Numerical Solution and Stability Analysis of Huxley Equation

The numerical solution of Huxley equation by the use of two finite difference methods is done. The first one is the explicit scheme and the second one is the Crank-Nicholson scheme. The comparison between the two methods showed that the explicit scheme is easier and has faster convergence while the Crank-Nicholson scheme is more accurate. In addition, the stability analysis using Fourier (von Neumann) method of two schemes is investigated. The resulting analysis showed that the first scheme is conditionally stable if, and the second scheme is unconditionally stable.


ABSTRACT
The numerical solution of Huxley equation by the use of two finite difference methods is done. The first one is the explicit scheme and the second one is the Crank-Nicholson scheme. The comparison between the two methods showed that the explicit scheme is easier and has faster convergence while the Crank-Nicholson scheme is more accurate. In addition, the stability analysis using Fourier (von Neumann) method of two schemes is investigated. The resulting analysis showed that the first scheme is conditionally stable if,

Introduction
It is probably not an overstatement to say that almost all partial differential equations (PDEs) that arise in a practical setting are solved numerically on a computer. Since the development of high-speed computing devices the numerical solution of PDEs has been in active state with the invention of new algorithms and the examination of the underlying theory. This is one of the most active areas in applied mathematics and it has a great impact on science and engineering because of the ease and efficiency it has shown in solving even the most complicated problems. The basic idea of the method of finite differences is to cast the continuous problem described by the PDE and auxiliary conditions into a discrete problem that can be solved by a computer in finitely many steps. The discretization is accomplished by restricting the problem to a set of discrete points. By systematic procedure, we then calculate the unknown function at those discrete points. Consequently, a finite difference technique yields a solution only at discrete points in the domain of interest rather than, as we expect for an analytical calculation, a formula or closed-form solution valid at all points of the domain [11]. Manoranjan et al [12] obtained estimates for the critical lengths of the domain at which bifurcation occurs in the cases , 2 and 1. Manoranjan [13] studied in detail the solutions bifurcating from the equilibrium state a u = . Eilbeck and Manoranjan [3] considered different types of basis functions for the pseudo-spectral method applied to the nonlinear reaction-diffusion equation in 1-and 2-space dimensions. Eilbeck [4] extended the pseudo-spectral method to follow steady state solutions as a function of the problem parameter, using path-following techniques. Fath and Domanski [6] studied the cellular differentiation in a developing organism via a discrete bistable reaction-diffusion model and they used the numerical simulation to support their expectations of the qualitative behavior of the system. Lewis and Keener [10] studied the propagation failure using the one -dimensional scalar bistable equation by a passive gap and they used the numerical simulation in their study. Binczak et al [1] compared the numerical predictions of the simple myelinated nerve fibers with the theoretical results in the continuum and discrete limits. Broadbridge et al [2] re-examined the derivation of the gene-transport equations and used the Gaussian clump of alleles by the use of a numerical method-oflines by using PDETWO program. Lefantzi et al [9] presented their findings for various orders of spatial discretizations as applied to SAMR (Structured Adaptive Mesh Refinement) simulations.
In this paper, the numerical solution of Huxley equation by using two finite difference methods and stability analysis of these two methods are analyzed.

The Mathematical Model
One of the famous non-linear reaction-diffusion equations is the generalized Burgers-Huxley (gBH) equation: and , equation (1) becomes the following Burgers-Huxley (BH) equation: Equation (2) shows a prototype model for describing the interaction mechanism, convection transport. When 1 , 0 = = δ β and , equation (1) is reduced to Burgers equation which describes the far field of wave propagation in nonlinear dissipative systems and , equation (1) is reduced to the Huxley equation which describes nerve pulse propagation in nerve fibers and wall motion in liquid crystal It is known that nonlinear diffusion equations (3) and (4) play important roles in nonlinear physics. They are of special significance for studying nonlinear phenomena [19]. Zeldovich and Frank-Kamenetsky formulated the equation (4) in 1938 as a model for flame front propagation and for this reason this equation sometimes named Zeldovich-Frank-Kamenetsky (ZF) equation, which has been extensively studied as a simple nerve model [1]. In 1952 Hodgkin and Huxley proposed their famous Hodgkin-Huxley model for nerve propagation. Because of the mathematical complexity of this model, it led to the introduction of the simpler Fitzhugh-Nagumo system. The simplified model of the Fitzhugh-Nagumo system is Huxley equation [18]. Because Huxley equation is a special case of Fitzhugh-Nagumo system, it is sometimes named Fitzhugh-Nagumo (FN) equation [5] or the reduced Nagumo equation or Nagumo equation [15]. In sixties, Fitzhugh used equation (4) as an approximate equation for the description of dynamics of the giant axon. This equation was among the first models of excited media [8].
In this paper, we shall take the Huxley equation as a model problem [12]: For the purpose of numerical calculations, we shall take: ( )

Derivation of the Explicit Scheme Formula of Huxley Equation
Assume that the rectangle and the initial condition is [12]: Where the grid points are: Neglecting the terms ( ) , and use approximation in equations (8) and (9), which are in turn substituted in equation (4), we get After some mathematical manipulation, we obtain (12) represents the explicit finite difference formula for equation (4). Equation (12) is employed to create ( )th j 1 + row across the grid, assuming that approximations in the jth row are known. Notice that this formula explicitly gives the value To apply von Neumann method to equation (4), we resort to the linearized stability analysis [7], we have ( )  Where ξ can be visualized as the amplification factor and we get As we advance the solution from a particular plane ( ) From equation (15), for boundedness of (15), we need In the above inequality, the right-side inequality is: and this is always true. Hence, in order that (16)  This precisely the conditions imposed on the explicit scheme to be stable.

Derivation of the Crank-Nicholson Scheme Formula of Huxley Equation
The is diffusion term xx u in this method is represented by central differences, with their values at the current and previous time steps averaged [17]: in equations (19) and (20), which are in turn substituted into equation (4), we have  (22) represents the Crank-Nicholson formula for equation (4).
The terms on the right hand side of equation (22)    When the Crank-Nicholson scheme is implemented with a computer, the linear system B AX = can be solved by either direct means or by iteration.In this paper, the Gaussian elimination method (direct method) has been used to solve the algebraic system AX = B.

Stability Analysis of the Crank-Nicholson Scheme Using Fourier (von Neumann) Method
The finite difference Crank-Nicholson formula for (13) Nicholson scheme which is more accurate than the explicit scheme and the results of this paper are affirming the analytical results which obtained by Manoranjan et al [12] as shown below: .
then as shown in figure (1) and table (1). In addition, from stability analysis, we concluded that the explicit scheme is conditionally stable if