Bidimensional Gegenbauer Polynomials for Variable‐Order Time‐Fractional Integro‐Partial Differential Equation With a Weakly Singular Kernel

ABSTRACT In this paper, a pseudo‐operational collocation method based on Gegenbauer polynomials is presented to solve a category of variable‐order time‐fractional integro‐partial differential equations with singular kernels. The applications of these functional equations can be revealed in the theory of elasticity, hydrodynamics, heat conduction, and nuclear reactor theory. The pseudo‐operational matrices are constructed utilizing bivariate Gegenbauer polynomials to approximate the solution of the mentioned equation. Then, using the collocation method and resultant matrices, the main equation is converted into a system of algebraic equations that can be solved by Newton's iteration method. Besides presenting a fast and accurate method, an error bound is determined in a Gegenbauer‐weighted space for the residual function obtained from the proposed approach. Finally, several test examples are performed to confirm the reliability and efficiency of the proposed method.