HYBRID FINITE DIFFERENCE METHODS FOR SOLVING MODIFIED BURGERS AND BURGERS-HUXLEY EQUATIONS

摘要

Most phenomena in the real world are described through non-linear equations. One of the most fascinating extensions of the Burgers: equation in the description of non-linear phenomena is the modified Burgers' equation and the Burgers-Huxley equation. Modified Burgers equation has varied applications in the field of Physics and particularly wherein dissipation is a significant aspect of wave propagation. On the other hand. Burgers-Huxley equation, under special choice of parameters, namely the Hodgkin-Huxley equation, describes how action potentials in neurons are initiated and propagated. This equation also shows the interplay between non-linear reaction and diffusive transport. Our objective in this paper is to devise and analyze robust numerical methods for numerically solving the modified Burgers' equation and the Burgers-Huxley equation. The methods are primarily based on monotone hybrid finite difference methods with piecewise uniform layer adaptive mesh. A rigorous analysis of the proposed methods for uniform convergence is given and the error estimates are derived. Several numerical experiments on benchmark problems are carried out and comparison of the numerical results made with the existing methods demonstrate the improvement and efficiency of the proposed methods.