Collocation Approach Based on an Extended Cubic $B$-Spline for a Second-Order Volterra Partial Integrodifferential Equation

Abstract

This paper focuses on an efficient spline-based numerical technique for numerically addressing a second-order Volterra partial integrodifferential equation. The time derivative is discretized using a finite difference scheme, while the space derivative is approximated using the extended cubic B -spline basis. The scheme is also tested for stability study to ensure that the errors do not accumulate. The convergence of the proposed scheme is also investigated. The scheme’s key benefit is that the approximate solution is produced as a smooth piecewise continuous function allowing us to approximate the solution at any location in the domain. Numerical study is performed, and the comparison of results is made to previously reported results in the literature to show the efficiency of the suggested scheme.

Besse Extended Cubic B-spline Collocation Method for Solving Benjamin-Bona-Mahony Equation

“…The BBME has been solved numerically by various methods based on finite difference, finite element, Petrov-Galerkin finite element, Fourier pseudo-spectral, collocation methods, and many other methods [4][5]. Lately, the B-spline collocation method with quadratic B-spline [4], cubic B-spline (CB) [6][7][8], cubic trigonometric B-spline (CTB) [9][10][11], and extended cubic B-spline (ECB) [12][13][14][15][16] have been formulated successfully on some differential equations with accurate results and high efficiency. There are several techniques used to handle the nonlinear term in the BBME namely Taylor series expansion [17], quasilinearization [18] and Adomian polynomials [19].…”