Solution of variable‐order partial integro‐differential equation using Legendre wavelet approximation and operational matrices

Abstract

AbstractThis article is concerned with developing a method to find a numerical solution of a one‐dimensional variable‐order non‐linear partial integro‐differential equation (PIDE) viz., reaction–advection–diffusion equation with initial and boundary conditions. The proposed numerical scheme shifted Legendre collocation method is based on operational matrices. The operational matrices using one‐dimensional wavelets are derived to solve the said variable‐order model. First operational matrices have been introduced for integration and variable‐order derivatives using one‐dimensional Legendre wavelets (LWs). After that, using the shifted Legendre collocation points, the model is reduced to a system of algebraic equations, which are solved using Newton–Cotes method. The error is then calculated by comparing the numerical solution obtained from the system of algebraic equations and the known exact solution of an existing problem to validate the efficiency of the proposed numerical scheme. The main contribution of the article is the graphical exhibitions of the solution profile for different variable order derivatives in presence of different values of the parameters.