Euler wavelets method for solving fractional-order linear Volterra-Fredholm integro-differential equations with weakly singular kernels

The main objective of this paper is to establish a fractional-order operational matrix method based on Euler wavelets for solving linear Volterra-Fredholm integro-differential equations with weakly singular kernel. First, the one-dimensional Euler wavelet is introduced, and then by using it, the operational matrix of fractional integration is constructed. Using the operational matrix of integration of fractional order, the weakly singular linear fractional Volterra-Fredholm integro-differential equations are reduced to a system of algebraic equations. The convergence of the proposed method has been analyzed and the numerical convergence rate for the presented scheme is established. Also, the error analysis of the proposed technique has been investigated. Furthermore, some numerical examples are solved to illustrate the applicability and efficiency of the proposed approach. Finally, a comparison of absolute error values obtained by different wavelet methods has been presented to clarify the error analysis of the proposed method.