Abstract
In this paper, we construct four numerical methods to solve the Burgers–Huxley equation with specified initial and boundary conditions. The four methods are two novel versions of nonstandard finite difference schemes (NSFD1 and NSFD2), explicit exponential finite difference method (EEFDM) and fully implicit exponential finite difference method (FIEFDM). These two classes of numerical methods are popular in the mathematical biology community and it is the first time that such a comparison is made between nonstandard and exponential finite difference schemes. Moreover, the use of both nonstandard and exponential finite difference schemes are very new for the Burgers–Huxley equations. We considered eleven different combination for the parameters controlling diffusion, advection and reaction, which give rise to four different regimes. We obtained stability region or condition for positivity. The performances of the four methods are analysed by computing absolute errors, relative errors, L 1 and L ∞ errors and CPU time.
Highlights
Numerical and mathematical analysis are of significant importance for the solution and understanding of problems in science and engineering
There are some non-linear partial differential equations that become integrable after some symbolic transformation
Recently in 2017, Burgers–Huxley equations were solved using an explicit exponential finite difference method constructed by İnan [46]
Summary
Numerical and mathematical analysis are of significant importance for the solution and understanding of problems in science and engineering. There are some non-linear partial differential equations that become integrable after some symbolic transformation In this case, the analytical solution becomes obtainable. Diaz et al [40] Another class of schemes known as exponential finite difference methods have been used to solve the Burgers type equations. Employed implicit and fully implicit exponential finite difference method to solve Burgers equation and their paper was published in 2013. Recently in 2017, Burgers–Huxley equations were solved using an explicit exponential finite difference method constructed by İnan [46]. The quite good accuracy of the exponential finite difference schemes and the nature of some of these alluring non-linear problems has made these methods quite popular for the Burgers-type equations. A very short version of this paper has been accepted for publication as [49]
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