Numerical Investigation of Two-Dimensional Nonlinear Transport and Diffusion Phenomena Using Bicubic B-Splines

Nonlinear partial differential equations, such as the two-dimensional Burgers–Fisher equation, arise in many applications in biology, engineering, and physics. Owing to the strong nonlinearity of this equation, obtaining analytical solutions is generally difficult, motivating the development of efficient numerical methods. In this paper, the numerical solution of the two-dimensional Burgers–Fisher equation is investigated using a collocation approach based on bicubic B-spline and bicubic trigonometric B-spline basis functions. The spatial domain is discretized using the corresponding B-spline basis functions, while the temporal derivative is approximated using a finite difference method with a θ-weighted Crank–Nicolson scheme. The nonlinear terms are treated using a quasilinearization technique, resulting in a linear system at each time level. A von Neumann stability analysis demonstrates that the proposed schemes are conditionally stable. To assess the accuracy and efficiency of the proposed methods, four numerical test problems are considered. The results indicate that high accuracy can be achieved with a relatively small number of grid points m,n, reducing computational cost. Consequently, the proposed numerical approachs are effective and reliable tools for solving two-dimensional nonlinear partial differential equations.