The quadratic B-spline method for approximating systems of Volterra integro fractional-differential equations involving both classical and fractional derivatives

The quadratic B-spline method is a widely recognized numerical technique for solving systems of Volterra integro-differential equations that involve both classical and fractional derivatives (SVIDE’s-CF). This study presents an improved application of the quadratic B-spline approach to achieve highly accurate and computationally efficient solutions. In the method developed in this paper, control points are treated as unknown variables within the framework of the approximate solution. The fractional derivative  is considered in the Caputo sense. First, we divide the domain into subintervals, then construct quadratic B-spline basis functions over each subinterval. The approximate solution is presented as a quadratic combination of these B-spline functions over each subinterval, where the control points act as variables. To simplify the system of (VIDE’s-CF) into a solvable set of algebraic equations, the collocation method is applied by discretizing the equations at chosen points within each subinterval. The Jacobian matrix method is employed to perform computations efficiently. In addition, a careful, step-by-step algorithm for employing the proposed method is presented to simplify its use, we implemented the method in a Python program and optimized it for efficiency. Experimental example demonstrates effectiveness and accuracy of the proposed technique and its comparison with present techniques in terms of accuracy and computational efficiency.