Linear B-spline functions are essential mathematical tools for deriving approximate solutions for both linear and nonlinear functional equations of arbitrary order due to their simplicity and computational efficiency and ability to approximate functions accurately. In this paper, a system of multi-fractional order nonlinear integro-differential equations of Volterra-Hammerstein Type (NIFDEs-VH) with variable coefficients, in which the multi-fractional derivatives are described in the Caputo sense, is solved using a first-degree B-spline curve. The proposed method utilizes linear B-splines functions and Gauss-Legendre quadrature for numerical integration to convert the system of NIFDEs-VH into a system of nonlinear algebraic equations containing unknown control points. Newton’s method is selected for its efficiency in iteratively solving nonlinear algebraic systems, specifically aiming to determine the control points and obtain linear approximate functions. Four algorithms are proposed and implemented in MatLab to operationalize these methods, offering general computational frameworks. The effectiveness and accuracy of the proposed approach are demonstrated through various illustrative examples. Each example includes graphical representations of both exact and approximate solutions. In addition, the results are presented in tabular form.

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.

Numerical approximations of multi-term nonlinear Volterra–Hammerstein fractional integro-differential systems (NVHFIDS) present considerable challenges in scientific computing and numerical analysis because of their sophisticated mathematical structure. The main novelties of this study are its ability to overcome a significant limitation of existing numerical algorithms and provide more accurate numerical results for a broader class of multi-term NVHFIDS with variable coefficients using piecewise continuous functions constructed with B-spline functions. First, to address these challenging problems, a continuous interval is discretized into a finite set of subintervals. The corresponding B-spline curve is then constructed over this partition by incorporating unknown control points. Appropriate collocation points are selected, the Caputo derivative of the piecewise continuous function is evaluated at these points, and the Gauss quadrature rule is employed to approximate all integral terms. Thus, the NVHFIDS is transformed into nonlinear algebraic systems. The resulting systems are then solved using Newton’s iterative method, which ensures high-precision computation and fast convergence. Four algorithms are formulated and implemented using MATLAB (version 9.2). Finally, to demonstrate the effectiveness of the method, a series of benchmark examples is examined, highlighting its stability, efficiency, and versatility. Each illustrative example is supported by detailed data tables, graphical figures, and piecewise continuous approximation functions. A comparative analysis with existing approaches is provided to demonstrate the accuracy of the proposed algorithms. The numerical results demonstrate that the proposed algorithms reduce the mean least squares error by up to three orders of magnitude compared with existing methods.