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A Haar wavelet operational matrix is applied to fractional integration, which has not been undertaken before. The Haar wavelet approximating method is used to reduce the fractional Volterra and Abel integral equations to a system of algebraic equations. A global error bound is estimated and some numerical examples with smooth, nonsmooth, and singular solutions are considered to demonstrate the validity and applicability of the developed method.

In this paper, the linear B-spline scaling functions and wavelets operational matrix of fractional integration are derived. A new approach implementing the linear B-spline scaling functions and wavelets operational matrices combining with the spectral tau method is introduced for approximating the numerical solutions of time-space fractional partial differential equations with initial-boundary conditions. They are utilized to reduce the main problem to a system of algebraic equations. The uniform convergence analysis for the linear B-spline scaling functions and wavelets expansion and an efficient error estimation of the presented method are also introduced. Illustrative examples are given and numerical results are presented to demonstrate the efficiency and accuracy of the proposed method. Special attention is given to a comparison between the numerical results obtained by our new technique and those found by other known methods.