In this study, we illustrate the quadratic B-spline functions as an efficient numerical approximation method to obtain the solution of the system of Volterra integro-differential equations involving the classical and fractional derivatives in the Caputo sense (SVIDE's-CF). The given technique starts by partitioning the problem domain into a finite number of intervals, followed by the construction of a quadratic B-spline basis functions on each sub-interval. Moreover, control points are used as unknown variables in the approximate solution, which is expressed as a quadratic combination of such B-spline functions. Collocation points indicate discretising the equation by enforcing it at certain locations inside each interval. Then, the (VIFDEs-CF) system given is reduced to a system of algebraic equations, which are solved with the method of a Jacobian matrix efficiently. Quadratic B-splines improve the accuracy of the solution, and, therefore, the complex dynamics of both the classical and fractional parts are captured. This method's implementation is supported by a Python program, which ensures its effective computational processing. Examples are used to illustrate the usefulness and resilience of this strategy. These illustrations demonstrate how the proposed method improves the current approaches in terms of accuracy and processing efficiency. To facilitate implementation, an itemised version of this approach's algorithm is provided.
Abdullah et al. (Thu,) studied this question.