Utilizing quadratic B-spline functions to solve nonlinear systems of Volterra integro-fractional differential equations with variable coefficients

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Abstract
The primary objective of this article is to develop new methods based on quadratic B-spline functions to numerically solve the general nonlinear system of Volterra integro-fractional differential equations of Hammerstein type (NSVIFDE-H) described in the Caputo sense. Using the Gauss quadrature method for numerical integration and quadratic B-spline functions, we propose four novel strategies for solving the considered equations. A key advantage of the proposed schemes is their ability to transform the NSVIFDE-H into a nonlinear system of ordinary algebraic equations, which can be effectively solved using appropriate techniques. By deriving the general form of the Caputo fractional derivative of the quadratic B-spline curve, we convert the system into a set of nonlinear ordinary algebraic equations involving unknown control points. Newton’s method is then applied to solve the resulting system and determine the control points. Numerical examples, along with corresponding graphs and tables, are presented to demonstrate the effectiveness and efficiency of the new algorithms. These results are compared with exact solutions, and most of the general programs were developed using MATLAB software 9.2 to implement the solutions for the test examples. Furthermore, these mathematical methods can be applied to solve complex physical problems, improve system performance, and address physics-related challenges such as optimizing optical systems, enhancing signal processing for antennas, and refining image analysis.

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