COMPACT FINITE DIFFERENCE METHOD FOR PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS

Abstract

In this study, we investigate the Generalized Burger’s Fisher and Generalized Burger’s Huxley equations using a numerical approach via the combination of Compact Finite Difference of order six and Strong Stability Preserving RK-43 Method. These equations are important in modeling many physical models in Traffic Flow Model, Wave Propagation, Nonlinear Optics, Financial Market Dynamics, Drug Diffusion in Tissues, etc. are governed by these two physical models. We outline the methodology involved, beginning with the application of CFDM6 to discretize the partial differential equations (PDEs) into a system of ordinary differential equations (ODEs). By implementing CFDM, we evaluate spatial derivatives (both first and second order) for both non-linear PDEs. Subsequently, we employ the Strong Stability Preserving Runge-Kutta (SSP RK) scheme to solve the resulting ODEs, thus obtaining the numerical solutions with SSP-RK Method ensuring accurate and stable computations in the simulation of complex systems. Additionally, stability analysis of the numerical schemes is conducted to ensure the reliability and accuracy of the computed solutions. The stability results are ensured by real eigenvalue plots with different parameters for Generalized Burger’s Fisher and Generalized Burger’s Huxley equation. These methods offers ease of implementation with less computational cost and high accuracy, making it a promising approach for solving complex differential equations encountered in various scientific and engineering disciplines. In terms of efficacy, CFDM6 has been found better than other methods like Adomain Decomposition Method(ADM) [14], Compact Finte Difference Method(CFDM) [28] , Exponential Time Differencing Method (ETDM) [5] , Variational Iteration Method [13] , Analytic Aprroximate Solution [3] , Laguerre Matrix Approach [11] to solve numerical schemes. Furthermore, we provide several examples with numerical and exact solution plots for different parameters to illustrate the efficacy of the proposed numerical approach in capturing the dynamics described by these non linear equations.