Theoretical and Numerical Study for Volterra−Fredholm Fractional Integro-Differential Equations Based on Chebyshev Polynomials of the Third Kind

Abstract

In this paper, we develop an efficient numerical method to approximate the solution of fractional integro-differential equations (FI-DEs) of mixed Volterra−Fredholm type using spectral collocation method with shifted Chebyshev polynomials of the third kind (S-Cheb-3). The fractional derivative is described in the Caputo sense. A Chebyshev−Gauss quadrature is involved to evaluate integrals for more precision. Two types of equations are studied to obtain algebraic systems solvable using the Gauss elimination method for linear equations and the Newton algorithm for nonlinear ones. In addition, an error analysis is carried out. Six numerical examples are evaluated using different error values (maximum absolute error, root mean square error, and relative error) to compare the approximate and the exact solutions of each example. The experimental rate of convergence is calculated as well. The results validate the numerical approach’s efficiency, applicability, and performance.

Numerical Solution of the Burgers’ Equation Using Chelyshkov Polynomials

“…These include the variational iteration method [6], automatic differentiation method [7], finite difference schemes [8], modified cubic B-splines collocation and cubic hyperbolic Bspline method [9,10], least-squares quadratic B-spline finite element method [11], finite element collocation method [12], mixed finite volume element methods [13], sinc-Galerkin method [14], and Laplace transform decomposition method [15]. Orthogonal polynomials play a vital role in solving a wide range of mathematical problems, including differential and integro-differential equations [16][17][18][19][20][21][22][23][24][25]. Chelyshkov polynomials, a class of orthogonal polynomials, possess properties similar to classical orthogonal polynomials and are associated with hypergeometric functions, orthogonal exponential polynomials, and Jacobi polynomials [26,27].…”