A Fourth Order Improved Numerical Scheme for the Generalized Burgers—huxley Equation

A fourth order finite-difference scheme in a two-time level recurrence relation is proposed for the numerical solution of the generalized Burgers-Huxley equation. The resulting nonlinear system, which is analyzed for stability, is solved using an improved predictor-corrector method. The efficiency of the proposed method is tested to the kink wave using both appropriate boundary values and conditions. The results arising from the experiments are compared with the relevant ones known in the available bibliography.


Introduction
A. Hodgkin and A. Huxley [1] proposed a model, known henceforth as the Huxley equation, in order to explain the ionic mechanisms underlying the initiation and propagation of action potentials in the squid giant axon.The most general form of the Huxley equation, known as the generalized Burgers-Huxley equation (BgH) [2,3], has the form [4]    where is a sufficiently often differentiable function,


. Equation (1.1), which models the interaction between reaction mechanisms, convection effects and diffusion transport, is the modified Burgers equation for = 0  (see [5] and the references therein), is also the Huxley equation [1] for = 0  and is the Fitzhugh-Nagoma equation [6] for = 0  .Many researchers have used various methods to solve the BgH equation.A theoretical study of the BgH equation was found in Wang et al. [4], while analytical solutions using various techniques in [7][8][9][10][11], etc., have been proposed.As far as the numerical methods are concerned among others the Adomian decomposition method was used by Ismail et al. [12] for the BgH and the Burgers-Fisher equation, and by Hashim et al. [13] for the BgH equation.Javidi [14] used the pseudospectral method, while Javidi [15], Javidi and Golbabai [16] the spectral collocation method.Batiha et al. [17] used the variational iteration method and Khattak [18] the collocation method with radial basis functions.Babolian and Saeidian [19] used the homotopy analysis method, etc.
The initial condition associated with Equation (1.1) will be

Theoretical Solution
It is known [4] that Equation (1.1) has the following kink wave solution are the wave number and the velocity respectively.

Grid and Solution Vector
To obtain numerical solutions the region with its boundary consisting of R  the lines , and is covered with a rectangular mesh of points, , with co-ordinates with .The theoretical solution of Equation (1.1) at the typical mesh point will be denoted by and the relevant of an approximating difference scheme by . .
Let the solution vector at time level be

Boundaries
The following were used: 1) The space derivatives at the left boundary were replaced with second order finite-difference replacements of the form ( [20] p. 17 and with analogous replacements to the right boundary .= 1 x 2) The boundary conditions were used, while at the other interior points of the grid G the well-known approximants based on the central-difference formulas.

The Proposed Method
Applying Equation (1.1) at each point of the grid G at time level ; leads to a first-order initial-value problem, which is written in a matrix-vector form as where   D t U is given by (2.6) and replacing the matrix-exponential term with the fourth order rational approximant ( [21] (2.14), when applied to the general mesh point of the grid G, gives

Stability Analysis
Following the Fourier method of analysing stability ( [21] p. 142) if = e    is the amplification factor and the numerical value of actually obtained, an error of the form where 0 a typical value of , ; U with the set of the complex numbers, so the von Neumann necessary criterion for stability which for = 0  holds, while for = 0   will be satis- fied when 1.

The Modified Predictor-Corrector Scheme
To avoid solving the nonlinear system (2.14) the following Modified Predictor-Corrector (MPC) scheme is proposed.

 
ˆt  U  is evaluated from the reccurence relation (2.12) replacing the matrix-exponential term with the following explicit second order rational approximant which is of the form (2.17 which again subject to (2.20) is always satisfied.

Corrector
The corrector arises from Equation (2.13) as follows odified predictor-corrector method (MPC) was applied [5].The MPC method, which is explicit and is applied once, consists of considering (2.27) component-wise and using an updated component in the corrector vector as soon as it becomes available.Hence, in computing in  , The stability analysis of the corrector is given in Section 2.2.1.

Numerical Results
For the linearization was given.Let the error at time level ; be and e x the x-coordinate at which e occurs.Then e (2.2) -(2.3) denotes the error arising when using the boundary values (2.2) -(2.3), while analogous notations for the other boundary conditions are used.In all experiments the initial condition (1.2) was given by the value   =   ,0 f x u x = 0.1 h with u the theoretical solution (1.3).Experiments proved that the most accurate results are obtained for and 4 = 10 .
  For reasons of comparison with the corresponding works in [12,13,16,17] the same parameter values were used.

Problem [12]
From the experiments the following are deduced: 1) when = 0   (Table 1) using: i) the boundary values (2.2) -(2.3) the method introduced gives more accurate results for all time levels used than the corresponding results in [12] and marginally more accurate than those in [13,17], ii) the boundary condition (2.5) gives more accurate results than those in [12] and approximately equivalent to those in [13,17].
From (i) -(ii) it is deduced that the boundary values (2.2) -(2.3) give more accurate results than the boundary condition (2.5).
2) when = 0  (Table 2) using the boundary values (2.2) -(2.3) the method introduced has given -for = 1  more accurate results for all time levels used than the corresponding in [12], and -for > 1  results with marginally inferior accuracy to those in [12].

Problem [17]
From Table 3 it is deduced that the method introduced using the boundary values (2.2) -( 2.3) has given more accurate results for all time levels and parameters used than the relevant method in [17].

Problem [16]
For reasons of comparison with the relevant results in [18] the boundary conditions (2.4) -(2.5) with   0 = g t and   0, u t     1 = 1, g t u t were used.From Table 4 it is deduced that the proposed method: -has given marginally more accurate results to those in [16] for all time levels and  ,  used, -for fixed  ,  and •  , the accuracy increases and U tends to identify with u at long time level as  is refined, •  , as  increases, the accuracy decreases.

Conclusions
An implicit finite difference scheme based on fourthorder rational approximants to the matrix exponential term was proposed for the numerical solution of the Burgers-Huxley equation.The resulting nonlinear scheme was solved using an improved predictor-corrector method.The computational efficiency of the proposed method given in detail in Section 3 was tested by comparing the numerical results to selected ones in [12,13,16,17] using both appropriate boundary values and conditions.Conclusions for the boundaries used were derived.
Since the real world problems lead to the numerical solution of nonlinear equations or systems of equations, the introduced low cost and easy-to-handle method enables us to obtain accurate solutions.
number and  real is considered.Then Equa- 23)