## 2 Governing Equations
\right\} \tag{2c}\]  
where \((i,j=r,\theta)\) is the stress tensor components in the FG core \((e=c)\) and piezoelectric layers \((e=p)\); \(u\) and \(v\) are the displacement components at an arbitrary material point of the panel along the \(r\) and \(\theta\)-directions, respectively; \(\alpha\) denotes the thermal expansion coefficient; \((\Delta T)^{c}=T^{c}-T_{0}\) is the temperature variation; \(T_{0}\) is the shell stress free temperature; \(C_{ij}^{c}(i,j=1,\ 2,\ 5)\) is the material elastic coefficients of the core and piezoelectric layers; \(D_{i}\) and \(E_{i}\)\((i=r,\theta)\) are, the electric displacement and electric field in the \(i\)th direction, respectively; and \(e_{i}\)\((i=2,3,4)\), \(K_{ii}\)\((i=2,3)\) and \(P_{3}\) refer to the piezoelectric, dielectric, and pyroelectric constants, respectively. The following relations are introduced between the electric field \(E_{i}\) and the electric potential \(\psi\) in piezoelectric layers:  
\[E_{r} =-\frac{\partial\psi}{\partial r}, \tag{3a}\] \[E_{\theta} =-\frac{\partial\psi}{r\partial\theta} \tag{3b}\]  
The FG cylindrical panel is subjected to an initial temperature \(T_{0}\) before the variation of the thermal boundary conditions at the two external surfaces occurs. Under the plane strain assumptions, the governing equations are independent of the axial coordinate variable \(z\). Thus, for the FG core layer and piezoelectric layers, the linearized coupled energy balance equation according to the L-S thermoelasticity theory together with the equation of motion can be defined as [1]. Energy balance equation:  
\[\frac{1}{r}\frac{\partial}{\partial r}\!\left(rk^{c}\frac{\partial T^{c}}{ \partial r}\right)+\frac{1}{r^{2}}\frac{\partial}{\partial\theta}\!\left(\! \kappa^{c}\frac{\partial T^{c}}{\partial\theta}\!\right)=\rho^{c}c\!\left( \frac{\partial T^{c}}{\partial t}+\tau_{0}^{c}\frac{\partial^{2}T^{c}}{ \partial t^{2}}\right)+\overrightarrow{\alpha}^{c}T_{0}\!\left(\!\tau_{0}^  
be more convenient and suitable for use. Due to the computational efficiency and accuracy of the DQM demonstrated in the literature for different heat transfer and structural problems [11; 20; 36; 43; 44; 45; 46; 47; 48; 49; 50; 51], we combined this method with the NURBS-based multi-step method in the current work to discretize the governing equations in the spatial and time domains, respectively. Based on the DQM, the FG core and piezoelectric layers are discretized into a set of \(N_{\xi}^{e}(e=c,p)\) and \(N_{\theta}\) grid points along the \(r\)- and \(\theta\)-directions, respectively. In this regard, the Gauss-Lobatto-Chebyshev grid generation rule is employed [11; 20; 36; 43; 44; 45; 46; 47; 48; 49; 50; 51]. Thus, the spatial derivatives involved in the differential equations are discretized together with the corresponding thermo-mechanical boundaries and compatibility conditions at the interface between the core and piezoelectric layers. For the sake of brevity, we describe only the discretized form for the of energy balance from Equation (4), as also repeated for all the other equations of the problem. \[ \begin{split}\rho_{i}^{e}c_{i}^{e}\bigg{[}\frac{dT_{ij}^{e}}{dt} +\Big{(}\tau_{0}^{e}\Big{)}_{i}\frac{d^{2}T_{ij}^{e}}{dt^{2}}\bigg{]}+\overline {\alpha}_{i}^{e}T_{0}\Big{[}\Big{(}\tau_{0}^{e}\Big{)}_{i}\frac{\partial^{2}} {\partial t^{2}}+\frac{\partial}{\partial t}\bigg{]}\bigg{(}\sum_{m=1}^{N_{\xi }^{e}}A_{im}^{\xi}u_{mj}^{e}+\frac{u_{ij}^{e}}{r_{ij}}+\frac{1}{r_{ij}}\sum_{n =1}^{N_{\theta}}A_{jn}^{\theta}v_{in}^{e}\bigg{)}\\ =k_{i}^{e}\sum_{m=1}^{N_{\xi}^{e}}B_{im}^{\xi}T_{mj}^{e}+\frac{2k_ {i}^{e}}{r_{ij}}\sum_{m=1}^{N_{\xi}^{e}}A_{im}^{\xi}T_{mj}^{e}+\frac{k_{i}^{e} }{r_{ij}}\sum_{n=1}^{N_{\theta}}A_{jn}^{\theta}T_{in}^{e}\end{split}