University of Tabriz Computational Methods for Differential Equations 2345-3982 13 3 2025 07 01 On the wavelet Galerkin method for solving the fractional Fredholm integro-differential equations 885 903 18609 10.22034/cmde.2024.62193.2725 EN Sharareh Ranjbari Department of Mathematics, Ta.C., Islamic Azad University, Tabriz, Iran. Mahdi Baghmisheh Department of Mathematics, Ta.C., Islamic Azad University, Tabriz, Iran. Mohammad Jahangiri Rad Department of Mathematics, Ta.C., Islamic Azad University, Tabriz, Iran. Behzad Nemati Saray Department of Mathematics, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan 45137-66731, Iran. Journal Article 2024 06 20 An effective scheme is presented to estimate the numerical solution of fractional integro-differential equations (FIDEs). In the present method, to obtain the solution of the FIDEs, they must first be reduced to the corresponding Volterra Fredholm integral equations (VFIEs) with a weakly singular kernel. Then, by applying the matrix that represents the fractional integral (FI) based on biorthogonal Hermite cubic spline scaling bases (BHCSSb), and using the wavelet Galerkin method, the reduced problem can be solved. The combination of singularity and the challenge related to nonlinearity poses a formidable obstacle in solving the desired equations, but our method overcomes them well. An investigation of the method's convergence is provided, and it verifies that the convergence rate is $O(2^{-J})$ where $J\in \mathbb{N}_0$ is the refinement level. The verification of convergence has also been demonstrated through the presentation of several numerical examples. Compared to other methods, the results obtained show better accuracy. https://cmde.tabrizu.ac.ir/article_18609_8ce686c49d69bcb7719d69decf72b974.pdf