Mathematical Modeling and Numerical Investigation of Reaction–Diffusion Systems

Abstract

Reaction diffusion systems have found broad applications in the study of spatiotemporal phenomena in biology, chemistry, physics, and other disciplines. But a major computational challenge is to model correctly the nonlinear interplay between the diffusion processes and reaction kinetics. This paper contains a mathematical modeling and numerical analysis of reaction-diffusion equations with a special focus on the design of a robust and efficient computing infrastructure. The equations of the model are derived as a generalized reaction-diffusion equation where the source terms are not linear, and the correct initial and boundary conditions are used. To achieve numerical stability and convergence, a finite difference discretization and an implicit time integration scheme is used. The stability and error analyses are performed to confirm the stability of the proposed method. The space and time development of the system at different parameters is analyzed, using numerical simulations. The outcomes show that the model is able to reproduce important dynamical phenomena, such as pattern formation through diffusion and nonlinear transients. Better accuracy and computational efficiency are also confirmed by comparative and sensitivity analyses. In general, the suggested framework is a promising method of researching reaction diffusions, which may have an application in the biological models, chemistry processes, and environmental systems.