Piecewise B-Spline Collocation Methods for Multi-Term Nonlinear Caputo-Type Volterra Integro-Differential Systems

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.