Cubic B-splines collocation method for solving nonlinear parabolic partial differential equations with Neumann boundary conditions

In this paper, a numerical method is proposed to approximate the solution of the nonlinear parabolic partial differential equation with Neumann’s boundary conditions. The method is based on collocation of cubic B-splines over finite elements so that 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 its derivatives, which produce a system of first order ordinary differential equations. We solve this system by using SSP-RK3 scheme. The numerical approximate solutions to the nonlinear parabolic partial differential equations have been computed without transforming the equation and without using the linearization. Four illustrative examples are included to demonstrate the validity and applicability of the technique. In numerical test problems, the performance of this method is shown by computing L∞andL2error norms for different time levels. Results shown by this method are found to be in good agreement with the known exact solutions.

► Numerical solutions of nonlinear parabolic PDE are given using cubic B-spline with SSP-RK3 scheme. ► Numerical solutions of equations have been evaluated without transformation and linearization. ► The presented method needs less storage space that causes to less accumulation of numerical errors. ► The results exhibited by this method are found to be in good agreement with the exact solutions. ► Easy implementation is the strength of this method.