Employing Quadratic B‐Spline Collocation Method for Solving Nonlinear Volterra‐Type Integro‐Fractional Differential Equations

Source abstract

ABSTRACT This article presents an efficient numerical solution for specific integro‐differential equations of all fractional orders in the Caputo sense within the interval . We propose a new numerical scheme to evaluate the approximate solution of a nonlinear system of integro‐fractional differential equations of the Volterra–Hammerstein type by utilizing a quadratic B‐spline curve, while a linear B‐spline curve is employed as a predictor at the initial step to initiate the scheme. First, we establish the Caputo‐type fractional‐order derivative of the quadratic B‐spline curve and apply it to transform the considered problem into a system of nonlinear algebraic equations, where the Gauss–Legendre quadrature formula is applied for integral evaluations. The resulting system is then solved using Newton's method. Additionally, the linear B‐spline curve is utilized as a predictor to determine the values of the unknown functions at a single node and to compute one of the unknown control points. The presented method enhances the accuracy and efficiency of numerical computations, making it a useful technique for addressing complex fractional‐order models in applied mathematics and engineering. Some illustrative examples are presented, with results displayed in graphs and tables, to validate the method's effectiveness and robustness. MATLAB 9.2 was used to execute and perform the computations for the proposed algorithms.