An Operational Matrix Technique Based on Chebyshev Polynomials for Solving Mixed Volterra-Fredholm Delay Integro-Differential Equations of Variable-Order

Abstract

In this work, an algorithm for finding numerical solutions of linear fractional delay-integro-differential equations (LFDIDEs) of variable-order (VO) is introduced. The operational matrices are used as discretization technique based on shifted Chebyshev polynomials (SCPs) of the first kind with the spectral collocation method. The proposed VO-LFDIDEs have multiterms of integer, fractional-order derivatives for delayed or nondelayed and mixed Volterra-Fredholm integral terms. The introduced model is a more general form of linear fractional VO pantograph, neutral, and mixed Fredholm–Volterra integro-differential equations with delay arguments. Caputo’s VO fractional derivative operator is used to generate the matrices of the derivative. Operational matrices are presented for all terms. The reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments. Also, some examples are included to improve the validity and applicability of the techniques. Finally, comparisons between the proposed method and other methods were used to solve this kind of equation.

Numerical Solution of Non-linear Integro-differential Equations using Operational Matrix based on the Hosoya Polynomial of a Path Graph

“…To determine the analytical solution to the linear and nonlinear FIDE and VIDE, A. S. Khan (3) has used a variation of parameter techniques. Numerous academics have recently worked on integro-differential equations and achieved superior results (4)(5)(6)(7)(8)(9)(10)(11) . The study of integral and IDEs, which contain two different types of integral operators, was the main emphasis of Hamoud and Ghadle (12) .…”