Computational Wavelet Method for Multidimensional Integro-Partial Differential Equation of Distributed Order

TL;DR

This paper introduces a novel computational algorithm using Legendre wavelets and tau methods to solve complex multidimensional distributed order integro-partial differential equations, with rigorous analysis and numerical validation.

Contribution

The paper presents a new computational approach combining Legendre wavelets and tau method for solving multidimensional distributed order integro-partial differential equations.

Findings

The algorithm effectively approximates solutions with proven convergence.

Error bounds and stability are rigorously established.

Numerical examples demonstrate the method's robustness and accuracy.

Abstract

This article provides an effective computational algorithm based on Legendre wavelet (LW) and standard tau approach to approximate the solution of multi-dimensional distributed order time-space fractional weakly singular integro-partial differential equation (DOT-SFWSIPDE). To the best of our understanding, the proposed computational algorithm is new and has not been previously applied for solving DOT-SFWSIPDE. The matrix representation of distributed order fractional derivatives, integer order derivatives and weakly singular kernel associated with the integral based on LWare established to find the numerical solutions of the proposed DOT-SFWSIPDE. Moreover, the association of standard tau rule and Legendre-Gauss quadrature (LGQ) techniques along with constructed matrix representation of differential and integral operators diminish DOT-SFWSIPDE into system of linear algebraic equations. Error bounds, convergence analysis, numerical algorithms and also error estimation of the DOT-SFWSIPDE are regorously investigated. For the reliability of the proposed computational algorithm, numerous test examples has been incorporated in the manuscript to ensure the robustness and theoretical results of proposed technique.