Spherical partial differential equation with non‐constant coefficients for modeling of nonlinear unsteady heat conduction in functionally graded materials

Abstract

The objective of this research paper is to propose an exact solution for resolving the transient conduction in a 2D sphere. The coefficients of governing equation are varied according to the material properties. The thermo‐physical properties are regarded as functions that follow a power‐law relationship concerning the radial direction. Both radial and angular thermal conductivity coefficients change with radius. Thermal boundary conditions are considered in a general state, which can cover different thermal conditions, including Dirichlet, Neumann, and Convection surface conditions. Laplace transform and Meromorphic function methods are used in the solution approach to the current unsteady problem. Two unsteady case studies with complex boundary conditions have been considered to show the credibility of the current solution. The results of both case studies have been successfully validated. The results confirm the high capability of the present solution in solving unsteady thermal problems of functionally graded materials in spherical coordinates.