A Bicubic trigonometric B-Spline Approach for Solving the Nonlinear Generalized 2D Burger's Equation

Nonlinear reaction-diffusion problems, such as the nonlinear generalized two-dimensional Burgers’ equation, play a crucial role in various fields, including developmental biology, population dynamics, engineering, and physics. This study focuses on the numerical solution of the two-dimensional Burgers’ equation using a collocation method based on bicubic trigonometric B-spline functions combined with a θ-weighted scheme. The spatial and temporal domains are discretized using bicubic trigonometric B-spline functions and a finite difference approach, respectively. The nonlinear terms in the equation are handled through quasilinearization. The effectiveness of the proposed method is demonstrated by simulating some test problems with different initial and boundary conditions. The influence of various reaction terms is analyzed and presented in both tabular and graphical formats. Moreover, using the Von Neumann stability analysis, the proposed scheme is shown to be conditionally stable. The results indicate that the present method is highly effective for solving nonlinear partial differential equations arising in a wide range of scientific and engineering applications.