ResearchArticle PseudospectralMethodBasedonM¨ untz–Legendre Wavelets for SolvingtheAbelIntegralEquation Ioannis Dassios, 1 FairouzTchier , 2 andF.M.O.Tawfiq 2 1 AMPSAS, University College Dublin, Dublin 4, Ireland 2 Department of Mathematics, King Saud University, P.O. Box 22452, Riyadh 11495, Saudi Arabia Correspondence should be addressed to Fairouz Tchier; ftchier@ksu.edu.sa Received 17 December 2021; Accepted 14 February 2022; Published 4 March 2022 Academic Editor: Azhar Hussain Copyright © 2022 Ioannis Dassios et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. is paper deals with the numerical solution of the Abel integral equation based on M¨ untz–Legendre wavelets. To this end, the Abel integral operator is represented by M¨ untz–Legendre wavelets as an operational matrix. To find this matrix, we use the similarity between the Abel integral operator and the fractional integral operator. e proposed method can be easily used to solve weakly singular Volterra integral equations. We have proved the convergence of the proposed method. To demonstrate the ability and accuracy of the method, some numerical examples are presented. 1.Introduction In this paper, we focus our attention on constructing and applying M¨ untz–Legendre (M-L) wavelets that will be used as the basis in the pseudospectral method to solve the fa- mous Abel integral equation u(x)− A α (k)(x)� f(x), (1) in which Abel’s integral operator A α (k)(x) of order 0 < α < 1 is defined in [1] as follows: A α (k)(x) ≔  x 0 k(x, s, u(s))(x − s) − α dx, x ∈ [0, 1]. (2) Here, given Ω �[0, 1], f(x), and k(x, s, u(s)) are as- sumed to be continuous functions on Ω and S with S �(x, s, u): x, s ∈ [0, 1],u ∈ R { }. Further, we suppose that the kernel function k(x, s, u(s)) is equal to the form g(x, s)u(s). In other words, the desired equation is assumed to be linear. e Abel equation is a special case of the integral equations with the weakly singular kernel that was first introduced by Abel. In investigating the generalization of the tautochrone problem, he introduced this equation [2]. is equation appears widely in modeling many physical prob- lems, such as nuclear physics, X-ray radiography, fluid flow [3], scattering theory, plasma diagnostics, semiconductors, physical electronics, and nonlinear diffusion[1, 4]. Given this equation’s wide application, solving this equation is very important. But one cannot always solve the equation ana- lytically, and we need to use numerical methods for it. Among the many papers that have considered the nu- merical solution of this equation, we can mention some of them. Saadatmandi and Dehghan [5] utilized the collocation method based on shifted Legendre polynomials. Piessens and Verbaeten [6] introduced a numerical method based on Chebyshev polynomials, and after approximating the un- known solution based on these bases, they obtained the solution as a sum of hypergeometric functions. Using the Bernstein operational matrix, Singh et al. [7] introduced a stable numerical method to solve this problem. In [8], we can find the integrable solution of the Abel integral equation under certain conditions, and also the sufficient and nec- essary conditions for the existence of this solution are presented. In [9], the authors proposed the Laplace trans- form method to solve the problem, where they assumed that the solution would be differentiable and continuous. Saray [10] introduced a novel and efficient method based on Hindawi Journal of Mathematics Volume 2022, Article ID 2251623, 8 pages https://doi.org/10.1155/2022/2251623