# Numerical Solutions of the Generalized Burgers-Huxley Equation by a Differential Quadrature Method

Authors: Murat Sari; Gürhan Gürarslan
Journal: Mathematical Problems in Engineering
 (2009)

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## Abstract

Numerical solutions of the generalized Burgers-Huxley equation are obtained using a polynomial differential quadrature method with minimal computational effort. To achieve this, a combination of a polynomial-based differential quadrature method in space and a low-storage third-order total variation diminishing Runge-Kutta scheme in time has been used. The computed results with the use of this technique have been compared with the exact solution to show the required accuracy of it. Since the scheme is explicit, linearization is not needed and the approximate solution to the nonlinear equation is obtained easily. The effectiveness of this method is verified through illustrative examples. The present method is seen to be a very reliable alternative method to some existing techniques for such realistic problems.


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## Body

## 1. Introduction

Nonlinear partial differential equations are
encountered in  various fields of science. Generalized
Burgers-Huxley equation
being a nonlinear partial differential equation is of high importance for describing  the interaction between reaction
mechanisms, convection effects, and diffusion transports. Since there exists no
general technique for finding analytical solutions of nonlinear diffusion
equations so far, numerical solutions of nonlinear differential equations are
of great importance in physical problems.There are many researchers
who used various numerical techniques to
obtain numerical solution of the  Burgers-Huxley equation.
Wang et al. [1] studied the solitary wave
solutions of the generalized Burgers-Huxley equation and Estevez [2]
presented nonclassical symmetries and the singular
modified Burgers and Burgers-Huxley equation. In the past few years, various
powerful mathematical methods such as spectral methods [3–5], Adomian
decomposition method [6–8], homotopy analysis
method [9], the tanh-coth method [10],
variational iteration method [11, 12],  and Hopf-Cole transformation [13] have been used in attempting to solve the equation.To the best knowledge of the authors, the idea of the differential
quadrature method (DQM), where approximations of the spatial derivatives have
been based on a polynomial of high degree, has not been implemented for the
problems in physical phenomena represented by the generalized
Burgers-Huxley equation so far. The DQM is an efficient
discretization technique in solving initial and/or boundary value problems
accurately using a considerably small number of grid points. Bellman et al. [14]
introduced the DQM in the early
seventies and, since then, the technique has been successfully employed in
finding the solutions of many problems in applied and physical sciences [15–21]. Recent comparative studies show
that the DQM provides highly accurate
and efficient solutions of the ordinary/partial differential equations taking
a noticeably small number of grid points. Due to the aforementioned advantages,
the DQM has been projected by its proponents as a potential alternative to the
conventional numerical solution techniques such as the finite difference and
finite element methods.In the DQM,
derivatives of a function with respect to a coordinate direction are expressed
as linear weighted sums of all the functional values at all grid points along
that direction. The weighting coefficients in that weighting sum are determined
using test functions. Among the many kinds of test functions, the Lagrange
interpolation polynomial is widely used since it has no limitation on the
choice of the grid points. This leads to polynomial-based differential
quadrature (PDQ) method which is suitable in most problems. For periodic
problems, Fourier series expansion can be the best approximation giving the
Fourier expansion-based differential quadrature (FDQ) method. To clearly
describe the DQ method, the readers can see that the PDQ method was first
presented in the work of Shu and Richards [18], and FDQ method was first
appeared in the works of Shu and Chew [22], and Shu and Xue [23]. The
determination of weighting coefficients in explicit formulations [24] for both
cases is based on the analysis of function approximation and analysis of a
linear vector space.Unlike some previous techniques using various
transformations to reduce the equation into more simple equation, the current
method does not require extra effort to
deal with the nonlinear terms. Therefore the equations are solved easily and
elegantly using the present method. This method has also additional advantages
over some rival techniques, ease in use and computational costeffectiveness
in order to find solutions of the given nonlinear equations. The combination of
the PDQ method in space with the low-storage third-order total
variation diminishing Runge-Kutta (TVD-RK3) scheme in time [25] provides an efficient explicit solution with high
accuracy and minimal computational effort for the problems represented by the
generalized Burgers-Huxley equation.The present method is useful for obtaining numerical
approximations of linear or nonlinear differential equations and it is also
quite straightforward to write codes in any programming languages. Also, round
off errors and necessity of large computer memory are not faced in this method.
The
computed results obtained by this way have been compared with the exact
solution to show the required accuracy of it. Furthermore,
the current method is of a general nature and can therefore be used for
solving the nonlinear partial differential equations arising in various areas. Therefore,
this paper suggests the use of this technique for solving the generalized Burgers-Huxley equation
problems.


## 2. The Model Equation

Behaviors of
many physical systems encountered in models of reaction mechanisms, convection effects,  and diffusion transports give rise to
the generalized Burgers-Huxley equation. The following generalized
Burgers-Huxley equation problem arising  in
various fields of science is considered:(2.1)ut+αuδux−uxx=βu(1−uδ)(uδ−γ),0≤x≤1,t≥0 with
the initial condition (2.2)u(x,0)=(γ2+γ2tanh(a1x))1/δ and the boundary conditions (2.3)u(0,t)=(γ2+γ2tanh(−a1a2t))1/δ,t≥0,u(1,t)=(γ2+γ2tanh[a1(1−a2t)])1/δ,t≥0. The exact solution of  (2.1) is
(2.4)u(x,t)=(γ2+γ2tanh[a1(x−a2t)])1/δ,t≥0, where
(2.5)a1=−αδ+δα2+4β(1+δ)4(1+δ)γ,a2=γα1+δ−(1+δ−γ)(−α+α2+4β(1+δ))2(1+δ),   where α,β,γ, and δ are parameters that β≥0,δ>0.
The role of the parameters on exact solutions was analyzed by Efimova and
Kudryashov[13]. If β=0,
(2.1) reduces to Burgers’ equation. When α=0,
it is the Fitzhugh-Nagoma equation [26, 27].The current work aims to demonstrate
that the proposed numerical algorithm is capable of achieving high accuracy for
the problems represented by the generalized Burgers-Huxley equation. The
computed results are compared with the exact solutions to verify the
effectiveness of the current method.


## 3. Polynomial-Based Differential Quadrature Method

The method
uses the basis of the quadrature method in driving the derivatives of a
function.  It follows that the partial
derivative of a function with respect to a space variable can be approximated
by a weighted linear combination of function values at some intermediate points
in that variable.The selection of locations of the sampling points plays an important
role in the accuracy of the solution of the differential equations. Using
uniform grids can be considered to be a convenient and easy selection method.
Quite frequently, the DQM delivers more accurate solutions using the so-called
Chebyshev-Gauss-Lobatto points [20, 24]. For a domain specified by a≤x≤b and discretized by a set of nonuniform grids,
then the coordinate of any point i can be evaluated by
(3.1)xi=a+12(1−cos(i−1N−1π))(b−a).
The values of
function u(x,t) at any time on the above grid  points are given
as u(xi,t),i=1,2,…,N.
Here N stands for the number of grid points. The
differential quadrature discretizations of the first- and second-order spatial
derivatives are given by, respectively:
(3.2)ux(xi,t)=∑j=1Naiju(xj,t),i=1,2,…,N,uxx(xi,t)=∑j=1Nbiju(xj,t),i=1,2,…,N,
where aij and bij are the weighting coefficients of the first-
and second-order derivatives, respectively [24]. Once the weighting
coefficients are determined, the bridge to link the derivatives in the
governing differential equation and the functional values at the mesh points is
established. The weighting coefficients of the first-order derivatives are as
follows [24]:
(3.3)aij=1xj−xi∏k=1,k≠i,jNxi−xkxj−xk,i≠j,aii=−∑j=1,j≠iNaij.
For weighting coefficients of the second = order derivative, the formulae are [24]:
(3.4)bij=2aij(aii−1xi−xj),i≠j,bii=−∑j=1,j≠iNbij.In order to attain the accurate numerical solution of differential
equations, proper implementation of the boundary is also very important. For
prescribing the Dirichlet boundary conditions (2.3), (2.1) should only
be applied at the interior points since the solution at the boundary grid
points is known. Thus, (2.1) can be written in discretized form
(3.5)duidt=βui(1−uiδ)(uiδ−γ)−αuiδ∑k=1Naikuk+∑k=1Nbikuk+si,i=2,3,…,N−1,
where
(3.6)si=−αuiδai1u1−αuNδaiNuN+bi1u1+biNuN.After the
discretization, using the PDQ method, (3.5) can be reduced into a set
of ordinary differential equations in time. Thus,
(3.7)duidt=Lui,where L shows a spatial nonlinear differential
operator. The low-storage explicit TVD-RK3 scheme integrates from time t0 (step m) to t0+Δt (step m+1) through the operations [26]
(3.8)ui(1)=uim+ΔtLuim,ui(2)=34uim+14ui(1)+14ΔtLui(1),uim+1=13uim+23ui(2)+23ΔtLui(2).
In this procedure,
each spatial derivative on the right hand side of      (3.7) was computed
with the use of the PDQ method and then the semidiscrete equation (3.7) was
solved with the help of the low-storage explicit TVD-RK3 scheme. Thus, the
solution is obtained without solving any algebraic system of equations, and requiring
neither linearization nor any transformation.


## 4. Numerical Illustrations

In order to see numerically whether the
present methodology leads to accurate solutions, the PDQ solutions will be
evaluated for some examples of the generalized Burgers-Huxley equations given above. To verify the efficiency, measure its
accuracy and the versatility of the PDQ method for the current problem in
comparison with the exact solution, absolute error for different values ofα, β, δ, and γ is
reported which is defined by
(4.1)|u(xi,tj)−U(xi,tj)|
in the point (xi,tj).
Here u(xi,tj) the solution portraying the behaviors of
physical systems is obtained by the present scheme while U(xi,tj)  stands  for the exact solution.Consider
the generalized  Burgers-Huxley equation in the form (2.1)
with the initial condition (2.2), boundary conditions (2.3),  and the exact
solution (2.4). The results are compared with the exact solution. The numerical
computations were performed using nonuniform grids. The current method is quite straightforward to write codes in any
programming languages. Here, all computations were carried out using some
codes produced in Visual Basic 6.0. N and Δt are taken to be 16 and 0.0001, respectively,
in all examples except in Example 4.7. The
differences between the computed solution and the exact solution for some
values of the constants δ,α, β, and γ are shown in
Tables 1–6. As various problems of science were modelled by nonlinear partial
differential equations and since therefore the generalized Burgers-Huxley equation is of great importance, various values of the
parameters have been considered in the following examples [4].Table 1
The absolute errors for various values ofδ, x,  and t with α=1, β=1, γ=0.001.
xitδ=1δ=2δ=4δ=8x30.206.841E-093.194E-072.239E-066.058E-060.607.733E-093.610E-072.531E-066.842E-060.807.748E-093.617E-072.535E-066.852E-06x70.203.644E-081.701E-061.193E-053.226E-050.604.226E-081.973E-061.383E-053.739E-050.804.236E-081.977E-061.386E-053.746E-05x130.201.420E-086.630E-074.649E-061.257E-050.601.615E-087.538E-075.284E-061.428E-050.801.618E-087.553E-075.294E-061.431E-05Table 2
The absolute errors for various values ofδ, x, and t with α=0.1, β=0.001,  γ=0.0001.
xitδ=1δ=2δ=4δ=8x30.104.059E-145.984E-127.554E-112.802E-100.505.888E-148.681E-121.096E-104.065E-101.005.924E-148.733E-121.102E-104.090E-10x70.102.021E-132.979E-113.760E-101.395E-090.503.215E-134.741E-115.984E-102.220E-091.003.239E-134.775E-116.027E-102.236E-09x130.108.301E-141.224E-111.545E-105.731E-100.501.229E-131.812E-112.288E-108.487E-101.001.237E-131.824E-112.302E-108.541E-10Table 3
The absolute errors for various values ofδ, x, and t with α=−0.1, β=0.1, γ=0.001.
xitδ=1δ=2δ=4δ=8x40.302.317E-091.013E-076.628E-071.682E-060.502.413E-091.055E-076.902E-071.751E-060.902.428E-091.062E-076.944E-071.762E-06x80.306.580E-092.878E-071.882E-064.776E-060.506.899E-093.017E-071.974E-065.007E-060.906.950E-093.039E-071.988E-065.043E-06x120.303.695E-091.616E-071.057E-062.681E-060.503.855E-091.686E-071.103E-062.798E-060.903.881E-091.697E-071.110E-062.816E-06Table 4
The absolute errors for various values ofβ, x, and t with α=1, δ=1, γ=0.001.
xitβ=1β=10β=50β=100x40.301.545E-081.854E-079.807E-071.989E-060.501.608E-081.930E-071.021E-062.071E-060.901.618E-081.942E-071.027E-062.082E-06x80.304.387E-085.264E-072.785E-065.649E-060.504.600E-085.520E-072.920E-065.922E-060.904.633E-085.560E-072.941E-065.960E-06x120.302.463E-082.956E-071.564E-063.172E-060.502.570E-083.084E-071.632E-063.309E-060.902.587E-083.105E-071.642E-063.328E-06Table 5
The absolute errors for various values ofγ, x, and t with α=5, β=10, δ=2.
xitγ=10−2γ=10−3γ=10−4γ=10−5x40.302.039E-046.505E-062.059E-076.512E-090.502.116E-046.771E-062.144E-076.780E-090.902.111E-046.808E-062.157E-076.823E-09x80.305.809E-041.848E-055.848E-071.849E-080.506.071E-041.937E-056.131E-071.939E-080.906.064E-041.950E-056.175E-071.953E-08x120.303.268E-041.038E-053.283E-071.038E-080.503.399E-041.083E-053.426E-071.083E-080.903.392E-041.089E-053.448E-071.091E-08Table 6
Comparison of PDQ-Lagrange and PDQ-Chebyshev methods: the absolute errors for various
values ofx and t with α=2, β=3, γ=0.001, δ=1.
xitPDQ-LagrangePDQ-Chebyshevx40.017.6338863E-097.6338860E-090.103.1524288E-083.1524290E-081.004.6979115E-084.6979110E-08x80.011.0877730E-081.0877730E-080.108.2905812E-088.2905810E-081.001.3450669E-071.3450670E-07x120.019.7652503E-099.7652500E-090.104.9131274E-084.9131270E-081.007.5113908E-087.5113910E-08Example 4.1.
In Table1, the absolute errors were
shown for various values of δ, x, and t with α=1, β=1, γ=0.001.
A comparison has been made between the computed
and the exact results for various values of the parameters.Example 4.2.
In Table2, the absolute errors have
been shown for various values of δ, x, and t with α=0.1, β=0.001, γ=0.0001.
A comparison between the exact and the current
results is given in Table 2. The obtained results are seen to be very accurate.Example 4.3.
For the computational work in this
example, the absolute errors have been shown for various values ofδ, x, and t with α=−0.1, β=0.1, γ=0.001. The
corresponding results of the parameters have been presented in Table 3. Again
accuracy of the present results is clearly seen for the values of the
parameters in Table 3.Example 4.4.
Here, the absolute errors have been
shown in Table4 for various values of β, x, and t with α=1, δ=1, γ=0.001.
As is the previous examples, the results of the combination of the PDQ method with
the low-storage explicit TVD-RK3 have been presented in Table 4. Comparisons of
the current results with the exact results showed that the presented results
are very accurate.Example 4.5.
Table5 shows absolute error for
various values of γ, x, and t with α=5, β=10, δ=2.
The
results of the present method for the above values of the parameters are shown
in Table 5. It is important to see from the computed results in Table 5 that the
method is very accurate.Example 4.6.
The PDQ method based on the Lagrange
interpolation functions and the PDQ method based on the Chebyshev interpolation
functions are compared in terms of the absolute errors for various values ofx and t with α=2, β=3, γ=0.001, δ=1 in Table 6. The results in both cases are
nearly the same. However, it is important to know that the grid points can be
chosen arbitrarily in the case of Lagrange interpolation. This is not the case
for the Chebyshev interpolation function, when the problems which are of
asymptotic behaviors  and need good mesh refinement near the sudden changes take
place.  This can be easily  carried out in the first case while that is not the case
in the latter one. Note that the Chebyshev polynomial interpolation function is a good choice
for
periodic problems.Example 4.7.
The maximum absolute errors
and convergence rate (CR) of the proposed method are produced for various
values ofN with α=5, β=10, γ=2, δ=1, Δt=0.00001, t=0.1 in Table 7. Accuracy of the present method is
shown by calculating the pointwise rate of convergence. Numerical rate of convergence
(CR) has also been studied  to know about the convergency of the scheme. The
rate of convergence for the scheme is calculated using the following formula: (4.2)rate  of  convergence  (CR)≈log(E(N2)/E(N1))log(N1/N2),where E(Nj) is the maximum absolute error when using the
number of grid points Nj.
Also computational orders in Table 7 show the high-order accuracy of the
present method for solving such problems. To see the behaviors at various
times, the PDQ solutions are exhibited for different times with α=5, β=10, γ=2, δ=1, N=15, Δt=0.00001 in Figure 1.
In the examples above, although a very few number of grids are used and even when the parameters are taken to increase the nonlinearity of the problem, the present results are still seen to be very accurate. Tables1–6 show that a very good approximation to the actual solution of the equations was achieved by using the method. A very good agreement between the results of the combination of the PDQ with the low-storage explicit TVD-RK3
scheme and exact solution was observed, which confirms the validity of the present method. This method is a very reliable alternative technique to some existing methods which face well-known difficulties.Table 7
The maximum absolute errors and convergence
rate (CR) of the DQM-TVD-RK3 withα=5, β=10, γ=2, δ=1, Δt=0.00001, t=0.1 in
Example 4.7.
NMaximum absolute errorCR52.103906E-03101.019676E-0714.33154.738276E-1118.93204.665157E-1316.06Figure 1
Solutions of Example4.7 at different times for α=5,β=10,γ=2,δ=1,N=15,Δt=0.00001.


## 5. Conclusions

In this paper, use of a combination of polynomial-based differential
quadrature method in space and a low-storage third-order total variation
diminishing Runge-Kutta method in time has been proposed for the generalized
Burgers-Huxley equation,
with high convergence. Comparisons of the computed results with exact
solutions showed that the method has the capability of solving the generalized
Burgers-Huxley equation
and is also capable of producing highly accurate solutions with minimal
computational effort for both time and space. It was seen that the polynomial-based differential quadrature technique approximates the exact solution very
well. Since the scheme is explicit, linearization is not needed. No requiring extra
effort to deal with nonlinear terms, ease in use, and computational
cost-effectiveness have made the current method an efficient alternative method for modelling these
nonlinear behaviors. For concrete problems where an exact solution does not
exist, the present method is a very good choice to achieve a high degree of
accuracy while dealing with the problems.



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