2 on the education in Nigeria. Ganiny and Nisar [4] discussed the spreading of the disease in Indian regions. Aidoo et al., [5] discussed the modelling of SARS-CoV-2 incidence in the African sub regions using smooth transition autoregressive model. Other studies applied SIRD (susceptible-infected-recovered-dead) epidemic model for the studying the forecasting and spreading of SARS-CoV-2 disese [6,7,8,9,10] in addition to finding the indicators of the model for the disease. Also, in other studies, the SEIR (susceptible-exposed-infected-recovered) epidemic model was applied for the forecasting of SARS-CoV-2 disease [11,12,13,14] for different scenarios.

In general, the epidemiology forecasting model was suggested by Kermack [15] for the first time as a simple SIR epidemic model and other epidemiology compartmental forecasting models were derived based on this model such as SEIR model [16,17,18,19,20,21,22], SIRD model [9,10] and SVEIS (susceptible-vaccinated-exposed-infected- susceptible) model [23] which takes the vaccination into account. In this work, we use the SEIR model for finding some important predictors of SARS-CoV-2 by simulating the previous model using the numerical analysis methods. The first predicator which we focus on in this work is the period of incubation, which is one of the most important indicators of the spreading of a specific pandemic such as SARS-CoV-2, this period represents the average time of the incubation from the exposing and this period gives the ratio between the exposed population to the rate of the infectious population when we eliminate other infected reasons.

There are two methods for determining the values of the incubation period. The first one is observation of exposed persons by medical observer and the other method is theoretical method which estimates incubation period values based on one of compartmental models in epidemiology and collected data of total cases of a specific pandemic. The theoretical method is more preferable because it eliminates the contacts between patients and doctors or nurses. In this work, the period of incubation values of SARS-CoV-2 are estimated based on the numerical analysis methods for different location countries. The other considered predictors which we focus on in this study are the population steady state, which represents the ratio between the average age of infection and the average age at which every individual in the model is assumed to die, the critical immunisation threshold, which is the minimum of the proportion of the population that is immune, and the basic reproduction number, which represents the expected cases which is generated by one infectious case in a certain population with a specific disease. We calculate all of the previous indicators based on the same method via the SEIR model. The first two equations of the SEIR model are non-linear equations [24] and describe the change of the susceptible population and the change of the exposed population with respect to the time and the others equations describe the rate of the infections population and the recovery population in respect to the time. The four equations of the SEIR epidemiological model are given as follows:

where \(N\) is the population number, \(\xi_{1}\), \(\xi_{2}\), \(\xi_{3}\) and \(\xi_{4}\) are parameters and \(S(t)\), \(I(t)\), \(R(t)\) and \(E(t)\) are the susceptible population of the pandemic individual, the infectious population of the pandemic individual, the recovered cases pandemic individual and the exposed population of the pandemic individual respectively. The previous populations are governed under the following conservation condition: In this study, we apply previous epidemiology model for estimating period of incubation, population steady state, critical immunization threshold and basic reproduction number of the new corona virus disease in the United States, where the first case of the pandemic was observed in January 2020, Russia, where the first case of the pandemic was observed in January 2020, the United Kingdom, where the first case of the pandemic was observed in January 2020, Brazil, where the first case of the pandemic was observed in February 2020, Spain, where the first case of the pandemic was observed in January 2020, Bahrain, where the first case of the pandemic was observed in February 2020, Egypt, where the first case of the pandemic was observed in February 2020, Iran, where the first case of the pandemic was observed in February 2020, Cyprus, where the first case of the pandemic was observed in March 2020 and the Syrian Arab Republic, where the first case of the pandemic was observed in March 2020. The aforementioned countries are considered located in different locations in all of the worldwide. In the Section 2, we illustrate the principle of the method used to find the predicators using the applied epidemiological model and in the Section 3, we illustrate the results and the discussion of the results while in the last section, we illustrate the conclusion of the results.

2. Computational Implementations

We collected the observed data of new corona virus disease recorded in the United States, Russia, the United Kingdom, Brazil, Spain, Bahrain, Egypt, Iran, Cyprus, France, India and the Syrian Arab Republic up to the binging of 2021. After that, We applied Runge-Kutta (R-K) method on the SIER epidemiologic model where in the light of this numerical method, the compartmental populations individual are written as follows [25]: with the following R-K functions: where \(\tau\) is the step of the time and \(S(t_n)\), \(I(t_n)\), \(R(t_n)\) and \(E(t_n)\) are the susceptible cases of pandemic individual, infectious cases of pandemic individual, recovered cases of pandemic individual and exposed cases of pandemic individual respectively at the moment \(t_n\) and \(S(t_{n+1})\), \(I(t_{n+1})\), \(R(t_{n+1})\) and \(E(t_{n+1})\) are susceptible cases of pandemic individual, infectious cases of pandemic individual, recovered cases of pandemic individual and exposed cases of the pandemic individual respectively at the moment \(t_{n+1}\). The coefficients lmj and cuj, appeared in the equation of the Runge-Kutta method (Equations (6)-(21)), are the weights and the nodes of the function expansion and \(a_{qu}^{j}\) are the Runge-Kutta matrix coefficients, all the previous coefficients can be found using Butcher tableau and the weights are under the following condition \(\sum_{m=1}^{m=z}l^{j}m=1\). In the flowchart presented in Figure 1, we illustrate the procedures followed for estimating the predictors of SARS-CoV-2 using the previous method. First, we write the collected data of the populations (susceptible, exposed, infectious and recovery) as matrices, then, we find the coefficient of the model and after that we find the SEIR predictors. Here, we note that we have to apply the conservation condition (Equation (5)) at each step of the calculations.

3. Results and Discussion

We calculated coefficient of exposing, coefficient of infection, coefficient of recovery and coefficient of mortality of the new coronavirus disease for the United States, which is located in north America, Russia which is located between Asia and Europe, the United Kingdom, which is located at north-west Europe, Brazil, which is located in south America, Spain, which is located at south-west Europe, Bahrain, which is located in Arabian Gulf, Egypt, which is located in Africa, Iran, which is located in west Asia, Cyprus, which is located in Mediterranean, India, which located in south Asia, France, which is located in west Europe and the Syrian Arab Republic, which is located at East of Mediterranean based on the reported data of the all cases of the new coronavirus disease in each country. After that we found the incubation periods of SARS-CoV-2. We illustrate the period of incubation values for the United States, Russia, the United Kingdom, Brazil, Spain, Bahrain, Egypt, Iran, Cyprus, India, France and the Syrian Arab Republic In Table 1, besides, the climate type of each country is illustrated in the same table. In addition to the incubation period values, we estimated the values of the steady state population of SARS-CoV-2 for the previous countries and we illustrated the results of this estimations in Table 2. Finally, the values of the critical immunisation threshold of SARS-CoV-2 were estimated for the previous countries and the results of this threshold were illustrated in Table 3 with the basic reproduction number values of SARS-CoV-2 for these countries.

Table 1. The values of the incubation period of SARS-CoV-2 in the United States, Russia, the United Kingdom, India, France, Brazil, Spain, Bahrain, Egypt, Iran, Cyprus and the Syrian Arab Republic and the geographical location of each country.

The country	The location of the country	\(P_{i}d\)
The United States	North America	4.643
India	South-Asia	8.000
Russia	Easter Europe-Northern Asia	4.000
The United Kingdom	North-West Europe	6.406
France	West-Europe	2.560
Brazil	South America	7.143
Spain	South-Western Europe	8.117
Bahrain	Arabian Gulf	2.500
Egypt	North Africa	10.00
Iran	West Asia	5.000
Cyprus	Mediterranean	10.00
The Syrian Arab Republic	East of Mediterranean	10.00

As we see from Table 1, the largest value of the incubation periods of SARS-CoV-2 is 10.00 days which returns to some middle east countries and the smallest value is 2.500 days which returns to Bahrain. Besides, we find that the average value of the incubation period of SARS-CoV-2 is 6.531 days.

Table 2. The values of the steady state population of SARS-CoV-2 in the United States, Russia, the United Kingdom, Brazil, Spain, Bahrain, Egypt, Iran, Cyprus and the Syrian Arab Republic and the climate type of each country.

The country	The climate	\(S_{M}\)
The United States	Changeable	0.310
Russia	Continental	0.448
The United Kingdom	Temperate	0.416
Brazil	Tropical	0.366
Spain	Temperate	0.666
Bahrain	Arid	0.329
Egypt	Arid	0.623
Iran	Arid	0.497
Cyprus	Mediterranean	0.636
The Syrian Arab Republic	Mediterranean	0.450

We see from Table 2 that the values of the steady state population of the new corona virus disease are located between 0.310 for the united states and 0.666 for Spain and the average value of the steady state population equals to 0.474 for different location countries.

Table 3. The values of the critical immunization threshold and the basic reproduction number of SARS-CoV-2 in the United States, Russia, the United Kingdom, Brazil, Spain, Bahrain, Egypt, Iran, Cyprus and the Syrian Arab Republic.

The country	\(n_{c}\)	\(R_{o}\)
The United States	0.690	3.224
Russia	0.552	2.231
The United Kingdom	0.584	2.403
Brazil	0.634	2.731
Spain	0.334	1.501
Bahrain	0.671	3.040
Egypt	0.337	1.606
Iran	0.503	2.012
Cyprus	0.364	1.573
The Syrian Arab Republic	0.550	2.222

As we see from Table 3, the value of the basic reproduction number of the new coronavirus pandemic for the United States and Bahrain are the greatest values in the previous different countries and the value of the basic reproduction number of the new coronavirus pandemic for Spain and Cyprus are the smallest between the previous different countries which returns to the high numbers of the infectious cases in the United States and the number of the cases with the new corona virus disease in Bahrain comparing to the number of people to the begging of 2021. Also, we see that the values of the reproduction numbers of the new coronavirus pandemic are in the range [1.5-3.5] for the pervious different countries with the climate and the geographic locations. Alternatively, we see from the same table that the critical immunization threshold are located between 0.334 which returns to Spain and 0.690 which returns to the United States. In addition, we see that the average values of the SARS-CoV-2 critical immunization threshold and the SARS-CoV-2 basic reproduction number are 0.526 and 2.254 respectively.

4. Conclusions

We employed one of the epidemiological models, namely, the SEIR model for estimating some important indicators values of the new coronavirus pandemic in twelve different countries, namely, the United States, Russia, the United Kingdom, Brazil, Spain, Bahrain, Egypt, Iran, Cyprus, India, France and the Syrian Arab Republic. The choice of countries was based on the different geographical location of countries on the earth and different with the type of the weather in each one. We used the recorded data of the collected cases of SARS-CoV-2 in the aforementioned countries, up to the beginning of 2021, to find the coefficient of exposing, the coefficient of infection, the coefficient of recovery and the coefficient of mortality of the new coronavirus pandemic for every country. First, we calculated the incubation period values of the new coronavirus pandemic in the aforementioned countries based on the coefficients of the SEIR model and the Rungee-Kutta simulation method. We found that the average value of the incubation period was about 6.531 days and the values of the period of infection (Table 1) were located in the interval [2.5-10] where the largest values return to Egypt, Cyprus and the Syrian Arab Republic and the smallest one returns to Bahrain. Also, we calculated the steady state population values of the new corona virus disease (Table 2) and found that the values of this population ratio are located in the interval [0.3-0.7] for the different location countries.

Finally, we found the critical immunization threshold values and the basic reproduction number values of the new coronavirus disease for ten of the previous countries. We found that the basic reproduction number values of the new coronavirus pandemic (Table 3) are in the interval [1.5-3.5] for the different countries. Besides, we found that the values of the basic reproduction number of the new coronavirus pandemic for the United States and Bahrain were the greatest values while the smallest values were for Spain and Cyprus. Alternatively, we found that the values of the critical immunization threshold are located in the interval [0.3-0.7] for those countries. We can use the same method for estimating the predictors of SARS-CoV-2 for other countries with different numbers of the new coronavirus pandemic, however, we chose the previous countries to clarify the relation between the incubation periods of SARS-CoV-2 with the geographical location and the climate of the country.

Abbreviations

• SARS: Severe acute respiratory syndrome.
• SARS-CoV-2: Severe acute respiratory syndrome-Corona Virus-2.
• SIRD: The susceptible-infected-recovered-dead.
• SEIR: The susceptible-exposed-infected-recovered.
• SVEIS: The susceptible-vaccinated-exposed-infected-susceptible.
• RK: Rungee-Kutta.

Conflicts of Interest: 

''The author declares no conflict of interest.''

1. Introduction

The country such as Bangladesh has very much potential in the textile sector. In the garments sectors of our country, mainly woven and knit fabrics are used. Nowadays, Bangladesh has attained the top most leading position in the garments business. To help the garment industry, a large number of the other textile sectors have been developed and there will be more in the near future [1].

However, the textile industry includes many sections with operating operations, one of which is the dyeing industry for coloring fabrics. The textile dyeing industry's contribution as a whole is essential to make fabrics with different shades. For this reason, to produce different shades on fabrics, textile dyes play a significant role. Moreover, textile dyes are a potentially important component of the industry as a whole. As Bangladesh is abundant in natural resources, the production of dyes in Bangladesh is relatively simple and easier. The textile dyes industry is booming in Bangladesh because of the textile industry's growth in the emerging economies [2]. Additionally, it is estimated that the Textile Dyes Market, which was accounted for USD 7.34 billion in 2017, will increase to USD 9.82 billion by 2022 [3]. That's why fabric dyeing processing has become the talked of topics today in our country.

In textile manufacturing, dyes are derived from two primary sources: natural dyes and synthetic dyes. Natural dyes can be characterized as organic materials that have the potential to add color to any substance for that they may have had an affinity. Natural dyes are environmentally friendly and very compliant with the environment, as they can be derived either from plants, animals, or minerals. They have beauty, depth of color, and less bright than synthetic dyes, but these natural color components were sustainable and biodegradable [4,5,6,7].

Compared to natural dyes, synthetic dyes have become commonly used due to cheaper costs [7,8] and a wide variety of vibrant colors with dramatically enhanced color fastness properties [6,9]. For example, reactive dyes have a chromophore group that is responsible for reacting with the substrate. These dyes have good color fastness properties due to the covalent bond established while dyeing [9,10]. The use of synthetic dyes became more appropriate in the food, cosmetic, and textile industry. Although most of the synthetic dyes were prepared from chemical compounds recently, those were not always welcoming for the human. However, all the natural dyes were suitable for humans, and these dyes did not harm the skin [11,12,13]. Furthermore, they have reduced toxicity and adverse reactions than that of synthetic dyes. Needless to say, many synthetic dyes have been forbidden since they induce allergy or carcinogenicity like symptoms.

Due to increased consciousness of the therapeutic properties of colors, global demand for natural dyes is of significant concern today. Though natural dyes applied in food are evaluated for protection, the details are not available for other natural dyes used in craft dyeing and possibly for boarder use. There is a reason to suppose that natural consumables are better than manufactured goods because they naturally have been made. If they are more commonly and commercially used, the safety of natural dyes must be demonstrated [5]. With respect to environmental considerations, the use of renewable non toxic and gentle natural colors needs to be reconsidered. Environment friendly natural dyes are a well-known phenomenon which produces innovative ideas. This is because it may be a viable choice for our dyeing requirements.

Highlighting certain of such problems, the purpose of this study was to determine the dye ability of henna dye towards cotton fabrics in contrast with the reactive dye using reactive dyeing technique.

2. Design of experiment

The test design is shown in the following Figure 1.

3. Method and materials

3.1. Materials

In this paper's experiment, 100% single jersey cotton knit fabric was used. The yarn count of the fabric was 28s meant that the single-ply yarn of 28 Ne had been used to make the fabric. The GSM (Gram per Square Meter) of the fabric was 160.

3.2. Chemicals

The Four H Dyeing & Printing Ltd supplied caustic scoured as well as bleached cotton knit fabric for the dyeing purposes. Four H Dyeing & Printing Ltd. was responsible for the supply of chemical auxiliaries, including sequestering agent, wetting agent, detergent, leveling agent, glauber salt, sodium carbonate, and acetic acid.

3.3. Additional materials and equipments

The instruments and ingredients for the research are as follows: electric balance, open dye bath, water, henna leaves, oven dryer, pot, reactive dye, rota wash fastness tester and crock meter.

3.4. Henna leaves preparation for test

Firstly, the leaves of Henna were taken from the plant. The leaves were cleaned off so as to eliminate dirt and adjunct material from the leaves. Then, with a view to removing the leaves' water content, the leaves were dried for several days under room temperature. For removal of the dye from the leaves, the dried leaves were pulverized with a grinding machine where the dried leaves were turned into powder form. After that, the powder was screened with a strainer to erase some stalk that is not well-pulverized.

3.5. Dye extraction process of henna dyes

A half kilogram of henna powder, along with the pH level of 9, was weighed in soda ash solution and kept for 24 hours under room temperature. Indeed, the solution of henna was reddish-orange color. Until the color was separated from the powder to the aqueous media, the mixture was impregnated with soda ash solutions. Reddish orange color has been formed when the liquid is alkaline. In the alkaline reddish-orange henna solution, acetic acid has been added to make pH neutral. The solution was mixed into a funnel with 5 ml of chloroform since it was natural dyes compound. After that, the solution was desiccated with glauber salt; as a result, the orange-yellow solution was produced. Until chloroform was volatilized, the solution was kept out of the sun. In the end, the mixture was then stored separately for storage. Henna leaves are shown in Figure 2 and henna powder is shown in Figure 3.

3.6. Operation for dyeing

The following techniques have been utilized in the study are explicated in the below.

Process: For 5% and 10% shade, 5% and 10% of dyestuffs, i.e., henna dye and reactive dye, has been used. Moreover, 10 gram per liter of glauber salt, 5 gram per liter of Sodium Carbonate, 2 gram per liter of Leveling Agent, 2 gram per liter of Sequestering Agent, 1 gram per liter of Wetting Agent, temperature \(70^{o}C\), time 60 minutes, 20 gm of fabric weight, and 1:30 of Materials to Liquor ratio were used. Firstly, samples and stock solutions were prepared according to the recipe. The required amounts were then taken from the stock solution by burette and measuring cylinder into an open bath. The bath was placed over an oven and stirred continuously. A thermometer was used to measure and control the temperature of the dye bath. When the temperature reached \(40^{o}C\), the soaked and squeezed sample was given to the bath with continuous stirring. After that, the temperature rose to \(70^{o}C\) within 15 minutes and maintained the temperature for 60 minutes. Then the fabric sample was taken out of the bath and rinsed with cold water. To perform after treatment, the washed sample was treated with 1 gram per liter of acetic acid at \(40^{o}C\) for 2-3 min to remove unfixed dye molecules from the fabric surface. In an oven dryer, samples are dried by hot air.

3.7. Analysis of colour fastness to rubbing

Assessment of Color Fastness to Rubbing ISO 105-X12: 1993 method was used to conduct the fabric test. Using crock meter, \(20cm\times5cm\) of sample size, \(5cm\times 5cm\) of bleached fabric, and given rotation of 10 cycles for 10 seconds to carry out the test.

3.8. Analysis of colour fastness to washing

ISO 105-C06 B2S procedure was used to execute the fabric test. 1 gram per liter of \(BNaO_3·4H_{2}O\) 4 gram per liter of ECE phosphate detergent were used. In addition, \(10cm \times 4cm\) of sample size, \(10cm \times 4cm\) of multi-fiber size, and 25 numbers of steel balls were used at 50°C for 30 minutes to perform the test.

3.9. I. Analysis of Colour Fastness to Perspiration

ISO 105-E04:2013 method was used to operate the fabric test. For the acidic solution, 5 gram per liter of sodium chloride, 2.2 gram per liter of sodium dihydrogen orthophosphate dihydrate, and 0.5 gram per liter of L-Histidine monohydrochloride monohydrate were used. Besides, \(10cm \times 4cm\) of sample size and \(10cm \times 4cm\) of multi-fiber size were used at \(38^{o}C\) for 4 hours 5 minutes to carry out the test. The solution's pH value was 5.5 that were maintained entirely in the perspiration (acid) test.

Furthermore, for the alkaline solution, 5 gram per liter of sodium chloride, 2.5 gram per liter of disodium hydrogen phosphate dehydrate, and 0.5 gram per liter of L-Histidine monohydrochloride monohydrate were used. Moreover, \(10cm \times 4cm\) of sample size and \(10cm \times 4cm\) of multi-fiber size were used at \(38^{o}C\) for 4 hours 5 minutes to operate the test. The solution's pH value was 8.0 that were maintained entirely in the perspiration (alkaline) test. The graphical representation of dyeing curve is given in Figure 4.

4. Result and discussion

4.1. Colour fastness to rubbing

From Table 1, it can say that the result of rubbing fastness to dry values of the cotton fabric dyed with henna dye was as same as the cotton fabric dyed with reactive dye, which was shown excellent results. Besides, it can be said that the result of rubbing fastness to wet values of the cotton fabric dyed with henna dye was shown very good results for 5% and 10% shade. Conversely, the result of rubbing fastness to wet values of the cotton fabric dyed with reactive dye was shown good and fair results for 5% and 10% shade, respectively. Hence, from the above discussion, we can say that in comparison with the cotton fabric dyed with reactive dye specimens, henna dye treated cotton fabrics deliver greater values for 5% and 10% shade.

Table 1. Effect of color fastness to rubbing.

Dye Name	% of Shade	Type	Result
Henna	5.0%	Dry	5
		Wet	4/5
	10.0%	Dry	5
		Wet	4/5
Reactive	0.5%	Dry	5
		Wet	4
	10.0%	Dry	5
		Wet	3

4.2. Colour fastness to washing

From Table 2, Color fastness to wash demonstrates the effects in terms of color staining. These findings indicate that the wash materials vary in the cotton fabric dyed with reactive dye specimens and the cotton fabric dyed with henna dye specimens depending on the nature of the color's fastness. Yet, henna dye treated cotton fabrics showed more affinity towards protein fiber than cellulosic fiber, understood from the multi-fiber grading in wash fastness.

Table 2. Effect of color fastness to washing.

% of Dye	Color Staining	Grade
5% Reactive	Cotton	4/5
	Nylon	5
	Polyester	5
	Acrylic	5
	Wool	3
55 % Henna	Cotton	4/5
	Nylon	5
	Polyester	5
	Acrylic	5
	Wool	3/4

4.3. Colour fastness to perspiration

From Table 3, it has been seen that caustic soda-treated fabric with reactive dye has been shown lower values than that of henna dye since the henna dye is more responsive to cotton fabric binding.

Table 3. Effect of color fastness to perspiration.

Dye Name	% of Shade	Type	Result
Henna	5.0%	Dry	5
		Wet	4/5
	10.0%	Dry	5
		Wet	4/5
Reactive	0.5%	Dry	5
		Wet	4
	10.0%	Dry	5
		Wet	3

4.4. Swatches

Figures 5-7, showed that henna provides an outstanding color with a strong coloration for 50% shade.

5. Conclusion

From the above discussion, it is understood that henna dye shows poor to moderate dye ability towards cotton fabrics in comparison with reactive dye if the reactive dyeing procedure is followed for henna dyeing. As opposed to the reactive dye, henna dye showed satisfactory fastness properties. In case of henna dye for 50% shade, it gives an excellent color tone with good level dyeing. This study's findings also show that natural henna derived dyes are not intended for decorating finger-nails, dyeing hair, or cosmetic products, but can also be used to impart textile colors, including cotton and protein fiber, to a certain degree. The outcome of this experiment would also introduce a rising need for organic garments to environmentally conscious customers. The mordant uses can be reduced by using natural dye, which ultimately results in showing eco-friendly dyeing. This study was practice-based, and the results are useful for the workers in the textile industries, who have been accountable for the color-fastness control and dyeing of textile materials with natural dyes.

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

A function \(f:\mathcal{K}\rightarrow \mathbb{R}\), where \(\mathcal{K}\) is an interval of real numbers, is called convex if the following inequality holds:

for all \(u_1,u_2\in \mathcal{K}\) and \(r\in[0,1]\). Function \(f\) is called concave if \(-f\) is convex.

The Hermite-Hadamard inequality [1] for convex functions \(f:\mathcal{K}\rightarrow \mathbb{R}\) on an interval of real line is:

where \(u_1,u_2\in \mathcal{K}\) with \(u_1< u_2\). Then Fejér [2] introduced the weighted generalization of (2) as follows where \(g:[u_1,u_2]\rightarrow \mathbb{R}\) is nonnegative, integrable and symmetric to \((u_1+u_2)/2\). For more results and details see [3,4,5,6,7,8,9,10,11,12,13,14].

Definition 1 ([15]). Let \(\mathcal{K}\subset\mathbb{R}\setminus\{0\}\) be a real interval. A function \(f:\mathcal{K}\rightarrow \mathbb{R}\) is said to be harmonically convex, if

for all \(u_1,u_2\in \mathcal{K}\) and \(r\in[0,1]\). If the inequality in (4) is reversed, then \(f\) is said to be harmonically concave.

Dragomir [16] gave the Hadamard's inequality for convex functions on the co-ordinate which is defined as:

Definition 2 ([16]). A function \(f:\Delta=[u_1,u_2]\times [v_1,v_2]\subseteq \mathbb{R}^{2} \rightarrow \mathbb{R}\) is called convex on the co-ordinate with \(u_1< u_2\) and \(v_1< v_2\) if the partial functions

\(f_{y}:[u_1,u_2] \rightarrow \mathbb{R}\), \(f_{y}(a)=f(a,y)\) and \(f_{x}:[v_1,v_2] \rightarrow \mathbb{R}\), \(f_{x}(c)=f(x,c)\) are convex for all \(x\in[u_1,u_2]\) and \(y\in [v_1,v_2]\).

Definition 3 ([17]). A function \(f:\Delta=[u_1,u_2]\times [v_1,v_2]\subseteq \mathbb{R}^{2} \rightarrow \mathbb{R}\) is called co-ordinate convex on \(\Delta\) with \(u_1< u_2\) and \(v_1< v_2\), if \begin{align*} &f(rx+(1-r)z,\tau y+(1-\tau)w)\leq r\tau f(x,y)+r(1-\tau)f(x,w)+(1-r)\tau f(z,y)+(1-r)(1-\tau)f(z,w), \end{align*} for all \(r,\tau\in [0,1]\) and \((x,y),(z,w)\in\Delta\). For more results and details see [16,17,18,19].

Definition 4 ([20]). A function \(f:\Delta=[u_1,u_2]\times [v_1,v_2]\subseteq (0,\infty)\times (0,\infty) \rightarrow \mathbb{R}\) is called co-ordinated harmonically convex on \(\Delta\) with \(u_1< u_2\) and \(v_1< v_2\), if \begin{align*} &f\left(\frac{xz}{rx+(1-r)z},\frac{yw}{\tau y+(1-\tau)w}\right)\leq r\tau f(x,y)+r(1-\tau)f(x,w)+(1-r)\tau f(z,y)+(1-r)(1-\tau)f(z,w), \end{align*} for all \(r,\tau\in [0,1]\) and \((x,y),(z,w)\in\Delta\).

Clearly, a function \(f:\Delta=[u_1,u_2]\times [v_1,v_2]\subseteq (0,\infty)\times (0,\infty) \rightarrow \mathbb{R}\) is called harmonically convex on the co-ordinate with \(u_1< u_2\) and \(v_1< v_2\) if the partial functions \(f_{y}:[u_1,u_2] \rightarrow \mathbb{R}\), \(f_{y}(a)=f(a,y)\) and \(f_{x}:[v_1,v_2] \rightarrow \mathbb{R}\), \(f_{x}(b)=f(x,b)\) are harmonically convex for all \(x\in[u_1,u_2]\) and \(y\in [v_1,v_2]\), see [21] for more details.

Definition 5 ([22]). Let \([u_1,u_2]\subset\mathbb{R}\) be a finite interval. The left- and right-side Katugampola fractional integrals of order \(\alpha(>0)\) of \(f\in X^{p}_{c}(u_1,u_2)\) are defined by,

\begin{equation*} ^{\rho}I^{\alpha}_{u_1+}f(x)=\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_{u_1}^{x}(x^{\rho}-t^{\rho})^{\alpha-1}t^{\rho-1}f(t)dt, \end{equation*} and \begin{equation*} ^{\rho}I^{\alpha}_{u_2-}f(x)=\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_{x}^{u_2}(t^{\rho}-x^{\rho})^{\alpha-1}t^{\rho-1}f(t)dt, \end{equation*} with \(u_1< x< u_2\) and \(\rho>0\), where \(X^{p}_{c}(u_1,u_2)\) \((c \in \mathbb{R}, 1\leq p\leq \infty)\) is the space of those complex valued Lebesgue measurable functions \(f\) on \([u_1,u_2]\) for which \(\|f\|_{X^{p}_{c}}< \infty\), where the norm is defined by \begin{equation*} \|f\|_{X^{p}_{c}}=\left( \int_{u_1}^{u_2}|t^{c}f(t)|^{p}\frac{dt}{t}\right)^{1/p}< \infty, \end{equation*} for \(1\leq p< \infty\), \(c\in \mathbb{R}\) and for the case \(p=\infty\), \begin{equation*} \|f\|_{X^{\infty}_{c}}= ess\ sup_{u_1\leq t\leq u_2}[t^{c}|f(t)|]. \end{equation*}

Definition 6 ([23]). Let \(f\in L_{1}([u_1,u_2]\times [v_1,v_2])\). The Katugampola fractional integrals \(^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{u_1+,v_1+}\), \(^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{u_1+,v_2-}\), \(^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{u_2-,v_1+}\) and \(^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{u_2-,v_2-}\) of order \(\alpha,\beta>0\) with \(a,c\geq 0\) are defined by

\begin{equation*} ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{u_1+,v_1+}f(x,y)=\frac{\rho^{1-\alpha}_{1}\rho^{1-\beta}_{2}}{\Gamma(\alpha)\Gamma(\beta)}\int_{u_1}^{x}\int_{v_1}^{y}(x^{\rho_{1}}-t^{\rho_{1}})^{\alpha-1}(y^{\rho_{2}}-s^{\rho_{2}})^{\beta-1}t^{\rho_{1}-1}s^{\rho_{2}-1}f(t,s)dsdt, \end{equation*} with \(x>u_1, \ y>v_1\), \begin{equation*} ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{u_1+,v_2-}f(x,y)=\frac{\rho^{1-\alpha}_{1}\rho^{1-\beta}_{2}}{\Gamma(\alpha)\Gamma(\beta)}\int_{u_1}^{x}\int_{y}^{v_2}(x^{\rho_{1}}-t^{\rho_{1}})^{\alpha-1}(s^{\rho_{2}}-y^{\rho_{2}})^{\beta-1}t^{\rho_{1}-1}s^{\rho_{2}-1}f(t,s)dsdt, \end{equation*} with \(x>u_1, \ y< v_2\), \begin{equation*} ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{u_2-,v_1+}f(x,y)=\frac{\rho^{1-\alpha}_{1}\rho^{1-\beta}_{2}}{\Gamma(\alpha)\Gamma(\beta)}\int_{x}^{u_2}\int_{v_1}^{y}(t^{\rho_{1}}-x^{\rho_{1}})^{\alpha-1}(y^{\rho_{2}}-s^{\rho_{2}})^{\beta-1}t^{\rho_{1}-1}s^{\rho_{2}-1}f(t,s)dsdt, \end{equation*} with \(x< u_2, \ y>v_1\), and \begin{equation*} ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{u_2-,v_2-}f(x,y)=\frac{\rho^{1-\alpha}_{1}\rho^{1-\beta}_{2}}{\Gamma(\alpha)\Gamma(\beta)}\int_{x}^{u_2}\int_{y}^{v_2}(t^{\rho_{1}}-x^{\rho_{1}})^{\alpha-1}(s^{\rho_{2}}-y^{\rho_{2}})^{\beta-1}t^{\rho_{1}-1}s^{\rho_{2}-1}f(t,s)dsdt, \end{equation*} with \(x< u_2, \ y< v_2\), respectively, where the Gamma function \(\Gamma\) is defined as \(\Gamma(\alpha)=\int_{0}^{\infty}e^{-t}t^{\alpha-1}dt\).

In the next section, we give result for harmonically convex functions in one dimension.

2. Hermite-Hadamard-Fejér type inequalities

In this section, we give Hermite-Hadamard-Fejér type inequalities for harmonically convex functions via Katugampola fractional integral in one dimension which will play a key role for the results in the next section. Latif et al., [24] defined following useful definition:

Definition 7 ([24]). A function \(h:[u_1,u_2]\subseteq\mathbb{R}\backslash \{0\}\rightarrow \mathbb{R} \) is said to be harmonically symmetric with respect to \(2u_1u_2/(u_1+u_2)\) if

\begin{equation*} h(x)=h\left(\frac{1}{\frac{1}{u_1}+\frac{1}{u_2}-\frac{1}{x}}\right) \end{equation*} holds for all \(x\in[u_1,u_2]\).

Lemma 1. Let \(\rho>0\). If \(h:[u_1^{\rho},u_2^{\rho}]\subseteq(0,\infty)\rightarrow \mathbb{R}\) is integrable and harmonically symmetric with respect to \(2u_1^{\rho}u_2^{\rho}/(u_1^{\rho}+u_2^{\rho})\), then

with \(\alpha>0\) and \(g(x^{\rho})=1/x^{\rho}\).

Proof. Since \(h\) is harmonically symmetric with respect to \(2u_1^{\rho}u_2^{\rho}/(u_1^{\rho}+u_2^{\rho})\), then by definition we have \(h(\frac{1}{x^{\rho}})=h\left(\frac{1}{\frac{1}{u_1^{\rho}}+\frac{1}{u_2^{\rho}}-x^{\rho}}\right)\), for all \(x^{\rho}\in\left[\frac{1}{u_2^{\rho}},\frac{1}{u_1^{\rho}}\right] \). In the following integral, by setting \(t^{\rho}=\frac{1}{u_1^{\rho}}+\frac{1}{u_2^{\rho}}-x^{\rho}\), we get \begin{align*} ^{\rho}I^{\alpha}_{1/u_2+}(h\circ g)(1/u_1^{\rho})&=\frac{\rho^{1-\alpha}}{\Gamma(\alpha)} \int^{\frac{1}{u_1}}_{\frac{1}{u_2}}\left(\frac{1}{u_1^{\rho}}-t^{\rho}\right)^{\alpha-1}t^{\rho-1}h\left( \frac{1}{t^{\rho}}\right) dt \\ &=\frac{\rho^{1-\alpha}}{\Gamma(\alpha)} \int^{\frac{1}{u_1}}_{\frac{1}{u_2}}\left(x^{\rho}-\frac{1}{u_2^{\rho}}\right)^{\alpha-1}x^{\rho-1}h\left( \frac{1}{\frac{1}{u_1^{\rho}}+\frac{1}{u_2^{\rho}}-x^{\rho}}\right) dx \\ &=\frac{\rho^{1-\alpha}}{\Gamma(\alpha)} \int^{\frac{1}{u_1}}_{\frac{1}{u_2}}\left(x^{\rho}-\frac{1}{u_2^{\rho}}\right)^{\alpha-1}x^{\rho-1}h\left( \frac{1}{x^{\rho}}\right) dx\\ &=\ ^{\rho}I^{\alpha}_{1/u_1-}(h\circ g)(1/u_2^{\rho}). \end{align*} This completes the proof.

Remark 1. In Lemma 1, if we take \(\rho\longmapsto 0\), we get Lemma 2 in [25].

Theorem 8. Let \(\rho>0\). Let \(f:[u_1^{\rho},u_2^{\rho}]\subseteq (0,\infty)\rightarrow\mathbb{R}\) be a harmonically convex with \(u_1< u_2\) and \(f\in L_1[u_1,u_2]\). If \(h:[u_1^{\rho},u_2^{\rho}]\subseteq(0,\infty)\rightarrow \mathbb{R}\) is nonnegative and harmonically symmetric with respect to \(2u_1^{\rho}u_2^{\rho}/(u_1^{\rho}+u_2^{\rho})\), then the following inequalities hold:

with \(\alpha>0\) and \(g(x^{\rho})=1/x^{\rho}\).

Proof. Since \(f\) is harmonically convex on \([u_1^{\rho},u_2^{\rho}]\), we have for all \(r\in[0,1]\)

Multiplying (7) by \(r^{\alpha\rho-1}h\left(\frac{u_1^{\rho}u_2^{\rho}}{r^{\rho}u_2^{\rho}+(1-r^{\rho})u_1^{\rho}}\right)\) on both sides and integrate with respect to \([0,1]\), we get \begin{align*} \begin{split} 2f\left(\frac{2u_1^{\rho}u_2^{\rho}}{u_1^{\rho}+u_2^{\rho}}\right)&\int_{0}^{1} r^{\alpha\rho-1}h\left(\frac{u_1^{\rho}u_2^{\rho}}{r^{\rho}u_2^{\rho}+(1-r^{\rho})u_1^{\rho}}\right)dr \\ &\leq \int_{0}^{1}r^{\alpha\rho-1}f\left(\frac{u_1^{\rho}u_2^{\rho}}{r^{\rho}u_1^{\rho}+(1-r^{\rho})u_2^{\rho}}\right) h\left(\frac{u_1^{\rho}u_2^{\rho}}{r^{\rho}u_2^{\rho}+(1-r^{\rho})u_1^{\rho}}\right)dr\\ &\;\;\;\;+\int_{0}^{1}r^{\alpha\rho-1}f\left(\frac{u_1^{\rho}u_2^{\rho}}{r^{\rho}u_2^{\rho}+(1-r^{\rho})u_1^{\rho}}\right) h\left(\frac{u_1^{\rho}u_2^{\rho}}{r^{\rho}u_2^{\rho}+(1-r^{\rho})u_1^{\rho}}\right)dr. \end{split} \end{align*} Since \(h\) is harmonically symmetric with respect to \(2u_1^{\rho}u_2^{\rho}/(u_1^{\rho}+u_2^{\rho})\). By setting \(x^{\rho}=\frac{r^{\rho}u_2^{\rho}+(1-r^{\rho})u_1^{\rho}}{u_1^{\rho}u_2^{\rho}}\), we get \begin{align*} \begin{split} 2\left(\frac{u_1^{\rho}u_2^{\rho}}{u_2^{\rho}-u_1^{\rho}}\right)^{\alpha}&f\left(\frac{2u_1^{\rho}u_2^{\rho}}{u_1^{\rho}+u_2^{\rho}}\right) \int_{1/u_2}^{1/u_1}\left(x^{p}-\frac{1}{u_2^{\rho}}\right)^{\alpha-1}h\left( \frac{1}{x^{\rho}}\right) dx \\ \leq&\left(\frac{u_1^{\rho}u_2^{\rho}}{u_2^{\rho}-u_1^{\rho}}\right)^{\alpha}\left[\int_{1/u_2}^{1/u_1}\left(x^{p}-\frac{1}{u_2^{\rho}}\right)^{\alpha-1} f\left(\frac{1}{\frac{1}{u_1^{\rho}}+\frac{1}{u_2^{\rho}}-x^{\rho}}\right)h\left( \frac{1}{x^{\rho}}\right) dx\right.\\&\left.+\int_{1/u_2}^{1/u_1}\left(x^{p}-\frac{1}{u_2^{\rho}}\right)^{\alpha-1} f\left( \frac{1}{x^{\rho}}\right) h\left( \frac{1}{x^{\rho}}\right) dx \right] \\ =&\left(\frac{u_1^{\rho}u_2^{\rho}}{u_2^{\rho}-u_1^{\rho}}\right)^{\alpha}\left[\int_{1/u_2}^{1/u_1}\left(\frac{1}{u_1^{\rho}}-x^{p}\right)^{\alpha-1} f\left( \frac{1}{x^{\rho}}\right) h\left(\frac{1}{\frac{1}{u_1^{\rho}}+\frac{1}{u_2^{\rho}}-x^{\rho}}\right)dx\right. \\ &\left.+\int_{1/u_2}^{1/u_1}\left(x^{p}-\frac{1}{u_2^{\rho}}\right)^{\alpha-1} f\left( \frac{1}{x^{\rho}}\right) h\left( \frac{1}{x^{\rho}}\right) dx \right]. \end{split} \end{align*} Then by Lemma 1, we have This completes the first inequality. For second inequality, we first note that if \(f\) is harmonically convex function, then we have Multiplying (8) by \(r^{\alpha\rho-1}h\left(\frac{u_1^{\rho}u_2^{\rho}}{r^{\rho}u_2^{\rho}+(1-r^{\rho})u_1^{\rho}}\right)\) on both sides and integrate with respect to \(r\in[0,1]\), we get \begin{align*} \begin{split} &\int_{0}^{1}r^{\alpha\rho-1}f\left(\frac{u_1^{\rho}u_2^{\rho}}{r^{\rho}u_1^{\rho}+(1-r^{\rho})u_2^{\rho}}\right) h\left(\frac{u_1^{\rho}u_2^{\rho}}{r^{\rho}u_2^{\rho}+(1-r^{\rho})u_1^{\rho}}\right)dr \\ &\;\;\;+\int_{0}^{1}r^{\alpha\rho-1}f\left(\frac{u_1^{\rho}u_2^{\rho}}{r^{\rho}u_2^{\rho}+(1-r^{\rho})u_1^{\rho}}\right) h\left(\frac{u_1^{\rho}u_2^{\rho}}{r^{\rho}u_2^{\rho}+(1-r^{\rho})u_1^{\rho}}\right)dr \\ &\leq (f(u_1^{\rho})+f(u_2^{\rho}))\int_{0}^{1}r^{\alpha\rho-1}h\left(\frac{u_1^{\rho}u_2^{\rho}}{r^{\rho}u_2^{\rho}+(1-r^{\rho})u_1^{\rho}}\right)dr, \end{split} \end{align*} i.e., \begin{align*} \begin{split} \left(\frac{u_1^{\rho}u_2^{\rho}}{u_2^{\rho}-u_1^{\rho}}\right)^{\alpha}&\rho^{\alpha-1}\Gamma(\alpha) \left[^{\rho}I^{\alpha}_{1/u_1-}(fh\circ g)(1/u_2^{\rho})+^{\rho}I^{\alpha}_{1/u_2+}(fh\circ g)(1/u_1^{\rho})\right] \\ &\leq\left(\frac{u_1^{\rho}u_2^{\rho}}{u_2^{\rho}-u_1^{\rho}}\right)^{\alpha}\rho^{\alpha-1}\Gamma(\alpha)\frac{f(u_1^{\rho})+f(u_2^{\rho})}{2} \left[^{\rho}I^{\alpha}_{1/u_1-}(h\circ g)(1/u_2^{\rho})+^{\rho}I^{\alpha}_{1/u_2+}(h\circ g)(1/u_1^{\rho})\right]. \end{split} \end{align*} This completes the proof.

Remark 2.

• 1)   In Theorem 8, if we take \(\rho\rightarrow 1\), we get Theorem 5 in [25].
• 2)   In Theorem 8, if we take \(\rho\rightarrow 1\) and \(\alpha=1\), we get Theorem 8 in [26].

3. Hermite-Hadamard-Fejér type inequalities on co-ordinates

In this section, we established some new results by using Katugampola fractional integrals on co-ordinates. First we give the following result:

Theorem 9. Let \(\alpha,\beta>0\) and \(\rho_{1},\rho_{2}>0\). Let \(f:\Delta=[u_1^{\rho_{1}},u_2^{\rho_{1}}]\times [v_1^{\rho_{2}},v_2^{\rho_{2}}]\subseteq (0,\infty)\times(0,\infty) \rightarrow \mathbb{R}\) be a co-ordinated harmonically convex on \(\Delta\), with \(0< u_1< u_2\), \(0< v_1< v_2\). If \(h:\Delta\rightarrow \mathbb{R}\) is nonnegative and harmonically symmetric with respect to \(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}}\), \(\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\) on \(\Delta\). Then

holds, where \(g(x^{\rho_{1}},y^{\rho_{2}})=\left(\frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}} \right) \).

Proof. Since \(f\) is co-ordinated harmonically convex on \(\Delta\), we have

Multiplying (11) by \(r^{\rho_{1}\alpha-1}\tau^{\rho_{2}\beta-1}h\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{r^{\rho_{1}}u_2^{\rho_{1}}+(1-r^{\rho_{1}})u_1^{\rho_{1}}},\frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{\tau^{\rho_{2}} v_2^{\rho_{2}}+(1-\tau^{\rho_{2}})v_1^{\rho_{2}}}\right)\) on both sides and then integrating with respect to \((r,\tau)\) over \([0,1]\times [0,1]\), we get \begin{align*} f\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right)& \int_{0}^{1}\int_{0}^{1}r^{\rho_{1}\alpha-1}\tau^{\rho_{2}\beta-1} h\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{r^{\rho_{1}}u_2^{\rho_{1}}+(1-r^{\rho_{1}})u_1^{\rho_{1}}},\frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{\tau^{\rho_{2}} v_2^{\rho_{2}}+(1-\tau^{\rho_{2}})v_1^{\rho_{2}}}\right)drd\tau \\ \leq& \frac{1}{4}\bigg[\int_{0}^{1}\int_{0}^{1} f\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{r^{\rho_{1}}u_1^{\rho_{1}}+(1-r^{\rho_{1}})u_2^{\rho_{1}}},\frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{\tau^{\rho_{2}} v_1^{\rho_{2}}+(1-\tau^{\rho_{2}})v_2^{\rho_{2}}}\right) \\ &\times h\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{r^{\rho_{1}}u_2^{\rho_{1}}+(1-r^{\rho_{1}})u_1^{\rho_{1}}},\frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{\tau^{\rho_{2}} v_2^{\rho_{2}}+(1-\tau^{\rho_{2}})v_1^{\rho_{2}}}\right)r^{\rho_{1}\alpha-1}\tau^{\rho_{2}\beta-1}drd\tau \\ & +\int_{0}^{1}\int_{0}^{1}f\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{r^{\rho_{1}}u_1^{\rho_{1}}+(1-r^{\rho_{1}})u_2^{\rho_{1}}},\frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{\tau^{\rho_{2}} v_2^{\rho_{2}}+(1-\tau^{\rho_{2}})v_1^{\rho_{2}}}\right) \\ &\times h\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{r^{\rho_{1}}u_2^{\rho_{1}}+(1-r^{\rho_{1}})u_1^{\rho_{1}}},\frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{\tau^{\rho_{2}} v_2^{\rho_{2}}+(1-\tau^{\rho_{2}})v_1^{\rho_{2}}}\right)r^{\rho_{1}\alpha-1}\tau^{\rho_{2}\beta-1}drd\tau \\ &+ \int_{0}^{1}\int_{0}^{1} f\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{r^{\rho_{1}}u_2^{\rho_{1}}+(1-r^{\rho_{1}})u_1^{\rho_{1}}},\frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{\tau^{\rho_{2}} v_1^{\rho_{2}}+(1-\tau^{\rho_{2}})v_2^{\rho_{2}}}\right) \\ &\times h\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{r^{\rho_{1}}u_2^{\rho_{1}}+(1-r^{\rho_{1}})u_1^{\rho_{1}}},\frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{\tau^{\rho_{2}} v_2^{\rho_{2}}+(1-\tau^{\rho_{2}})v_1^{\rho_{2}}}\right)r^{\rho_{1}\alpha-1}\tau^{\rho_{2}\beta-1}drd\tau \\ & +\int_{0}^{1}\int_{0}^{1}f\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{r^{\rho_{1}}u_2^{\rho_{1}}+(1-r^{\rho_{1}})u_1^{\rho_{1}}},\frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{\tau^{\rho_{2}} v_2^{\rho_{2}}+(1-\tau^{\rho_{2}})v_1^{\rho_{2}}}\right) \\ &\times h\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{r^{\rho_{1}}u_2^{\rho_{1}}+(1-r^{\rho_{1}})u_1^{\rho_{1}}},\frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{\tau^{\rho_{2}} v_2^{\rho_{2}}+(1-\tau^{\rho_{2}})v_1^{\rho_{2}}}\right)r^{\rho_{1}\alpha-1}\tau^{\rho_{2}\beta-1}drd\tau \bigg]. \end{align*} By change of variables \(x^{\rho_{1}}=\frac{r^{\rho_{1}}u_2^{\rho_{1}}+(1-r^{\rho_{1}})u_1^{\rho_{1}}}{u_1^{\rho_{1}}u_2^{\rho_{1}}}\) and \(y^{\rho_{2}}=\frac{\tau^{\rho_{2}} v_2^{\rho_{2}}+(1-\tau^{\rho_{2}})v_1^{\rho_{2}}}{v_1^{\rho_{2}}v_2^{\rho_{2}}}\) and using the symmetric property of \(h\), we find \begin{align*} &\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_2^{\rho_{1}}-u_1^{\rho_{1}}}\right)^{\alpha}\left( \frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_2^{\rho_{2}}-v_1^{\rho_{2}}}\right)^{\beta}f\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right) \\ &\;\;\;\;\times \int_{1/v_2}^{1/v_1}\int_{1/u_2}^{1/u_1} \left(x^{\rho_{1}}-\frac{1}{u_2^{\rho_{1}}} \right)^{\alpha-1}\left(y^{\rho_{2}}-\frac{1}{v_2^{\rho_{2}}} \right)^{\beta-1}x^{\rho_{1}-1}y^{\rho_{2}-1} h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dxdy \\ & \leq \frac{1}{4}\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_2^{\rho_{1}}-u_1^{\rho_{1}}}\right)^{\alpha}\left( \frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_2^{\rho_{2}}-v_1^{\rho_{2}}}\right)^{\beta}\left[\int_{1/v_2}^{1/v_1}\int_{1/u_2}^{1/u_1} \left(x^{\rho_{1}}-\frac{1}{u_2^{\rho_{1}}} \right)^{\alpha-1}\left(y^{\rho_{2}}-\frac{1}{v_2^{\rho_{2}}} \right)^{\beta-1}\right. \end{align*} \begin{align*} &\;\;\;\times x^{\rho_{1}-1}y^{\rho_{2}-1}f\left( \frac{1}{\frac{1}{u_1^{\rho_{1}}}+\frac{1}{u_2^{\rho_{1}}}-x^{\rho_{1}}},\frac{1}{\frac{1}{v_1^{\rho_{2}}}+\frac{2}{v_2^{\rho_{2}}}-y^{\rho_{2}}}\right) h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dxdy\\ &\;\;\;+\int_{1/v_2}^{1/v_1}\int_{1/u_2}^{1/u_1} \left(x^{\rho_{1}}-\frac{1}{u_2^{\rho_{1}}} \right)^{\alpha-1}\left(y^{\rho_{2}}-\frac{1}{v_2^{\rho_{2}}} \right)^{\beta-1} x^{\rho_{1}-1}y^{\rho_{2}-1}f\left( \frac{1}{\frac{1}{u_1^{\rho_{1}}}+\frac{1}{u_2^{\rho_{1}}}-x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right) h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dxdy \\ &\;\;\;+\int_{1/v_2}^{1/v_1}\int_{1/u_2}^{1/u_1} \left(x^{\rho_{1}}-\frac{1}{u_2^{\rho_{1}}} \right)^{\alpha-1}\left(y^{\rho_{2}}-\frac{1}{v_2^{\rho_{2}}} \right)^{\beta-1} x^{\rho_{1}-1}y^{\rho_{2}-1}f\left( \frac{1}{x^{\rho_{1}}},\frac{1}{\frac{1}{v_1^{\rho_{2}}}+\frac{2}{v_1^{\rho_{2}}}-y^{\rho_{2}}}\right) h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dxdy \\ &\;\;\;\left.+\int_{1/v_2}^{1/v_1}\int_{1/u_2}^{1/u_1} \left(x^{\rho_{1}}-\frac{1}{u_2^{\rho_{1}}} \right)^{\alpha-1}\left(y^{\rho_{2}}-\frac{1}{v_2^{\rho_{2}}} \right)^{\beta-1} x^{\rho_{1}-1}y^{\rho_{2}-1}f\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right) h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dxdy \right] \\ &= \frac{1}{4}\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_2^{\rho_{1}}-u_1^{\rho_{1}}}\right)^{\alpha}\left( \frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_2^{\rho_{2}}-v_1^{\rho_{2}}}\right)^{\beta}\left[\int_{1/v_2}^{1/v_1}\int_{1/u_2}^{1/u_1} \left(\frac{1}{u_1^{\rho_{1}}}-x^{\rho_{1}} \right)^{\alpha-1}\left(\frac{1}{v_1^{\rho_{2}}}-y^{\rho_{2}} \right)^{\beta-1}\right. \\ &\;\;\;\times x^{\rho_{1}-1}y^{\rho_{2}-1}f\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)h\left( \frac{1}{\frac{1}{u_1^{\rho_{1}}}+\frac{1}{u_2^{\rho_{1}}}-x^{\rho_{1}}},\frac{1}{\frac{1}{v_1^{\rho_{2}}}+\frac{2}{v_2^{\rho_{2}}}-y^{\rho_{2}}}\right)dxdy \\ &\;\;\;+\int_{1/v_2}^{1/v_1}\int_{1/u_2}^{1/u_1} \left(\frac{1}{u_1^{\rho_{1}}}-x^{\rho_{1}} \right)^{\alpha-1}\left(y^{\rho_{2}}-\frac{1}{v_2^{\rho_{2}}} \right)^{\beta-1} x^{\rho_{1}-1}y^{\rho_{2}-1}f\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)h\left( \frac{1}{\frac{1}{u_1^{\rho_{1}}}+\frac{1}{u_2^{\rho_{1}}}-x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dxdy \\ &\;\;\;+\int_{1/v_2}^{1/v_1}\int_{1/u_2}^{1/u_1} \left(x^{\rho_{1}}-\frac{1}{u_2^{\rho_{1}}} \right)^{\alpha-1}\left(\frac{1}{v_1^{\rho_{2}}}-y^{\rho_{2}} \right)^{\beta-1} x^{\rho_{1}-1}y^{\rho_{2}-1}f\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{\frac{1}{v_1^{\rho_{2}}}+\frac{2}{v_2^{\rho_{2}}}-y^{\rho_{2}}}\right)dxdy \\ &\;\;\;\left.+\int_{1/v_2}^{1/v_1}\int_{1/u_2}^{1/u_1} \left(x^{\rho_{1}}-\frac{1}{u_2^{\rho_{1}}} \right)^{\alpha-1}\left(y^{\rho_{2}}-\frac{1}{v_2^{\rho_{2}}} \right)^{\beta-1} x^{\rho_{1}-1}y^{\rho_{2}-1}f\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right) h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dxdy\right]. \end{align*} Thus, we get \begin{align*} \begin{split} \frac{\Gamma(\alpha)\Gamma(\beta)}{\rho_{1}^{1-\alpha}\rho_2^{1-\beta}}&\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_2^{\rho_{1}}-u_1^{\rho_{1}}}\right)^{\alpha}\left( \frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_2^{\rho_{2}}-v_1^{\rho_{2}}}\right)^{\beta}f\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right) \\ &\times\left[\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_1-,1/v_1-}h\circ g\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) +\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_1-,1/v_2+}h\circ g\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right)\right. \\ &\left.+\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_2+,1/v_1-}h\circ g\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) +\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_2+,1/v_2+}h\circ g\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right] \\ \leq& \frac{\Gamma(\alpha)\Gamma(\beta)}{4\rho_{1}^{1-\alpha}\rho_2^{1-\beta}}\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_2^{\rho_{1}}-u_1^{\rho_{1}}}\right)^{\alpha}\left( \frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_2^{\rho_{2}}-v_1^{\rho_{2}}}\right)^{\beta} \\ &\times \left[\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_1-,1/v_1-}(fh\circ g) \left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) +\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_1-,1/v_2+}(fh\circ g)\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right)\right. \\ &\left.+\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_2+,1/v_1-}(fh\circ g)\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) +\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_2+,1/v_2+}(fh\circ g)\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right]. \end{split} \end{align*} This completes the first inequality of (10). For the second inequality of (10) we use the co-ordinated harmonically convexity of \(f\) as: Thus multiplying (12) by \(r^{\rho_{1}\alpha-1}\tau^{\rho_{2}\beta-1}h\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{r^{\rho_{1}}u_2^{\rho_{1}}+(1-r^{\rho_{1}})u_1^{\rho_{1}}},\frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{\tau^{\rho_{2}} v_2^{\rho_{2}}+(1-\tau^{\rho_{2}})v_1^{\rho_{2}}}\right)\) and integrating with respect to \((r,\tau)\) over \([0,1]\times [0,1]\), we get the second inequality of (10). Hence the proof is completed.

Theorem 10. Let \(\alpha,\beta>0\) and \(\rho_{1},\rho_{2}>0\). Let \(f:\Delta=[u_1^{\rho_{1}},u_2^{\rho_{1}}]\times [v_1^{\rho_{2}},v_2^{\rho_{2}}]\subseteq (0,\infty)\times(0,\infty) \rightarrow \mathbb{R}\) be a co-ordinated harmonically convex on \(\Delta\), with \(0< u_1< u_2\), \(0< v_1< v_2\) and \(f\in L_1[\Delta]\). If \(h:\Delta\rightarrow \mathbb{R}\) is nonnegative and harmonically symmetric with respect to \(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}}\),\(\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\) on \(\Delta\). Then the following inequalities hold:

\begin{align*} f\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right)&\left[\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_1-,1/v_1-}h\circ g\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) +\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_1-,1/v_2+}h\circ g\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right)\right. \\ &\left.+\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_2+,1/v_1-}h\circ g\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) +\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_2+,1/v_2+}h\circ g\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right] \\ \leq& \ ^{\rho_{1}}I^{\alpha}_{1/u_1-}\left[ f\circ g_{1}\left( \frac{1}{u_2^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right)\ ^{\rho_{2}}I^{\beta}_{1/v_1-}h\circ g_2\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right] \\ &+\ ^{\rho_{1}}I^{\alpha}_{1/u_1-}\bigg[ f\circ g_{1}\left( \frac{1}{u_2^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right) \ ^{\rho_{2}}I^{\beta}_{1/v_2+}h\circ g_2\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \bigg] \\ &+\ ^{\rho_{1}}I^{\alpha}_{1/u_2+}\left[ f\circ g_{1}\left( \frac{1}{u_1^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right)\ ^{\rho_{2}}I^{\beta}_{1/v_1-}h\circ g_2\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right] \\ &+\ ^{\rho_{1}}I^{\alpha}_{1/u_2+}\left[ f\circ g_{1}\left( \frac{1}{u_1^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right)\ ^{\rho_{2}}I^{\beta}_{1/v_2+}h\circ g_2\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right] \\ &+\ ^{\rho_{2}}I^{\beta}_{1/v_1-}\bigg[ f\circ g_{2}\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}}, \frac{1}{v_2^{\rho_{2}}}\right) \ ^{\rho_{1}}I^{\alpha}_{1/u_1-}h\circ g_1\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \bigg] \\ &+ \ ^{\rho_{2}}I^{\beta}_{1/v_1-}\left[ f\circ g_{2}\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}}, \frac{1}{v_2^{\rho_{2}}}\right) \ ^{\rho_{1}}I^{\alpha}_{1/u_2+}h\circ g_1\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right] \\ &+\ ^{\rho_{2}}I^{\beta}_{1/v_2+}\left[ f\circ g_{2}\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}}, \frac{1}{v_1^{\rho_{2}}}\right) \ ^{\rho_{1}}I^{\alpha}_{1/u_1-}h\circ g_1\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right] \\ &+\ ^{\rho_{2}}I^{\beta}_{1/v_2+}\bigg[ f\circ g_{2}\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}}, \frac{1}{v_1^{\rho_{2}}}\right) \ ^{\rho_{1}}I^{\alpha}_{1/u_2+}h\circ g_1\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \bigg] \\ \leq& 2\left[\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_1-,1/v_1-}fh\circ g\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) +\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_1-,1/v_2+}fh\circ g\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right)\right. \\ &+\left.\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_2+,1/v_1-}fh\circ g\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) +\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_2+,1/v_2+}fh\circ g\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right] \\ \leq& \ ^{\rho_{1}}I^{\alpha}_{1/u_1-}\left[ f\circ g_{1}\left( \frac{1}{u_2^{\rho_{1}}},v_1^{\rho_{2}}\right)\ ^{\rho_{2}}I^{\beta}_{1/v_2+}h\circ g_2\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right] \\ &+\ ^{\rho_{1}}I^{\alpha}_{1/u_1-}\bigg[ f\circ g_{1}\left( \frac{1}{u_2^{\rho_{1}}},v_2^{\rho_{2}}\right) \ ^{\rho_{2}}I^{\beta}_{1/v_1-}h\circ g_2\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \bigg] \\ &+\ ^{\rho_{1}}I^{\alpha}_{1/u_2+}\left[ f\circ g_{1}\left( \frac{1}{u_1^{\rho_{1}}},v_1^{\rho_{2}}\right)\ ^{\rho_{2}}I^{\beta}_{1/v_2+}h\circ g_2\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right] \end{align*} \begin{align} \label{tt1e1} &+\ ^{\rho_{1}}I^{\alpha}_{1/u_2+}\left[ f\circ g_{1}\left( \frac{1}{u_1^{\rho_{1}}},v_2^{\rho_{2}}\right)\ ^{\rho_{2}}I^{\beta}_{1/v_1-}h\circ g_2\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right]\notag \\ &+\ ^{\rho_{2}}I^{\beta}_{1/v_1-}\left[ f\circ g_{2}\left(u_1^{\rho_{1}}, \frac{1}{v_2^{\rho_{2}}}\right) \ ^{\rho_{1}}I^{\alpha}_{1/u_2+}h\circ g_1\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right]\notag \\ &+ \ ^{\rho_{2}}I^{\beta}_{1/v_1-}\left[ f\circ g_{2}\left(u_2^{\rho_{1}}, \frac{1}{v_2^{\rho_{2}}}\right) \ ^{\rho_{1}}I^{\alpha}_{1/u_1-}h\circ g_1\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right]\notag \\ &+\ ^{\rho_{2}}I^{\beta}_{1/v_2+}\left[ f\circ g_{2}\left(u_1^{\rho_{1}}, \frac{1}{v_1^{\rho_{2}}}\right) \ ^{\rho_{1}}I^{\alpha}_{1/u_2+}h\circ g_1\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right]\notag \\ &+\ ^{\rho_{2}}I^{\beta}_{1/v_2+}\left[ f\circ g_{2}\left(u_2^{\rho_{1}}, \frac{1}{v_1^{\rho_{2}}}\right) \ ^{\rho_{1}}I^{\alpha}_{1/u_1-}h\circ g_1\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right]\notag \\ \end{align} where \(g(x^{\rho_{1}},y^{\rho_{2}})=\left(\frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}} \right) \), \(g_{1}(x^{\rho_{1}},y^{\rho_{2}})=\left(\frac{1}{x^{\rho_{1}}},y^{\rho_{2}} \right)\) and \(g_{2}(x^{\rho_{1}},y^{\rho_{2}})=\left(x^{\rho_{1}},\frac{1}{y^{\rho_{2}}} \right) \), respectively.

Proof. Since \(f\) is co-ordinated harmonically convex on \(\Delta\), then the function \(f_{1/x^{\rho_{1}}}:[v_1^{\rho_{2}},v_2^{\rho_{2}}]\rightarrow \mathbb{R}\), defined by \(f_{1/x^{\rho_{1}}}(y^{\rho_{2}})=f(\frac{1}{x^{\rho_{1}}},y^{\rho_{2}})\) is harmonically convex on \([v_1^{\rho_{2}},v_2^{\rho_{2}}]\) for all \(x^{\rho_{1}}\in \left[ \frac{1}{u_2^{\rho_{1}}},\frac{1}{u_1^{\rho_{1}}}\right] \). Then from (6), we have

Multiplying both sides of (14) by \(\frac{x^{\rho_1-1}\left(x^{\rho_1}-\frac{1}{u_2^{\rho_1}} \right)^{\alpha-1}}{\rho_1^{\alpha-1}\Gamma(\alpha)}\) and \(\frac{x^{\rho_1-1}\left(\frac{1}{u_1^{\rho_1}}-x^{\rho_1} \right)^{\alpha-1}}{\rho_1^{\alpha-1}\Gamma(\alpha)}\), and integrating with respect to \(x\) over \(\left[ \frac{1}{u_2},\frac{1}{u_1}\right] \), respectively, we get and \begin{align*} \frac{\rho_1^{1-\alpha}\rho_2^{1-\beta}}{\Gamma(\alpha)\Gamma(\beta)}&\left[\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(\frac{1}{u_1^{\rho_1}}-x^{\rho_1} \right)^{\alpha-1}\left(y^{\rho_2}-\frac{1}{v_2^{\rho_2}} \right)^{\beta-1} x^{\rho_{1}-1} y^{\rho_{2}-1}f\left(\frac{1}{x^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}} \right)\right. \\ &\times h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx +\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(\frac{1}{u_1^{\rho_1}}-x^{\rho_1} \right)^{\alpha-1}\left(\frac{1}{v_1^{\rho_2}}-y^{\rho_2} \right)^{\beta-1}x^{\rho_{1}-1}y^{\rho_{2}-1} \\ &\times \left.f\left(\frac{1}{x^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}} \right)h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx\right] \\ \leq &\frac{\rho_1^{1-\alpha}\rho_2^{1-\beta}}{\Gamma(\alpha)\Gamma(\beta)}\left[\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(\frac{1}{u_1^{\rho_1}}-x^{\rho_1} \right)^{\alpha-1}\left(y^{\rho_2}-\frac{1}{v_2^{\rho_2}} \right)^{\beta-1}x^{\rho_1-1} y^{\rho_{2}-1}\right. \\ &\times f\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx +\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(\frac{1}{u_1^{\rho_1}}-x^{\rho_1} \right)^{\alpha-1}\left(\frac{1}{v_1^{\rho_2}}-y^{\rho_2} \right)^{\beta-1} \\ &\left.\times x^{\rho_1-1} y^{\rho_{2}-1}f\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx\right] \end{align*} Using similar arguments for the mapping \(f_{\frac{1}{y^{\rho_2}}}:[u_1^{\rho_1},u_2^{\rho_1}]\rightarrow \mathbb{R}\), \(f_{\frac{1}{y^{\rho_{2}}}}(x^{\rho_{1}})=f(x^{\rho_{1}},\frac{1}{y^{\rho_{2}}})\), we have and By adding the inequalities (15)\(\sim\)(18), we get \begin{align*} ^{\rho_{1}}I^{\alpha}_{1/u_1-}&\left[ f\circ g_{1}\left( \frac{1}{u_2^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right)\ ^{\rho_{2}}I^{\beta}_{1/v_1-}h\circ g_2\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right] \\&+\ ^{\rho_{1}}I^{\alpha}_{1/u_1-}\left[ f\circ g_{1}\left( \frac{1}{u_2^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right) \ ^{\rho_{2}}I^{\beta}_{1/v_2+}h\circ g_2\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right] \\ &+\ ^{\rho_{1}}I^{\alpha}_{1/u_2+}\left[ f\circ g_{1}\left( \frac{1}{u_1^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right)\ ^{\rho_{2}}I^{\beta}_{1/v_1-}h\circ g_2\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right] \\ &+\ ^{\rho_{1}}I^{\alpha}_{1/u_2+}\left[ f\circ g_{1}\left( \frac{1}{u_1^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right)\ ^{\rho_{2}}I^{\beta}_{1/v_2+}h\circ g_2\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right] \\ &+\ ^{\rho_{2}}I^{\beta}_{1/v_1-}\left[ f\circ g_{2}\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}}, \frac{1}{v_2^{\rho_{2}}}\right) \ ^{\rho_{1}}I^{\alpha}_{1/u_1-}h\circ g_1\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right] \\ &+ \ ^{\rho_{2}}I^{\beta}_{1/v_1-}\left[ f\circ g_{2}\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}}, \frac{1}{v_2^{\rho_{2}}}\right) \ ^{\rho_{1}}I^{\alpha}_{1/u_2+}h\circ g_1\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right]\\ &+\ ^{\rho_{2}}I^{\beta}_{1/v_2+}\left[ f\circ g_{2}\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}}, \frac{1}{v_1^{\rho_{2}}}\right) \ ^{\rho_{1}}I^{\alpha}_{1/u_1-}h\circ g_1\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right] \end{align*} \begin{align*} &+\ ^{\rho_{2}}I^{\beta}_{1/v_2+}\left[ f\circ g_{2}\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}}, \frac{1}{v_1^{\rho_{2}}}\right) \ ^{\rho_{1}}I^{\alpha}_{1/u_2+}h\circ g_1\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right] \\ \leq &2\left[\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_1-,1/v_1-}fh\circ g\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) +\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_1-,1/v_2+}fh\circ g\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right)\right. \\ &\left.+\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_2+,1/v_1-}fh\circ g\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) +\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_2+,1/v_2+}fh\circ g\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right] \\ \leq &\ ^{\rho_{1}}I^{\alpha}_{1/u_1-}\left[ f\circ g_{1}\left( \frac{1}{u_2^{\rho_{1}}},v_1^{\rho_{2}}\right)\ ^{\rho_{2}}I^{\beta}_{1/v_2+}h\circ g_2\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right]+\ ^{\rho_{1}}I^{\alpha}_{1/u_1-}\left[ f\circ g_{1}\left( \frac{1}{u_2^{\rho_{1}}},v_2^{\rho_{2}}\right)\right. \\ &\left.\times ^{\rho_{2}}I^{\beta}_{1/v_1-}h\circ g_2\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right]+\ ^{\rho_{1}}I^{\alpha}_{1/u_2+}\left[ f\circ g_{1}\left( \frac{1}{u_1^{\rho_{1}}},v_1^{\rho_{2}}\right)\ ^{\rho_{2}}I^{\beta}_{1/v_2+}h\circ g_2\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right] \\ &+\ ^{\rho_{1}}I^{\alpha}_{1/u_2+}\left[ f\circ g_{1}\left( \frac{1}{u_1^{\rho_{1}}},v_2^{\rho_{2}}\right)\ ^{\rho_{2}}I^{\beta}_{1/v_1-}h\circ g_2\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right]+\ ^{\rho_{2}}I^{\beta}_{1/v_1-}\left[ f\circ g_{2}\left(u_1^{\rho_{1}}, \frac{1}{v_2^{\rho_{2}}}\right)\right. \\ &\left.\times ^{\rho_{1}}I^{\alpha}_{1/u_2+}h\circ g_1\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right]+ \ ^{\rho_{2}}I^{\beta}_{1/v_1-}\left[ f\circ g_{2}\left(u_2^{\rho_{1}}, \frac{1}{v_2^{\rho_{2}}}\right) \ ^{\rho_{1}}I^{\alpha}_{1/u_1-}h\circ g_1\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right] \\ &+\ ^{\rho_{2}}I^{\beta}_{1/v_2+}\left[ f\circ g_{2}\left(u_1^{\rho_{1}}, \frac{1}{v_1^{\rho_{2}}}\right) \ ^{\rho_{1}}I^{\alpha}_{1/u_2+}h\circ g_1\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right]+\ ^{\rho_{2}}I^{\beta}_{1/v_2+}\left[ f\circ g_{2}\left(u_2^{\rho_{1}}, \frac{1}{v_1^{\rho_{2}}}\right)\right. \\ &\left.\times ^{\rho_{1}}I^{\alpha}_{1/u_1-}h\circ g_1\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right]. \end{align*} This completes the second and third inequality of (13). Now, using the first inequality of (6), we find Adding (19) and (20) and using the fact that \(h\) is symmetric, we get \begin{align*} \begin{split} f\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right)&\left[\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_1-,1/v_1-}h\circ g\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) +\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_1-,1/v_2+}h\circ g\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right)\right. \\ &\left.+\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_2+,1/v_1-}h\circ g\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) +\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_2+,1/v_2+}h\circ g\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right] \\ \leq& \ ^{\rho_{1}}I^{\alpha}_{1/u_1-}\left[ f\circ g_{1}\left( \frac{1}{u_2^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right)\ ^{\rho_{2}}I^{\beta}_{1/v_1-}h\circ g_2\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right] \\ &+\ ^{\rho_{1}}I^{\alpha}_{1/u_1-}\left[ f\circ g_{1}\left( \frac{1}{u_2^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right) \ ^{\rho_{2}}I^{\beta}_{1/v_2+}h\circ g_2\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right] \\ &+\ ^{\rho_{1}}I^{\alpha}_{1/u_2+}\left[ f\circ g_{1}\left( \frac{1}{u_1^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right)\ ^{\rho_{2}}I^{\beta}_{1/v_1-}h\circ g_2\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right] \\ &+\ ^{\rho_{1}}I^{\alpha}_{1/u_2+}\left[ f\circ g_{1}\left( \frac{1}{u_1^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right)\ ^{\rho_{2}}I^{\beta}_{1/v_2+}h\circ g_2\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right] \\ &+\ ^{\rho_{2}}I^{\beta}_{1/v_1-}\left[ f\circ g_{2}\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}}, \frac{1}{v_2^{\rho_{2}}}\right) \ ^{\rho_{1}}I^{\alpha}_{1/u_1-}h\circ g_1\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right] \\ &+ \ ^{\rho_{2}}I^{\beta}_{1/v_1-}\left[ f\circ g_{2}\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}}, \frac{1}{v_2^{\rho_{2}}}\right) \ ^{\rho_{1}}I^{\alpha}_{1/u_2+}h\circ g_1\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right] \\ &+\ ^{\rho_{2}}I^{\beta}_{1/v_2+}\left[ f\circ g_{2}\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}}, \frac{1}{v_1^{\rho_{2}}}\right) \ ^{\rho_{1}}I^{\alpha}_{1/u_1-}h\circ g_1\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right] \\ &+\ ^{\rho_{2}}I^{\beta}_{1/v_2+}\left[ f\circ g_{2}\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}}, \frac{1}{v_1^{\rho_{2}}}\right) \ ^{\rho_{1}}I^{\alpha}_{1/u_2+}h\circ g_1\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right]. \end{split} \end{align*} This completes the first inequality of (13). Now, to achieve the last inequality of (13), applying the second inequality of (6) as: and By adding the inequalities (21)\(\sim\)(24), we get the last inequality of (13).

Remark 3.

• 1)   From Theorems 9 and 10, we can get new Hermite-Hadamard-Fejér type inequalities for co-ordinated harmonically convex functions via Riemann-Liouville fractional integral by taking \(\rho_1=\rho_2=1\).
• 2)   From Theorems 9 and 10, we can get new Hermite-Hadamard-Fejér type inequalities for co-ordinated harmonically convex functions via classical integral by taking \(\rho_1=\rho_2=1\) and \(\alpha=\beta=1\).

4. Conclusion

In this paper, firstly we established the Hermite-Hadamard-Fejér type inequalities for harmonically convex function in one dimension which is further used to establish the Hermite-Hadamard-Fejér type inequalities for harmonically convex function via Katugampola fractional integral. The results provided in our paper are the generalizations of some earlier results.

Acknowledgments

This research is supported by National University of Science and Technology(NUST), Islamabad, Pakistan.

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

Marriage is a socially accepted union or legal contract between two persons that established rights and obligations between them and their in-laws as well as the society. It can sometimes be referred to as the union of matrimony or wedlock, where interpersonal sexual relationship is biologically acknowledged [1]. The institution of marriage originally was meant to be a union of no bitterness. However, ugly experiences have emerged in which divorce and separation remain the most re-occurring decimals. The later phenomena are likely to happen when the ratio of negative to positive behaviour is equal or greater than unity. Divorce is the dissolution of marriage between two persons [2] which has a deformable mark on the family structure in our present era. This bitter trend continues to persist across Africa and the developed nations with no stigmatization. The immediate causes of divorce may not be limited to early marriage, less education, low income, premarital cohabitation and pregnancy, infidelity and lack of religion affiliation [3].

Divorce is an endemic issue that seriously affects the social and economic structure of contemporary society as much as any disease. Approximately one – half of all first marriages in the world end up in divorce or separation [4], with even higher rates of divorce for second marriages [5]. Separation leads to divorce as 75% of separation eventually result in divorce [6]. Divorce terms to increase childhood poverty and illiteracy rates [7]. It becomes important to eliminate this deadly social virus from our society.

Mathematical modelling has been used as a veritable tool for the control of epidemics in the past decades, and we can adopt it for the containment of the spread of divorce in marital institutions. In the past, concern has been focused on the influence of economy [8] and social contagion [9] on divorce models. Recently, [2] formulated a model for the spread of divorce in Ghana with three major compartments; Married, Divorced and Separated cases. Meanwhile, [10] extended their work to include the population of singles without any divorce prevention measure. Other mathematical studies on this subject matter can be found in [11,12,13]. Like other authors [2,9,10], we constructed a mathematical model for the spread of divorce epidemic. Apart from that, we incorporate a class of restored marital cases with anti-divorce therapy and reconciliation to the model of [10] using standard incidence function as the force of marital disunity, where the social family disorder is spread by divorced and separated persons over the married individuals. This is a major feature missing in the past works.

The organization of this paper is as follows: In Section 2, we formulate the model and establish the existence of equilibra and their local stability. In Section 3, we prove the global dynamics of each of the feasible steady states of the system (2) by constructing Lyapunov functions. In Section 4, we present the results and discussion of the numerical simulation. Finally, a brief conclusion is given in Section 5 to end this work.

2. Model construction

The present model derived its motivation from the work of [10], in which for the purpose of clarity, the basic model can be stated as; A more realistic model of divorce epidemic (1) with anti-divorce therapy is formulated based on the assumptions below and the schematic diagram in Figure 1.

1. People who are single or individuals who are ready for marriage but not yet married.
2. Some Separated cases may resort to divorced but not vice-versa.
3. Divorce or Separated cases restored can remain unbroken.
4. Only Separated cannot be remarried.
5. People who divorce can remarry or remain single.
6. The anti-divorce parameter changes from 0 to 1 \((0\leq\phi\leq1)\).

Table 1. Variables and parameters of the model (2).

<b>Variables</b>	<b>Description</b>
\(S_{1}(t)\)	Number of singles who are due for marriage at time \(t\)
\(M(t)\)	Number of Married cases at time \(t\)
\(D(t)\)	Number of Divorced cases at time \(t\)
\(S_{2}(t)\)	Number of Separated cases  at time \(t\)
\(R(t)\)	Number of Restored Marital cases at time \(t\)
Parameters	Description
\(Q\)	Recruitment rate for the Single individuals
\(\beta_{1}\)	Divorced rate of the married
\(\beta_{2}\)	Separated rate of the married
\(\alpha_{1}\)	Rate of getting married by the singles
\(\alpha_{2}\)	Rate of re-marriage after divorced
\(d_{3}\)	Separated rate of the married
\(d_{2}\)	Proportion of married individuals that divorced
\(d_{1}\)	Proportion of married cases that results into Separation
\(r_{1}\)	Rate of restoring divorced cases in marriage by reconciliation
\(r_{2}\)	Rate of restoring Separated cases in marriage through reconciliation
\(\gamma\)	Rate of remaining single after divorce
\(\phi\)	Anti-divorce parameter \((0\leq\phi\leq1)\)
\(\mu\)	Natural death rate of individuals
\(\lambda_{m}\)	Force of marital disunity

Note that, the first four Variables and the first seven Parameters in Table 1 have the same meaning as in model (1) [10]. However, \(\beta\) which is the rate of Separated getting married again in model (1) is disputed and so ignored in our study.

From the model flow-diagram of divorce epidemic and assumptions, the following differential equations are derived;

where Equation (3) is the force of marital disunity. The sum of the entire system (2) yield with \(N=S_{1} +M+D+S_{2}+R.\) The system (2) can be studied within the feasible domain \begin{equation*} D_{m}=\left\lbrace S_{1}>0, M\geqslant0, D\geqslant0, S_{2}\geqslant0, R\geqslant0\mid N=\frac{Q}{\mu}\right\rbrace. \end{equation*}

2.1. Model steady states

At the steady state, the system (2) takes the form where \(\omega_{1}=\alpha_{1} +\mu, \omega_{2}=\mu+\gamma+r_{1}+\alpha_{2}\) and \(\omega_{3}=\mu+r_{2}+d_{3}.\) The solution of the system (5) gives the endemic equilibra in terms of \( \lambda_{m}^{**}\). with Fundamentally, the divorce-free equilibrium (DFE) can be obtained from the relation (7) above. Thus, introducing the expressions (6) into (7) and simplifying accordingly we arrive at such that Clearly from (8) \(\lambda_{m}^{**}=0 \), which is significant for evaluating DFE. Consequently substituting \(\lambda_{m}^{**}=0 \) into the expressions (6) give \begin{equation*} E^{0}=(S_{1}^{0}, M^{0}, D^{0}, S_{2}^{0}, R^{0})=\left(\frac{Q}{\alpha_{1}+\mu},\frac{Q \alpha_{1}}{\alpha_{1}+\mu}, 0, 0, 0\right). \end{equation*}

2.2. Divorce control reproduction number

The computation of the divorce controlled reproduction number denoted here by \(R_{ed} \) is done following the standard next generation matrix explained in [14] with the usual notation being given as; \[R_{ed}=\rho \left(F V^{-1}\right)=(1-\phi)\frac{\alpha_{1}}{\omega_{1}} \left(\frac{\beta_{1} (d_{1} \omega_{3}+ d_{2} d_{3})+\beta_{2} d_{2} \omega_{2}}{\omega_{2} \omega_{3}}\right),\] with \[F=(1-\phi)\frac{\alpha_{1}}{\omega_{1}}\begin{pmatrix} d_{1}\beta_{1} & d_{1}\beta_{2} \\ d_{2}\beta_{1} & d_{2}\beta_{2}\\ \end{pmatrix}\quad \text{and}\quad V^{-1}=\begin{pmatrix} \frac{1}{\omega_{2}} & \frac{d_{3}}{\omega_{2} \omega_{3}}\\ 0 & \frac{1}{\omega_{3}}\\ \end{pmatrix}.\] In the absence of anti-divorce therapy and reconciliation, \( \phi=0, r_{1}=r_{2}=0 \), we derive divorce reproduction number as; \[R_{d}=\frac{\alpha_{1}}{\omega_{1}} \left[\frac{\beta_{1} [d_{1} (\mu+d_{3})+d_{2} d_{3}]+\beta_{2} d_{2} (\mu+\gamma+\alpha_{2})}{(\mu+d_{3})(\mu+\gamma+\alpha_{2})}\right].\] However, for single intervention in terms of anti-divorce therapy and reconciliation only we have the respective reproduction numbers as; \begin{align*} R_{ad}&=(1-\phi)\frac{\alpha_{1}}{\omega_{1}} \left[\frac{\beta_{1}[d_{1} (\mu+d_{3})+d_{2} d_{3}]+\beta_{2} d_{2} (\mu+\gamma+\alpha_{2})}{(\mu+d_{3})(\mu+\gamma+\alpha_{2})}\right],\\ R_{ec}&=\frac{\alpha_{1}}{\omega_{1}} \left(\frac{\beta_{1} (d_{1} \omega_{3}+ d_{2} d_{3})+\beta_{2} d_{2} \omega_{2}}{\omega_{2} \omega_{3}}\right).\end{align*} Interestingly, it can be deduced from Theorem 2 in [14] that:

Claim 1. The DFE is locally asymptotically stable in the case \( R_{ed}< 1\) and unstable when \( R_{ed}>1\).

2.3. Existence and local stability of divorce-endemic equilibrium

The existence of divorce endemic equilibrium follows immediately from (8) where \(\lambda_{m}^{**}\neq0\). Therefore, solving the remaining part of the equation leads to Thus, the divorce-endemic equilibrium denoted by DE after substituting (10) into (6) is given by \(DE=(S_{1}^{**},M^{**},D^{**},S_{2}^{**},R^{**} )\).

Recall from (10) that \(\lambda_{m}^{**}>0\) iff \(R_{ed}>1\) and \(\alpha_{1}>\alpha_{2}\), which are the necessary conditions for divorce to persist in marriage institution.

The method of linearization process of the system (2) at DE gives the following Jacobian Matrix

From the Jacobian matrix in (11), \( -\mu \) is an eigenvalue and the remaining ones are gotten from the matrix below with the corresponding characteristics equation defined as follows where \begin{align*} a_{4} =&1,\\ a_{3} =& \mu+\left(\sum_{i=1}^{3}\omega_{i}\right)+\lambda_{m}^{**}-(1-\phi) (d_{1} \beta_{1}+d_{2} \beta_{2})\left(\frac{M^{**}}{N^{**}}\right),\\ a_{2}=&\lambda_{m}^{**}\left[\left(\sum_{i=1}^{3}\omega_{i}\right)-d_{1} \alpha_{2}\right]+\mu\left(\sum_{i=1}^{3}\omega_{i}\right)+\omega_{1}\left(\sum_{i=2}^{3}\omega_{i}\right)-d_{2} (1-\phi)^{2} \left(\frac{M^{**}}{N^{**}}\right)^{2} \beta_{1}\beta_{2}(1-d_{1})\\ &+\omega_{2} \omega_{3}-(1-\phi)\left(\frac{M^{**}}{N^{**}}\right)\left[\beta_{2} d_{2} \omega_{2} +\beta_{!} (d_{1}\omega_{3}+d_{2} d_{3})+(\mu+\omega_{1})(d_{1}\beta_{1}+d_{2} \beta_{2})\right],\\ a_{1} =&\omega_{1}\omega_{2}\omega_{3} +\mu\left[\omega_{1}\left(\sum_{i=2}^{3}\omega_{i}\right)+\omega_{2}\omega_{3}\right]+\lambda_{m}^{**}\left[\omega_{1}\left(\sum_{i=2}^{3}\omega_{i}\right)-d_{2}\alpha_{2}(1-\phi) \left(\frac{M^{**}}{N^{**}}\right)\right.\beta_{2}(1-d_{1})\\ &+\omega_{2}\omega_{3}-d_{1}(\gamma\alpha_{1}+\alpha_{2}\omega_{1})-\alpha_{2}(d_{1}\omega_{3}+d_{2} d_{3})\Bigg]+(1-\phi)^{2} \left(\frac{M^{**}}{N^{**}}\right)^{2} \beta_{1}\beta_{2}\left[2\mu+\alpha_{2}(1-d_{1})\right]\\ &-(1-\phi)\left(\frac{M^{**}}{N^{**}}\right)\left[\beta_{1}(\alpha_{1}d_{2}d_{3}+d_{1}(\omega_{1}\omega_{3}+\mu(\omega_{1}+\omega_{3})))+\beta_{2}d_{2}(\mu\left(\sum_{i=1}^{2}\omega_{i}\right)+\omega_{2}\omega_{3})\right],\\ a_{0} =&\lambda_{m}^{**}\left[\omega_{1}\omega_{2}\omega_{3}-(\gamma\alpha_{1}+\alpha_{2}\omega_{1})d_{2}(1-\phi)\left(\frac{M^{**}}{N^{**}}\right)\beta_{2}(1+d_{1})+(d_{1}\omega_{3}+d_{2}d_{3})\right]\\ &+\mu\omega_{1}\left[\omega_{2}\omega_{3}-(1-\phi)\left(\frac{M^{**}}{N^{**}}\right)(\beta_{2}d_{2}\omega_{2}+\beta_{1}(d_{1}\omega_{3}+d_{2}d_{3}))-d_{2}(1-\phi)^{2}\left(\frac{M^{**}}{N^{**}}\right)\beta_{1}\beta_{2}(1-d_{1})\right]. \end{align*} To guarantee that all roots of (13) are real and negative, the Routh-Hurwitz stability criterion [15] requires; Thus, it is evident to see that all the coefficient of (13) are strictly positive if \[\lambda_{m}^{**}>0\;\; (\Leftrightarrow R_{ed}>1)\ \ \text{and}\ \ \frac{M^{**}}{N^{**}} < 1.\] Therefore, the inequalities (14) hold, and we cam claim that:

Claim 2. The divorce endemic equilibrium DE is locally asymptotically if \( R_{ed}>1\) and \(\frac{M^{**}}{N^{**}}< 1\).

3. Global dynamics

3.1. Global stability of DFE

This section discusses the global behavior of the model (2) at the divorce free equilibrium state following the well-known stability method of Lyapunov functions [16].

Claim 3. The system (2) admits a globally asymptotically DFE iff \( R_{ed}< 1\).

Proof. By matrix theoretic approach as done in [16] we derive the following Lyapunov function \[L(t)=\frac{\beta_{1}}{\omega_{2}} D +\left(\frac{\beta_{i} d_{3}}{\omega_{2} \omega_{3}}+\frac{\beta_{2}}{\omega_{3}}\right) S_{2},\] whose time derivative is given by \begin{align*} L^{'}(t) =&\frac{\beta_{1}}{\omega_{2}}\left[d_{1} \left((1-\phi) \frac{\beta_{1} D + \beta_{2} S_{2}}{N}\right)M + d_{3} S_{2}-\omega_{2} D\right]\\ &+\left(\frac{\beta_{i} d_{3}}{\omega_{2} \omega_{3}}+\frac{\beta_{2}}{\omega_{3}}\right)\left[d_{2}\left((1-\phi) \frac{\beta_{1} D + \beta_{2} S_{2}}{N}\right)M -\omega_{3} S_{2}\right]. \end{align*} Collecting like terms and simplifying leads to \begin{align*} L^{'}(t) & =(\beta_{1} D + \beta_{2} S_{2})\left[(1-\phi) \left(\frac{\beta_{1} (d_{1} \omega_{3}+ d_{2} d_{3})+\beta_{2} d_{2} \omega_{2}}{\omega_{2} \omega_{3}}\right)\frac{M}{N}-1\right]. \end{align*} Recall that at divorce-free equilibrium we have \(\frac{M}{N}=\frac{M^{0}}{N^{0}}=\frac{\alpha_{1}}{\omega_{1}}\) and so \begin{align*} L^{'}(t) & =(\beta_{1} D + \beta_{2} S_{2})\left[(1-\phi)\frac{\alpha_{1}}{\omega_{1}} \left(\frac{\beta_{1} (d_{1} \omega_{3}+ d_{2} d_{3})+\beta_{2} d_{2} \omega_{2}}{\omega_{2} \omega_{3}}\right)-1\right]. \end{align*} Thus, \( L^{'}(t) =(\beta_{1} D + \beta_{2} S_{2})(R_{ed}-1), \), \(L^{'}(t)\leqslant0\) if \(R_{ed}\leqslant1\) and \( D=S_{2}=0\). Therefore, by Lyapunov's stability theory [15], the equilibrium point \(E^{0} \) is globally asymptotically stable.

3.2. Global stability of DE

Claim 4. The divorce-endemic equilibrium of the system (2) at \(\alpha_{2}=0\) is globally stable if \(R_{ed}>1\).

Proof. Taking the Lyapunov function \( W_{1}(t)\) as in [17]

with \( m_{1}, m_{2}, . . . , m_{5} \) are positive constants to be properly chosen.

The differential coefficient of \(W_{1} (t)\) in (15) with the substitution of model (2) at \( \alpha_{2}=0\) gives

At divorce endemic equilibrium, we have \begin{align*} &\left(\alpha_{1} +\mu\right)S_{1}^{**} =Q+\gamma D^{**},\\ &\alpha_{1}S_{1}^{**}= \mu M^{**}+\left(\frac{\beta_{1} D^{**} + \beta_{2} S_{2}^{**}}{N}\right)M^{**},\\ &\left(\mu+\gamma+r_{1}\right)D^{**}=d_{3} S_{2}^{**}+d_{1} \left(1-\phi\right) \left(\frac{\beta_{1} D^{**} + \beta_{2} S_{2}^{**}}{N}\right)M^{**},\\ &\left(\mu+r_{2}+d_{3}\right)S_{2}^{**}=d_{2}\left(1-\phi\right) \left(\frac{\beta_{1} D^{**} + \beta_{2} S_{2}^{**}}{N}\right)M^{**},\\ &\mu R^{**} =r_{1} D^{**}+ r_{2} S_{2}^{**}. \end{align*} Therefore, where \begin{align*} G \left(S_{1}, M, D, S_{2}, R\right) =&-m_{1}\left(1-\frac{S_{1}^{**}}{S_{1}}\right)\left(1-\frac{D}{D^{**}}\right)\gamma D^{**}+m_{2}\left(1-\frac{M^{**}}{M}\right)\left(\frac{S_{1}}{S_{1}^{**}}-\frac{M}{M^{**}}\right)\mu M^{**}\\ & +m_{2}\left(1-\phi\right)\beta_{1}\left(1-\frac{M^{**}}{M}\right) \left(\frac{S_{1}}{S_{1}^{**}}-\frac{D M}{D^{**} M^{**}}\right)\frac{D^{**} M^{**}}{N}\\ & +m_{2}\left(1-\phi\right)\beta_{2}\left(1-\frac{M^{**}}{M}\right) \left(\frac{S_{1}}{S_{1}^{**}}-\frac{S_{2} M}{S_{1}^{**} M^{**}}\right)\frac{S_{2}^{**} M^{**}}{N}\\ & +m_{3} d_{1}\left(1-\phi\right)\beta_{1}\left(1-\frac{D^{**}}{D}\right) \left(\frac{D M}{D^{**} M^{**}}-\frac{D}{D^{**}}\right)\frac{D^{**} M^{**}}{N}\\ & +m_{3} d_{1}\left(1-\phi\right)\beta_{2}\left(1-\frac{D^{**}}{D}\right) \left(\frac{S_{2} M}{S_{2}^{**} M^{**}}-\frac{D}{D^{**}}\right)\frac{S_{2}^{**} M^{**}}{N}\\ & +m_{3} d_{1}\left(1-\phi\right)\beta_{2}\left(1-\frac{D^{**}}{D}\right) \left(\frac{S_{2}}{S_{2}^{**}}-\frac{D}{D^{**}}\right)d_{3} S_{2}^{**}\\ & +m_{4} d_{2}\left(1-\phi\right)\beta_{1}\left(1-\frac{S_{2}^{**}}{S_{2}}\right) \left(\frac{D M}{D^{**} M^{**}}-\frac{S_{2}}{S_{2}^{**}}\right)\frac{D^{**} M^{**}}{N}\\ & +m_{4} d_{2}\left(1-\phi\right)\beta_{2}\left(1-\frac{S_{2}^{**}}{S_{2}}\right) \left(\frac{S_{2} M}{S_{2}^{**} M^{**}}-\frac{S_{2}}{S_{2}^{**}}\right)\frac{S_{2}^{**} M^{**}}{N}\\ & +m_{5}\left(1-\frac{R^{**}}{R}\right) \left(\frac{D}{D^{**}}-\frac{R}{R^{**}}\right) r_{1} D^{**} +m_{5}\left(1-\frac{R^{**}}{R}\right) \left(\frac{S_{2}}{S_{2}^{**}}-\frac{R}{R^{**}}\right) r_{2} S_{2}^{**}, \end{align*} and \(G\) is a non-positive expression as supported by the Lemmas 2.2, 2.3 [18] and [17].

Since \((S_{1},M, D, S_{2},R)\neq(S_{1}^{**},M^{**},D^{**},S_{2}^{**},R^{**})\), \(W_{1}^{'} (t)< 0\), \(W_{1}^{'} (t)=0\) if \(2-\frac{S_{2}}{S_{2}^{**}}=\frac{S_{2}^{**}}{S_{2}}\leqslant0\) and \(S_{1}=S_{1}^{**}, M=M^{**}, D=D^{**}, S_{2}=S_{2}^{**}, R=R^{**} \). Hence, DE is globally stable by Lyapunov's stability theory [15].

4. Results and discussions

To aid the understanding of the analyzed results, we performed the numerical simulation of the proposed model using MATLAB ode45 solver and the parameter values given in Table 2.

Table 2. Values of variables and parameters of the model (2) used for numerical simulations.

<b>Initial Variables</b>	<b>Value</b>	<b>Source</b>
\(S_{1}(0)\)	50	Assumed
\(M(0)\)	40	Assumed
\(D(0)\)	15	Assumed
\(S_{2}(0)\)	15	Assumed
\(R(0)\)	10	Assumed
<b>Parameter</b>	<b>Value</b>	<b>Source</b>
\(Q\)	2	Assumed
\(\beta_{1}\)	0.022	\cite{j}
\(\beta_{2}\)	0.031	\cite{j}
\(\alpha_{1}\)	0.101	\cite{j}
\(\alpha_{2}\)	0.061	\cite{j}
\(d_{3}\)	0.021	\cite{j}
\(d_{2}\)	0.7	Assumed
\(d_{1}\)	0.3	Assumed
\(r_{1}\)	0.1	Assumed
\(r_{2}\)	0.3	Assumed
\(\gamma\)	0.01	Assumed
\(\phi\)	0.2	Assumed
\(\mu\)	0.02	Estimated

In Figure 2(a), we observe that married couples develop the potentials to stay or remain in marriage longer when they attained marriage seminars/courses and exhibit the spirit of reconciliation in handling their differences. On the other hand, the union can easily break apart when the above mentioned controls are ignored.

The impact of anti-divorce control is depicted on the divorced cases in Figure 2(b). Here, the cases of marriage breaking apart (divorce) persist uniformly in the family circle where reconciliation and anti-divorce therapy are absent. Furthermore, the results supports that with the controls, divorced cases can be eliminated in marriage institutions and marriage couples who stay in marriage up to 30 years may have a stable family structure and no longer divorced their partners. This agrees with the result of high commission for planning of Morocco on marriage and divorce as contained in [11] which says that, couples with 20 years of experiences has a decreased divorce rate at 3%.

In the case of separated in Figure 2(c), the number rises higher for the scenario where the couples have not had proper premarital education and do not reconcile their misunderstanding on continuous bases. More so, the study reveals that marriage couples that live for over 15 years without separation are less venerable to divorce tendencies in marriage.

Figure 3(a) is a simple demonstration that increasing the rate of reconciliation impacts positively on the broken families by reuniting and hence increasing the number of restored cases. By implication, separated or divorced families will remain broken without adequate reconciliation process. Meanwhile, the effect of \( \alpha_{2}\) on the divorced is illustrated in Figure 3(b). The positive variation in \( \alpha_{2}\) reduces the cases of divorce in marriage. Therefore, it is recommendable to say that the re-marriage among the divorced helps to re-unite broken families.

The separated/divorced cases grow exponentially with respect to the married cases in Figure 4(a). This means that as long as married couples lived together, marital disorder in terms of separation or divorce may always be experienced among them. In other words, divorce cannot occur without marriage and the more marriage cases, the large scale of divorced population expected.

In Figure 4(b) we observed the behavioral trends of the model reproduction numbers that follows the inequality \(R_{ed}< R_{ec}< R_{ad}< R_{d}\). This symbolizes that the combination of reconciliation and anti-divorced protocols is beneficial in stabilizing marriages than when a single intervention is used. Furthermore, it is important to emphasize that, reconciliation in family settlement is better than marriage seminar/courses as proposed in [10]. This is also in line with the result of [19] that says; therapy is only a support but not a major component in solving marriage problems since \(R_{ec}< R_{ad} \). However, \(R_{d}\) standing above the rest of the reproduction ratios means that marriages without controls are fragile to crack or break apart in future.

5. Conclusion

A non-linear ordinary differential system of equations for examining the spread of divorce epidemic with anti-divorce therapy has been proposed and analyzed. Important results from the qualitative analysis of the modified model reveal that, the model has globally stable equilibrium states, namely the divorce-free and endemic equilibrium. Numerical results of this study suggest that the presence of correctional practices such as marriage seminar/courses and reconciliation efforts can ensure prolonged stable marriages and prevents cracks and repairs the breaking points (separation or divorce) in family structure. In addition, the marriage couples that escape separations in early 15 years of marriage and stay together in the next 15 years cannot divorce any longer.

Author Contribution: 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.