A Numerical Approach to Solve the System of Fractional Fredholm-Volterra Integro-Differential Equations Using Rationalized Haar Wavelets

This study introduces a novel numerical methodology for approximating solutions to a system of Fredholm-Volterra integro-differential equations (FVIDEs) of fractional order. The method depends on the rationalized Haar wavelets defined over the semiopen interval [0,1)$$ \left[0,1\right) $$, which exhibit orthogonality within this interval. Initially, the rationalized form of Haar functions is introduced, alongside their properties and their relationship with the Heaviside step function. Subsequently, the operational matrix of Haar functions product and the generalized operational matrix of fractional-order integration are derived. These matrices, along with collocation techniques, facilitate the transformation of the continuous system into a set of nonlinear algebraic equations, which can be solved efficiently by using Newton's iterative method. The error bound and convergence analysis are given. The method is computationally attractive, and some numerical examples demonstrate its accuracy and applicability in solving the system of fractional-order FVIDEs.