Employing Cubic B-Spline Functions to Solve Linear Systems of Volterra Integro-Fractional Differential Equations with Variable Coefficients

Abstract

In this study, we present a collocation method based on cubic B-spline functions for solving systems of Volterra integro-differential equations involving both classical and fractional derivatives in the Caputo sense (LSVIDEs-CF). The approach begins by dividing the problem domain into a finite number of subintervals, followed by the construction of cubic B-spline basis functions within each segment. Control points are introduced as the unknowns in the approximate numerical solution, which is expressed as a cubic combination of these basis functions. The given system of VIFDEs-CF is then reduced to a system of algebraic equations, which are efficiently solved using the Jacobian matrix method. In practice, the integrals are approximated using the Clenshaw–Curtis quadrature rule. The implementation of this method is supported by Python software, ensuring effective computational processing. Numerical examples are provided to demonstrate the efficiency of the proposed method. An itemized version of the algorithm is also presented to facilitate its implementation.