The Müntz–Legendre Wavelet Collocation Method for Solving Weakly Singular Integro-Differential Equations with Fractional Derivatives

Abstract

We offer a wavelet collocation method for solving the weakly singular integro-differential equations with fractional derivatives (WSIDE). Our approach is based on the reduction of the desired equation to the corresponding Volterra integral equation. The Müntz–Legendre (ML) wavelet is introduced, and a fractional integration operational matrix is constructed for it. The obtained integral equation is reduced to a system of nonlinear algebraic equations using the collocation method and the operational matrix of fractional integration. The presented method’s error bound is investigated, and some numerical simulations demonstrate the efficiency and accuracy of the method. According to the obtained results, the presented method solves this type of equation well and gives significant results.

Contributions to the Numerical Solutions of a Caputo Fractional Differential and Integro-Differential System

“…Indeed, various efficient approaches for examining fractional integro-differential and differential problems have been developed, including the matrix collocation method [8], finite integration method [9], dynamical analysis [10], reproducing kernel Hilbert space approximation [11], Euler wavelet approach [12], pseudo-operational matrix method [13], spline quasi-interpolants [14], Haar wavelet [15], homotopy analysis method [16], shifted Legendre projection method [17], airfoil collocation method and the iterated projection method [18]. In [19], the author proposed a collocation approach via the wavelet functions for weakly fractional integro-differential problems. In paper [20], the authors devised the sinc collocation strategy to deal with weakly fractional partial integro-differential problems.…”