# Legendre wavelet collocation method combined with the Gauss--Jacobi quadrature for solving fractional delay-type integro-differential equations

## TL;DR

This paper introduces a novel collocation method combining Legendre wavelets and Gauss--Jacobi quadrature to efficiently solve fractional delay-type integro-differential equations with high accuracy.

## Contribution

The work develops a new numerical approach that integrates Legendre wavelets with Gauss--Jacobi quadrature for solving fractional integro-differential equations, including error analysis.

## Key findings

- Demonstrates high accuracy in example problems
- Provides error bounds for Legendre wavelet approximation
- Shows efficiency of the method through numerical results

## Abstract

In this work, we present a collocation method based on the Legendre wavelet combined with the Gauss--Jacobi quadrature formula for solving a class of fractional delay-type integro-differential equations. The problem is considered with either initial or boundary conditions and the fractional derivative is described in the Caputo sense. First, an approximation of the unknown solution is considered in terms of the Legendre wavelet basis functions. Then, we substitute this approximation and its derivatives into the considered equation. The Caputo derivative of the unknown function is approximated using the Gauss--Jacobi quadrature formula. By collocating the obtained residual at the well-known shifted Chebyshev points, we get a system of nonlinear algebraic equations. In order to obtain a continuous solution, some conditions are added to the resulting system. Some error bounds are given for the Legendre wavelet approximation of an arbitrary function. Finally, some examples are included to show the efficiency and accuracy of this new technique.