Error-Corrected Shifted Legendre Polynomial Collocation Method for Solving Classical and Caputo Fractional Volterra-Fredholm Integro-Differential Equations

In this paper, the approximate solution of a mixed integro-differential equations of Volterra-Fredholm type with multiple classical and fractional orders in the Caputo sense (CF-VFIDEs) has been successfully found by applying the shifted Legendre polynomial technique with matrix operators and combined collocation points, along with applying the Clenshaw-Curtis quadrature formula for integral parts. The matrix form of the equations is represented by means of these collocation-shifted Legendre polynomials together with operational matrices. This method’s primary feature is that it considerably simplifies the problem by reducing CF-VFIDEs to an algebraic equation system. The classical and fractional-order integro-differential equations will be transformed by matrix operators into a system of linear algebraic equations, which will then be solved to get the approximate solution to the problem mentioned previously. Numerical experimental illustrate the validity and applicability of the proposed technique while the MATLAB software is used to solve and describe a few situations that show how accurate this approach is.