Applications of cubic B‐splines collocation method for solving nonlinear inverse parabolic partial differential equations

Abstract

In this article, we discuss a numerical method for solving some nonlinear inverse parabolic partial differential equations with Dirichlet's boundary conditions. The approach used, is based on collocation of cubic B-splines over finite elements, so we have continuity of the dependent variable and its first two derivatives throughout the solution range. We apply cubic B-splines for spatial variable and derivatives, which produce an ill-posed system. We solve this system using the Tikhonov regularization method. The accuracy of the proposed method is demonstrated by applying it on two test problems. The figures and comparisons have been presented for clarity. Also the stability of this method has been discussed. The main advantage of the resulting scheme is that the algorithm is very simple, so it is very easy to implement.

Split-step cubic B-spline collocation methods for nonlinear Schrödinger equations in one, two, and three dimensions with Neumann boundary conditions

“…Spline collocation methods are well known for their conceptual simplicity, wide applicability, and ease of implementation [1]. The cubic B-spline collocation (3BC) method is one of them (see [2][3][4][5] and references therein). The governing discrete system obtained from the 3BC method is always tridiagonal for a one-dimensional (1D) problem, so the fast and efficient algorithm like Thomas algorithm can be applied [2].…”