Low cost numerical solution for three-dimensional linear and nonlinear integral equations via three-dimensional Jacobi polynomials

Abstract

In recent years, numerical methods have been introduced to solve two-dimensional Volterra and Fredholm integral equations. In this study, a numerical scheme is constructed to solve classes of linear and nonlinear three-dimensional integral equations (Volterra, Fredholm, and mixed Volterra–Fredholm). This operational approach is proposed to easily and directly solve these equations at low computational costs. The scheme is based on the Jacobi polynomials on the interval [0, 1] where three-variable Jacobi polynomials are introduced and their operational matrices of integration and product are derived. Compared to other existing methods for multidimensional problems, the Jacobi operational method eliminates the time-consuming computations and solely employs the one-dimensional operational matrix to construct corresponding multidimensional operational matrices. The absolute error of the proposed method is almost constant on the studied interval even at higher dimensions, confirming the stability of the proposed operational Jacobi method. Required theorems on the convergence of the method are proved in Jacobi-weighted Sobolev space. It is established that the error function vanishes as N increases. The method is evaluated using several illustrative examples which indicate the proposed method with lesser computational size compared to the Block–Pulse functions, differential transform, and degenerate kernel methods.