DISCRETE WEIGHTED RESIDUAL METHODS WHICH ARE TECHNIQUES USED FOR NUMERICAL SOLUTION TO MIXED VOLTERRA-FREDHOLM FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS

Abstract

This study aims to solve the most general form of linear mixed Volterra-Fredholm integro-differential equations of multi-fractional order in the Caputo sense (MV-FIFDEs), which is solved by using orthogonal generalized Bernstein’s polynomial expansion with collocation and moment in the discrete weighted residual method under suitable conditions; then, the Clenshaw-Curtis formula is applied to approximate the integral terms of the equation numerically. In this work, the integro-fractional differential equations are reduced to algebraic linear equations and then to an operational matrix, and the solution of the resultant system yields the unknown Bernstein coefficients of approximation solutions. An algorithm has been created for each technique to handle the MV-FIFDEs using the described methods. Furthermore, numerical examples are presented to demonstrate and compare the techniques’ validity and applicability and comparisons with previous results. The majority of programs are performed on a computer using MATLAB v. 9.7. Keywords: Mixed Fractional Integro-Differential Equations, Caputo Derivative, Bernstein Polynomial, Collocation, Moment, Discrete Weighted Residual, Clenshaw-Curtis Formula, Volterra-Fredholm Equations DOI： https://doi.org/10.35741/issn.0258-2724.58.3.43