Spectral Collocation Method for Handling Integral and Integrodifferential Equations of n-th Order via Certain Combinations of Shifted Legendre Polynomials

Abstract

In this study, an accurate and efficient numerical method based on spectral collocation is presented to solve integral equations and integrodifferential equations of n -th order. The method is developed using compact combinations of shifted Legendre polynomials as a spectral basis and shifted Legendre–Gauss–Lobatto nodes as collocation points to construct the appropriate algorithm that makes simple systems easy to solve. The technique treats both types of equations: linear and nonlinear equations. The study aims to provide the relevant spectral basis by the use of compact combinations, which allows us to take advantage of shifted Legendre polynomials and to reduce the dimension of the space of approximation. The reliability of the proposed algorithms is proven via different examples of several cases and the results are discussed to confirm the effectiveness of the spectral approach.

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

“…Collocation methods are becoming increasingly popular for solving a wide variety of diferential equations. For instance, spectral collocation methods are used for integral and integro-diferential equations (21), for multiorder FDEs [22], for multiorder FEs with multiple delays [23], for twodimensional FIDs with weakly singular kernel [24], and also for systems of FDEs [25].…”

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].…”