Addressing Volterra Partial Integro-Differential Equations through an Innovative Extended Cubic B-Spline Collocation Technique

Abstract

This paper introduces a novel collocation scheme based on an extended cubic B-spline for approximating the solution of a second-order partial integro-differential equation. The proposed scheme employs new extended cubic B-splines to discretize the second-order derivatives in the spatial domain, while discretization of spatial derivatives of lower orders is achieved using extended cubic B-spline functions. Temporal derivatives are discretized using the forward difference formula. The stability of the algorithm is assessed using the von Neumann stability method to ensure that error magnification is avoided. Furthermore, convergence analysis of the scheme is provided. Numerical experiments are conducted to validate the effectiveness and efficiency of the proposed scheme. The free parameter is optimized using L2 and L∞ norms. The computed results are compared with those obtained from various standard numerical schemes found in the literature. Mathematical 12 is used to obtain numerical results.