On the wavelet Galerkin method for solving the fractional Fredholm integro-differential equations

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(۲^{-J}) where J\in \mathbb{N}_۰ 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.

Keywords:

Wavelet Galerkin method , Fractional integro-differential equation , Biorthogonal wavelet , Hermite cubic splines , Convergence analysis