A uniformly convergent numerical method on non-uniform mesh for singularly perturbed unsteady...

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Applied Mathematics and Computation 195 (2008) 688–706

www.elsevier.com/locate/amc

A uniformly convergent numerical methodon non-uniform mesh for singularly perturbed

unsteady Burger–Huxley equation q

Aditya Kaushik a,*, M.D. Sharma b

a INRIA, Mathematiques Appliquees de Bordeaux, University Bordeaux 1, 351 Cours de Libration F-33405, Cedex Talence, Franceb Department of Mathematics, Kurukshetra University, Kurukshetra 136 119, Haryana, India

Abstract

In this paper, we present a numerical method which deals with the solvability of initial boundary value problem andrelated general properties of singularly perturbed Burger–Huxley equation. Singular perturbation problems are the problemsin which a very small positive parameter is multiplied to the highest order derivative term. When this parameter approachesthe limiting value of interest, i.e. �! 0, the problem exhibits boundary layers and most of the conventional methods fails tocapture this effect. Thus, the quest for some new numerical techniques that may handle the difficulties occurring due to thepresence of perturbation parameter and nonlinearity in the problem earns relevance. In this paper, one such parameterrobust numerical scheme is constructed. The first step in this direction is the discretization of the time variable using the Eulerimplicit method with a constant time step. This produces a set of nonlinear stationary singularly perturbed semidiscrete prob-lem class which is linearized further using the quasi-linearization process followed by spatial discretization using upwindfinite difference operator on a piecewise uniform mesh. An extensive amount of analysis is carried out that uses a suitabledecomposition of the error into smooth and singular component and a comparison principle combined with appropriate bar-rier functions. The error estimates are obtained, which ensures uniform convergence of the method. A set of numerical exper-iments is carried out in support of the predicted theory which validates the theoretical results computationally.� 2007 Elsevier Inc. All rights reserved.

Keywords: Burger–Huxley equation; Piecewise uniform mesh; Uniform convergence; Singular perturbation; Quasi-linearization

1. Introduction

In this paper, we construct a numerical method for solving time dependent singularly perturbed Burger–Huxley equation,

0096-3

doi:10

q Thvide N

* CoE-m

L�½u� � ut þ auux � �uxx � bð1� uÞðu� cÞu ¼ 0; ð1:1aÞ

003/$ - see front matter � 2007 Elsevier Inc. All rights reserved.

.1016/j.amc.2007.05.067

is work was completed with financial support by the UGC jointly with CSIR and HRDG Unit under Junior Research Fellowshipo. F.17-39/98(SA-I).rresponding author.ail address: [email protected] (A. Kaushik).

mailto:[email protected]

A. Kaushik, M.D. Sharma / Applied Mathematics and Computation 195 (2008) 688–706 689

D � fðx; tÞ 2 X� ð0; T � � ð0; 1Þ � ð0; T �g; ð1:1bÞuðx; 0Þ ¼ UðxÞ; x 2 X � ð0; 1Þ; ð1:1cÞuð0; tÞ ¼ uð1; tÞ ¼ 0; t 2 ð0; T �; ð1:1dÞ

where a; b P 0 and c 2 ð0; 1Þ and 0 < �� 1 are the parameters. The equation describes the interaction be-tween convection, diffusion, and reaction. When a = 0 and � = 1, Eq. (1.1) reduces to Huxley equation [1].This equation describes nerve pulse propagation in nerve fibers and wall motion in liquid crystals [2–6]:

ut � uxxðx; tÞ ¼ bð1� uÞðu� cÞu: ð1:2Þ
In the case when the epsilon is very small, i.e. �! 0, qualitative analysis of the model, which closely mimics
the ionic processes at a real nerve membrane, is performed by the authors, by means of a singular perturbationtheory in [7]. On the other hand, when b = 0, Eq. (1.1) reduces to the Burgers equation at high Reynolds num-ber that establishes a balance between time evolution, nonlinearity and diffusion:

ut þ auux ¼ �uxxðx; tÞ: ð1:3Þ
This equation was first introduced by Burger [8] primarily to throw light on turbulence described by the
interaction of two opposite effects of convection and diffusion. Burger equation arises in many physical prob-lems including one dimensional turbulence, sound waves in viscous medium [9], shock waves in viscousmedium, waves in a medium with fluid filled viscoelastic tubes and magnetohydrodynamic waves in a mediumwith finite electrical conductivity. This paper deals with the numerical study of generalized time dependent Bur-ger–Huxley equation which is singularly perturbed from mathematical perspective. Parameter dependent dif-ferential equations such as (1.1) occur in diverse area of science and engineering. The concept of singularperturbation is not new, indeed, it has been a formidable tool in the solution of some important applied math-ematical problems. In accordance with the informal principle, the behavior of solutions is governed primarilyby the highest order terms, a solution u� of the perturbed problem will often behave analytically quite differentlyfrom a solution of the original equation. When the perturbation parameter specifying the problem tends tozero, a breakdown occurs in the solution of the problem, and gives rise to boundary or the transition layersin the outflow boundary regions [10]. For higher values of � a number of solution methodologies existing inthe literature, but for sufficiently small �, the existing solution methodology fails and a discrepancy occurs inthe literature. In order to approximate the solution of such type of problems, numerical and asymptotic aretwo principal approaches. There are a wide variety of asymptotic expansion methods available for solvingthe problems of above type. But there can be difficulties in applying these asymptotic expansion methods, suchas finding the appropriate asymptotic expansions in the inner and the outer regions, which are not routine exer-cises but require skill, insight and experimentation. Then one comes down ultimately to the numerical methods.

It is a well established fact that nonlinear diffusion equations (1.1)–(1.3) with a small parameter plays animportant role in nonlinear physics. Also it is of great practical interest to study the nonlinear phenomena.In order to solve Eq. (1.1), various numerical approaches were adopted by the researchers. Ismail et al. [11]studied the adomian decomposition method for Burger–Huxley and Burger–Fisher equation. Wang et al.[1] studied the solitary wave solutions of the generalized Burger–Huxley equation and Estevez [12] presentedthe non-classical symmetries and the singular modified solutions of the Burger and Burger–Huxley equation.More recently, Javidi [13] proposed a numerical solution of generalized Burger–Huxley equation by pseudo-spectral method and Darvishi’s preconditioning.

However, it is well known for singular perturbation problem, that the numerical methods consisting ofstandard finite difference operators lead to severe restriction on the step size in order to attain the stabilityfor small values of �. To avoid these restrictions on the step size, a parameter uniform numerical scheme isconstructed. The method can easily be extended to higher dimensions and for the nonlinear problems, butthe only thing required in the process is the location and the exact width of the layer.

2. Bounds on the solution and time derivatives

Since we shall use such a priori estimates later on in several different situations, we present in this sectionthe a priori information needed in sequel. Moreover, the estimates theorems will be seen to be useful as a guidein the effective construction of the convergence and stability of the time discretization.

690 A. Kaushik, M.D. Sharma / Applied Mathematics and Computation 195 (2008) 688–706

Lemma 2.1. Let Uðx; tÞ;Wðx; tÞ 2 C2;1, i.e., twice continuously differential in space and once in time, and satisfy

the relation

jL½U�j < �L½W�; ð2:4Þ
where L is the operator defined in Eq. (1.1). If
jUj 6 W ð2:5Þ
holds on the boundary, then jUj 6 W holds throughout the domain.
Proof. Assume that ½U�W� attains the positive maximum at an interior point say (x, t) of the domain. Thenaccording to the maximum principle ðL�½U�W�Þðx;tÞ 6 0, a contradiction to the assumption (2.4). Thus ourassumption that ½U�W� attains positive maximum in the interior of the domain is wrong and hence½U�W� does not have positive maximum within the domain, also from (2.5), along the boundary of thedomain U�W 6 0, it follows that

U�W 6 0 in D: ð2:6Þ
Now suppose that �U�W attains a positive maximum in an interior point of the domain, then similarly itfollows that
�U�W 6 0 in D: ð2:7Þ
Now from (2.7), (2.6) we have, jUj 6 W holding throughout the domain. Hence the result follows. h
For obvious reasons the function W is called a barrier function for U. We impose the compatibility condi-tions u0ð0; 0Þ ¼ u0ð1; 0Þ ¼ 0, so that the data match at the corners (0, 0) and (1,0) of the domain. As an appli-cation of this lemma we immediately obtain

Theorem 2.2. There exists a number C independent of the perturbation parameter � such that for all sufficientlysmall positive values of �; juðx; tÞ � UðxÞj 6 Ct; t 2 ð0; T �. Furthermore,

juðx; tÞj 6 C; ðx; tÞ 2 X� ð0; T �: ð2:8Þ

Proof. We set gðx; tÞ ¼ uðx; tÞ � UðxÞ, then g satisfies the differential equation

L�gðx; tÞ ¼ �UxxðxÞ � aUUx þ bð1� UÞðU� cÞ ¼ H �ðx; tÞ; ð2:9Þ
where gðx; 0Þ ¼ 0; 0 < x < 1,
gð0; tÞ ¼ uð0; tÞ � Uð0Þ ¼ �Uð0Þ; gð1; tÞ ¼ uð1; tÞ � Uð1Þ ¼ �Uð1Þ; 0 6 t 6 1:

Now, clearly

L�ð�CtÞ 6 C: ð2:10Þ
Now, it can be immediately obtained from Lemma 2.1 and Eq. (2.10) that Ct is the barrier function for g forall small � provided C is chosen so large that C P supðx;tÞ2DjH �ðx; tÞj and so that Ct > jgj on the boundary.Thus
jgðx; tÞj ¼ juðx; tÞ � UðxÞj 6 Ct ð2:11Þ
is uniform D and small �. Using the properties of modulus, Eq. (2.11) is written as
juðx; tÞj � jUðxÞj 6 juðx; tÞ � UðxÞj 6 Ct; t 2 ð0; T �:
Now since UðxÞ is sufficiently smooth and x and t also lie in the bounded interval, it is clear from the aboverelations that the solution u is bounded. h
Lemma 2.3. By keeping x fixed along the line fðx; tÞ : 0 6 t 6 T g, the bound ut is given as

jutðx; tÞj 6 C: ð2:12Þ


Proof. We assume that the solution uðx; tÞ is sufficiently smooth in the domain X� ½0; T � and by mean valuetheorem, there exits a t* in the interval ðt; t þ kÞ along the line fðx; tÞ : 0 6 t 6 Tg such that

utðx; t�Þ ¼uðx; t þ kÞ � uðx; tÞ

k;

jutðx; t�Þj 62juðx; tÞj

k:

ð2:13Þ

Using the Eq. (2.8), we get

jutðx; tÞj 6 C: ð2:14Þ
Similarly we get the bounds of the uttðx; tÞ and utttðx; tÞ along the line fðx; tÞ : 0 6 t 6 Tg, i.e. we have
oiuðx; tÞoti

�� 6 C for i ¼ 0; 1; 2; 3: � ð2:15Þ

3. Time semidiscretization and quasi-linearization

The first step in the numerical solution consists of discretizing the time variable with the Euler’s implicitmethod with constant time step Dt. This produces a set of stationary singularly perturbed problems of type

� �Dt ujþ1

� �xxþ aDt½ujþ1�½ujþ1�x þ ½ujþ1� � bDtð1� ½ujþ1�Þð½ujþ1� � cÞ½ujþ1� ¼ ½uj�; ð3:16aÞ

ujþ1ð0Þ ¼ 0 and ujþ1ð1Þ ¼ 0; 0 6 j 6 M � 1: ð3:16bÞ

Now for each j, (3.16) is a singularly perturbed nonlinear ordinary differential equation. To develop a numer-ical scheme for the boundary value problem Eq. (1.1), we first linearize the nonlinear problem (3.16) by usingthe quasi-linearization process. The nonlinear differential equation is linearized around a nominal solution ofthe nonlinear differential equation which satisfies the boundary conditions. Suppose uðkÞðxÞ is the nominalsolution of the problem (3.16). Their application of the quasi-linearization process to the nonlinear problem(3.16) yields a sequence huðkÞi of linear equations determined by the following recurrence relation:

�ðu00jþ1Þðkþ1Þ þ ukþ1

j ¼ f ðkÞ þ ððu0jþ1Þðkþ1Þ � ðu0jþ1Þ

ðkÞÞf ðkÞu0jþ1þ ððujþ1Þðkþ1Þ � ðujþ1ÞðkÞÞf ðkÞujþ1

; k ¼ 0; 1; . . .

Thus (3.16) followed by quasi-linearization leads to

Lx;�ujþ1 ¼ hðxÞ ð3:17aÞ
with boundary condition
ujþ1ð0Þ ¼ 0 and ujþ1ð1Þ ¼ 0; 0 6 j 6 M � 1: ð3:17bÞ
where
Lx;� ¼ ��o2

ox2þ aðxÞ o

oxþ bðxÞ;

aðxÞ ¼ ak ¼ a ukjþ1

h iP b > 0;

bðxÞ ¼ bk ¼ 1

Dtþ a uk

jþ1

h ix� b 2 uk

jþ1

h i� 3 uk

jþ1

h i2

� cþ 2c ukjþ1

h i� �P h > 0;

hðxÞ ¼½ukþ1

j �Dtþ a½uk

jþ1�½ukjþ1�x þ bð2½uk

jþ1�3 � ð1þ cÞ½uk

jþ1�2Þ:

Remark 3.1. At first sight, the parameter Dt seems to behave like another perturbation parameter into the time
semidiscretized problem, but under enough smoothness and compatibility requirements on the data of thecontinuous problem we have seen that this parameter does not have any severe effect as that of theperturbation parameter �, to the multiscale character of the semidiscrete solutions.


Now the boundary value problem (3.17) is linear in uðkþ1Þjþ1 and now instead of solving the original nonlinear

problem (3.16), we will solve the sequence of boundary value problems for singularly perturbed second orderlinear differential equations given by (3.17), for k ¼ 0; 1; 2; . . . and 0 6 j 6 M � 1. Analytically, we requires ask tends to 1; uðkÞjþ1ðxÞ to converge to the solution ujþ1ðxÞ of the original nonlinear differential equation whilenumerically, we require that

uðkþ1Þjþ1 ðxÞ � uðkÞjþ1ðxÞ

�� < Tol:; 0 6 x 6 1;

where Tol. is a small tolerance prescribed by us. If the above condition is satisfied, we terminate the iterationand the profile uðkþ1Þ

jþi ðxÞ is the numerical solution of the nonlinear boundary value problem (3.16).Now we prove the convergence of the sequence huðkÞjþ1i. For the sake of convenience, consider

eu00jþ1 ¼ f ðujþ1Þ; ð3:18aÞ

ujþ1ð0Þ ¼ ujþ1ðbÞ ¼ 0: ð3:18bÞ

After quasi-linearization, we obtain a sequence huðkÞjþ1i of linear equations determined by the recurrence

eðu00jþ1Þðkþ1ÞðxÞ ¼ f ðuðkÞjþ1Þ þ ðu

ðkþ1Þjþ1 � uðkÞjþ1Þf 0ðu

ðkÞjþ1Þ; ð3:19Þ

uðkþ1Þjþ1 ð0Þ ¼ uðkþ1Þ

jþ1 ðbÞ ¼ 0; ð3:20Þ

where f 0ðujþiÞ ¼ df ðujþiÞ=dujþi. Suppose uð0Þjþ1ðxÞ is an initial approximation. Now from Eq. (3.19), we have

eðu00jþ1ÞðkÞðxÞ ¼ f ðuðk�1Þ

jþ1 Þ þ ðuðkÞjþ1 � uðk�1Þ

jþ1 Þf 0ðuðk�1Þjþ1 Þ: ð3:21Þ

Subtracting Eq. (3.21) from Eq. (3.19), we obtain

eðuðkþ1Þjþ1 � uðkÞjþ1Þ

00 ¼ f ðuðkÞjþ1Þ � f ðuðk�1Þjþ1 Þ � ðu

ðkÞjþ1 � uðk�1Þ

jþ1 Þf 0ðuðk�1Þjþ1 Þ þ ðu


ðkÞjþ1Þ: ð3:22Þ

Eq. (3.22) is a second order differential equation in ðuðkþ1Þjþ1 � uðkÞjþ1Þ. Converting this into an integral equation

using Green’s function, we get

eðuðkþ1Þjþ1 � uðkÞjþ1Þ ¼

Z b

0

Gðx; sÞ f ðuðkÞjþ1Þ � f ðuðk�1Þjþ1 Þ � ðu


jþ1 Þf 0ðuðk�1Þjþ1 Þ þ ðu


ðkÞjþ1Þ

h ids;

ð3:23Þ
where Gðx; sÞ is the Green’s function and is defined by
Gðx; sÞ ¼ðx�bÞs

b ; 0 6 s 6 x 6 b;

xðs�bÞb ; 0 6 x 6 s 6 b;

(ð3:24Þ

and

maxx;sjGðx; sÞj ¼ b=4: ð3:25Þ

By the mean value theorem, we have

f ðuðkÞjþ1Þ � f ðuðk�1Þjþ1 Þ ¼ ðu


jþ1 Þf 0ðuðk�1Þjþ1 Þ þ

ðuðkÞjþ1 � uðk�1Þjþ1 Þ

2

2f 00ðnÞ; ð3:26Þ

where uðk�1Þjþ1 6 n 6 uðkÞjþ1. Using Eq. (3.26) for f ðuðkÞjþ1Þ � f ðuðk�1Þ

jþ1 Þ in Eq. (3.23) followed by a simplification yields

eðuðkþ1Þjþ1 � uðkÞjþ1Þ ¼

Z b

0

Gðx; sÞ½ðuðkÞjþ1 � uðk�1Þjþ1 Þ

2f 00ðnÞ=2þ ðuðkþ1Þjþ1 � uðkÞjþ1Þf 0ðu

ðkÞjþ1Þ�ds: ð3:27Þ

Let

maxkujþ1k61

f 0ðujþ1Þ�� ¼ p; max

kujþ1k61f 00ðujþ1Þ�� ¼ q: ð3:28Þ


Now taking the maximum of the moduli over the domain of consideration on both the sides of Eq. (3.27),we get

e max06x6b

juðkþ1Þjþ1 � uðkÞjþ1j 6

Z b

0

maxx;sjGðx; sÞj max

xðuðkÞjþ1 � uðk�1Þ

jþ1 Þ2 max jf 00ðnÞj=2þmax

xjuðkþ1Þ

jþ1 � uðkÞjþ1jmax jf 0ðuðkÞjþ1Þj� �

ds:

ð3:29Þ
Using Eqs. (3.25) and (3.28) in Eq. (3.29), we obtain
max06x6b

juðkþ1Þjþ1 � uðkÞjþ1j 6

b2

4eq2

maxxðuðkÞjþ1 � uðk�1Þ

jþ1 Þ2 þ p max

xjuðkþ1Þ

jþ1 � uðkÞjþ1j� �

:

A rearrangement of terms yields

maxxjuðkþ1Þ

jþ1 � uðkÞjþ1j 6b2q

8eð1� b2p=4eÞmax

xðuðkÞjþ1 � uðk�1Þ

jþ1 Þ2: ð3:30Þ

This proves that the sequence hukjþ1i of linear equations converges quadratically provided

b2q

8eð1� b2p=4eÞ< 1: ð3:31Þ

The inequality (3.31) holds for sufficiently small b. If the interval ½0; b� is too large, we can still keepjuð1Þjþ1ðxÞ � uð0Þjþ1ðxÞj sufficiently small by choosing a judicious initial approximation uð0Þjþ1ðxÞ, which satisfies thesufficient condition for the convergence, i.e., maxxjuðkþ1Þ

jþ1 � uðkÞjþ1j is small enough for some k.

Remark 3.2. Thus we have shown that the sequence given by (3.17) of linear boundary value problemsobtained after linearizing the nonlinear problem (3.16) converges to the original nonlinear boundary valueproblem. Therefore to obtain the approximate solution of the nonlinear problem (3.16), it is sufficient toapproximate the linearized boundary value problem given by (3.17).

3.1. Convergence and stability

Lemma 3.3. If j oiuðx; tÞj 6 C, for ðx; tÞ 2 X� ð0; T � and 0 6 i 6 2. Then the local truncation error satisfies the

oti

estimate

kejþ1k1 6 CðDtÞ2:

Proof. Now, since the solution of (3.17) is smooth enough, it holds
Z t 2
uðtjÞ ¼ uðtjþ1Þ � Dtouotðtjþ1Þ þ

jþ1

tj

ðtj � nÞ o uot2ðnÞdn

¼ uðtjþ1Þ � Dtð�uxx � aðxÞux � bðxÞuðx; tÞ þ f ðx; tÞÞðtjþ1Þ þOðDt2Þ: ð3:32Þ

Subtracting (3.16) from (3.32), and note that the local truncation error ejþi � uðtjþ1Þ � ujþ1 at ðjþ 1Þth timestep is the solution of boundary value problem of type

Lx;�ejþ1 ¼ OðDt2Þ; ð3:33aÞejþ1ð0Þ ¼ ejþ1ð1Þ ¼ 0; ð3:33bÞ

where ujþ1 is the solution of the boundary value problem (3.17). Now, the operator Lx;� clearly satisfies themaximum principle: consequently

kL�1x;� k1 6 C

which ensures the stability of the semidiscretized scheme, where C is the positive constant independent of Dt.Thus we have

kejþ1k1 6 CðDtÞ2: � ð3:34Þ


Now, using the local error estimates up to the ðjþ 1Þth time step, followed by classical combination of sta-bility and consistency property of the scheme, leads us to

kEjk1 ¼Xj

k¼1

ek

1

6 CDt; ð3:35Þ

since jDt 6 T , Here C is a positive constant independent of � and Dt. Thus we have proved the following:

Theorem 3.4. Under the hypothesis of the preceding lemma, the global truncation error satisfies the estimate

kEjk1 6 CDt: ð3:36Þ

4. A priori estimates for semidiscretized problem

In this section of the paper, we meet the important and recurring concept of an a priori estimate. Also wediscuss qualitative properties of the solution of the semidiscretized problem. We now give to the maximumprinciple a mathematical basis, that can be a useful tool for deriving a priori bounds on the solutions ofthe differential equations and their derivatives. Rewriting the semidiscretized equation

Lx;� � ��½ukþ1jþ1 �xx þ akðxÞ½ukþ1

jþ1 �x þ bkðxÞ½ukþ1jþ1 � ¼ f kðxÞ;

ujþ1ð0Þ ¼ 0 and ujþ1ð1Þ ¼ 0; 0 6 j 6 M � 1

and the differential operator satisfies the following continuous maximum principal:

Lemma 4.1 (Maximum Principle). If a function U, twice continuously differentiable in space and once in time,

i.e. U 2 C2;1ðXÞ be such that Uðx; tÞP 0, for all ðx; tÞ 2 oD, then Lx;�Uðx; tÞP 0 for all ðx; tÞ 2 D implies that

Uðx; tÞP 0 for all ðx; tÞ 2 D.

Proof. Let ðx�; t�Þ be such that Uðx�; t�Þ ¼ minx2DUðx; tÞ, and suppose that Uðx�; t�Þ < 0. It is then clear fromthe hypothesis that ðx�; t�Þ 62 oD, from this it follows ðx�; t�Þ 2 D, hence Uxðx�; t�Þ ¼ 0, and Uxxðx�; t�ÞP 0,further

Lx;�Uðx�; t�Þ ¼ ��Uxxðx�; t�Þ þ akðxÞUxðx�; t�Þ þ bkðxÞUðx�; t�Þ 6 0; ð4:37Þ
which is a contradiction to our assumption, hence our assumption Uðx�; t�Þ < 0 is wrong. Hence, Uðx; tÞP 0for all x 2 D. Hence the required result holds. h
An immediate and important consequence of the maximum principle is the uniqueness and continuousdependence of the solution on the boundary values. For let hvi;jþ1i and hwi;jþ1i be the two sets of solutionsof the semidiscretized difference equations satisfying the boundary conditions. Suppose zi;jþ1 ¼ hvi;jþ1i�hwi;jþ1i, then since the semidiscrete operator is linear, we have zi;jþ1 ¼ hvi;jþ1i � hwi;jþ1i satisfies Lx;�zi;jþ1 ¼Lx;�vi;jþ1 � Lx;�wi;jþ1 ¼ 0 and obviously, z0;jþ1 ¼ 0; zN ;jþ1 ¼ 0. Thus the mesh function zi;jþ1 satisfies the hypoth-esis of discrete maximum principle that leads to

zi;jþ1 P 0 for all i ¼ 1; 2; . . . ;N : ð4:38Þ

On the other hand, if we set zi;jþ1 ¼ �ðhvi;jþ1i � hwi;jþ1iÞ, then a similar treatment to hzi;jþ1i with an applicationof maximum principle leads us to

�zi;jþ1 P 0: ð4:39Þ
Combining Eqs. (4.38) and (4.39), we have zi;jþ1 ¼ 0 and uniqueness follows immediately. Further since theproblem under consideration is linear, the existence follows from uniqueness.
The maximum principle can also be used to derive estimates on the derivatives of the solution providedsome additional conditions are placed on the equation. In the next two lemmas we establish a prioribound on the exact solution and its derivatives, of time semidiscretized problem class (3.17). It is conve-nient to use the terminology suggested by the comparison principle combined with appropriate barrierfunctions.


Lemma 4.2. Let u� be the solution of any problem from semidiscretized problem class (3.17), then

ku�k1 6 C: ð4:40Þ

Proof. Consider the barrier functions, f� ¼ 1h khk1 � u�ðx; tÞ. Note that the functions f� are non-negative at

x ¼ 0; 1 for all t 2 ð0; T �, and that for all ðx; tÞ

Lx;�f�ðx; tÞ ¼ bðxÞh�1khk1 � hðxÞP 0; ð4:41Þ

since bðxÞP h > 0 and khk1 P hðxÞ. Thus maximum principle gives f�ðx; tÞP 0 and so,

ku�k1 61

hkhk1 ¼ C for all ðx; tÞ 2 D: �

Lemma 4.3. The derivatives uðkÞ of the solution u of semidiscretized problem satisfy the following bounds:

kukkX 6 Cðffiffi�pÞ�kð1þ ð

ffiffi�pÞ�kÞmaxfkuk; khkg; k ¼ 1; 2; ð4:42Þ

ku3kX 6 Cðffiffi�pÞ�3ð1þ ð

ffiffi�pÞ�3Þmaxfkuk; khk; kh0kg: ð4:43Þ

Proof. Given any x 2 ð0; 1Þ, we can construct the neighborhood Nx ¼ ðp; p þ rÞ such that x 2 Nx andNx ð0; 1Þ. The mean value theorem implies that there exists x� 2 Nx such that

u0ðx�Þ ¼ uðp þ rÞ � uðpÞr

and so

ju0ðx�Þj 6 2kukr

: ð4:44Þ

Now we have

u0ðxÞ ¼ u0ðx�Þ þZ x

x�u00ðsÞds

and therefore from the time semidiscretized differential Eq. (3.17), we have

u0ðxÞ ¼ u0ðx�Þ � ��1

Z x

x�hðsÞdsþ ��1

Z x

x�aðsÞu0ðsÞ þ bðsÞuðsÞ½ �ds: ð4:45Þ

Now integration by parts leads us to
Z x
x�aðsÞu0ðsÞds ¼ aðsÞuj�x� �

Z �

x�a0ðsÞuðsÞ

� �: ð4:46Þ

Now using the fact that the maximum norm of a function is always greater than the value of the over the do-main of consideration, followed by calculation, yields
Z x
x�aðsÞu0ðsÞds

�� 6 ð2kak þ ka0kÞkuk: ð4:47Þ

Substituting (4.48) in (4.45), we have

u0ðxÞ ¼ u0ðx�Þ � ��1

Z x

x�hðsÞdsþ ��1

Z x

x�bðsÞuðsÞdsþ ��1 aðsÞujxx� �

Z x

x�a0ðsÞuðsÞds

� �: ð4:48Þ

Now since jx� x�j 6 r, and from (4.44) followed by (4.47) on simplification leads to

ju0ðxÞj 6 C1

rþ r�þ 1

�

� �kuk þ r

�kf k 6 ju0ðxÞj 6 C

1

rþ r�þ 1

�

� �maxfkuk; kf kg:


If we choose r ¼ffiffi�p

, then the right-hand side of the above expression is minimized with respect to r, and weobtain the required result for k = 1. Using the differential Eq. (3.17) for u, we can obtain the required boundsfor k = 2 and on differentiating again (3.17) the result for k = 3 follows. h

In order to have parameter uniform convergence results one needs to prove the stronger pointwise errorbounds. To find these sharper pointwise bounds the solution of the semidiscretized problem has to be decom-posed into smooth and singular components. The solution u� of the semidiscrete problem is decomposed into asmooth component v� and a singular component w� as follows:

u� ¼ v� þ w�;

where v� can be written in the form

v� ¼ v0 þ �v1 þ �2v2;

and v0; v1 and v2 are defined, respectively, to be the solutions of the problems

aðxÞv00ðxÞ þ bðxÞv0ðxÞ ¼ hðxÞ; x 2 X; v0ð1Þ ¼ u�ð1Þ ¼ 0; ð4:49aÞ

aðxÞv01ðxÞ þ bðxÞv1ðxÞ ¼ �v000; v1ð1Þ ¼ 0; ð4:49bÞ

� �v002ðxÞ þ aðxÞv02ðxÞ þ bðxÞv2ðxÞ ¼ �v001; v2ð0Þ ¼ v2ð1Þ ¼ 0: ð4:49cÞ

Thus the smooth component is the solution of

Lx;�v� ¼ hðxÞ; v�ð0Þ ¼ v0ð0Þ þ �v1ð0Þ; v�ð1Þ ¼ u�ð1Þ ¼ 0 ð4:50Þ
and consequently singular component satisfies
Lx;�w� ¼ 0; w�ð0Þ ¼ u�ð0Þ � v�ð0Þ; w�ð1Þ ¼ 0: ð4:51Þ

Lemma 4.4. For 0 < k 6 3; v satisfies the following estimates:
0
kvðkÞ0 k 6 C:

Proof. The problem (4.49a) can be written as

v00ðtÞ þ pðtÞv0ðtÞ ¼ f ðtÞpðtÞ; v0ð1Þ ¼ 0; ð4:52Þ

where pðtÞ ¼ bðxÞaðxÞ�1. The problem (4.52) is a first order linear differential equation in v0, in order to obtainthe solution we multiply Eq. (4.52) by exp

RpðtÞdt

� �, which followed by integration from x to 1, for some

x 2 ð0; 1Þ, yields

v0ðtÞ exp

ZpðtÞdt

� �� 1

x

¼Z 1

xf ðtÞ exp

ZpðtÞdt

� �pðtÞdt

which on simplification reduces to

v0ðxÞ ¼ �1

expR

pðtÞdt� �

t¼x

Z 1

xf ðtÞ exp

ZpðtÞdt

� �pðtÞdt: ð4:53Þ

Now since aðtÞ is sufficiently smooth and we have aðtÞ > h > 0 for all t 2 ½0; 1� also f ðtÞ is bounded for allt 2 ½0; 1�. Therefore this boundedness of v0 follows immediately. Again from Eq. (4.49a), we havev00ðxÞ ¼ f ðxÞpðxÞ � pðxÞv0ðxÞ and the boundedness of v00ðxÞ follows directly from the boundedness of v0. Boundson v000ðxÞ and v0000 ðxÞ can be easily obtained by successively differentiating Eq. (4.49a). Thus for 0 < k 6 3; v0 sat-isfies the estimates kvðkÞ0 k 6 C. h

Lemma 4.5. For 0 < k 6 3; v1 satisfies the following estimates:

kvðkÞ1 k 6 C:


Proof. Since v1 is the solution of the first order linear differential Eq. (4.49b) and since v0 is bounded, using theclassical argument used in Lemma 4.4 it is straightforward to prove that for 0 < k 6 3; v1 satisfies the requiredestimates. h

Theorem 4.6. For 0 < k 6 3; v2 satisfies the following estimates:

kvðkÞ2 k 6 C 1þ ��k exp�hð1� xÞ

�

� �� for all x 2 X: ð4:54Þ

Now we need to obtain the bound on the singular component of the solution. For if, introducing the twobarrier functions

w� :¼ C exp � hð1� xÞ�

� �� w�;

then we have w�ð0Þ ¼ C exp � hð1Þ�

�� 0 P 0, and we choose C so that it holds w�ð1Þ ¼ C � w� P 0. Further

L�W�ðxÞ ¼ h

��hþ aðnÞðxÞ� �

exp�hð1� xÞ

�

� �þ bðnÞðxÞ exp

�hð1� xÞ�

� �> 0; ð4:55Þ

since aðnÞðxÞP h > 0 and bnðxÞP b > 0. Therefore from Lemma 4.1 we obtain w� P 0 for all x 2 X whichgives us

kw�ðxÞk 6 C exp�hð1� xÞ

�

� �for all x 2 �X; ð4:56ÞZ x

0

ðaðw�ÞxÞðsÞds

�� ¼ aðnÞx w�ðxÞ �

Z x

0

aðnÞx ðw�Þx� �

ðsÞds

��; 6 C exp

�hð1� xÞ�

� �: ð4:57Þ

By mean value theorem in the interval ð0; �Þ, there exist z 2 ð0; �Þ such that

jw�ðzÞj ¼ w�ð�Þ � w�ð0Þ�

�� 6 C��1: ð4:58Þ

Integrating Eq. (4.51) w.r.t. x from 0 to x, we have

��ðw�ÞxðxÞ þ �ðw�Þxð0Þ þZ x

0

aðnÞðw�ÞxðsÞdsþZ x

0

bðnÞw�ðsÞds ¼ 0: ð4:59Þ

The above equation gives

j�ðw�Þxð0Þj ¼ �ðw�ÞxðzÞ �Z z

0

aðnÞðw�ÞxðsÞds�Z z

0

bðnÞw�ðsÞds

��: ð4:60Þ

Making use of Eqs. (4.57) and (4.58), we have

j�ðw�Þxð0Þj 6 C exp � hð1� �Þ�

� �6 C: ð4:61Þ

Now Eq. (4.59) gives the following estimate with the help of the estimates given by (4.57) and (4.61)

jðw�ðxÞÞxj 6 C��1; x 2 �X: ð4:62Þ
Using the above estimate in Eq. (4.51), we get the estimate for ðw�ðxÞÞxx, i.e.
jðw�Þxxj 6 C��2 ð4:63Þ
Similarly by differentiating Eq. (4.51) w.r.t. x and using the estimates given by Eqs. (4.62) and (4.63), we getthe estimate for ðw�Þxxx
jðw�ÞðkÞj 6 C��k for k ¼ 1; 2; 3: ð4:64Þ
Now, the above results proven by us are summarized in the following theorem:


Theorem 4.7. The solution u� of the semidiscretized problem admits the decomposition

u� :¼ v� þ w�;

where for all k; 0 6 k 6 3 and for all x 2 X, the smooth component of the solution v� satisfies

jðv�ÞðkÞðxÞj 6 Cð1þ �2�kÞ ð4:65Þ
and the singular component of the error w� satisfies
jðw�ÞðkÞj 6 C��k ð4:66Þ
for some constant C independent of the perturbation parameter.
5. The spatial discretization

5.1. Classical difference scheme

In order to have a better insight into the problem, we first consider the classical finite difference method thatconstitutes the base of the construction of the parameter uniform scheme later on. So let us first introduce anon-uniform mesh of the nodal points xi in the spatial direction. Let K1 be the number of the intervals. Define

hi ¼ xiþ1 � xi ¼ maxi

hi:

For the continuous problem we use the following difference scheme [14]:

Lj ¼ f ðxÞ: ð5:67Þ
Here,
Lj ¼ ��DþD�ui;jþ1 þ aðxiÞDþui;jþ1 þ bðxiÞui;jþ1; ð5:68aÞDþD�ui;jþ1 ¼ 2ðDþui;jþ1 � D�ui;jþ1Þ=ðhi þ hiþ1Þ; ð5:68bÞD�ui;jþ1 ¼ ðui;jþ1 � ui�1;jþ1Þ=hi; ð5:68cÞDþui;jþ1 ¼ ðuiþ1;jþ1 � ui;jþ1Þ=hi þ 1: ð5:68dÞ

The difference scheme so defined is monotone. By means of the maximum principle and taking into accountthe a priori estimates on the derivative and the solution of the semidiscretized scheme, we find that the solutionof the difference scheme converges for a fixed value of �

kuðx; tÞ � jðx; tÞk 6 Cð��2N�1 þM�1Þ: ð5:69Þ
This error bound for the classical scheme is clearly not �-uniform. The proof of (5.69) follows the lines of theclassical convergence proof for the difference schemes [14].
5.2. The �-uniformly convergent scheme

In this paper, we are considering the upwind discretization on piecewise uniform fitted meshes. Since inpractice we often want to construct a numerical method whose system matrix is a M-matrix. In this section,we discretize the boundary value problem (3.17) using the fitted mesh finite difference method composed of astandard upwind finite difference operator on a piecewise uniform mesh condensing at the boundary points.The fitted piecewise uniform mesh XN

x ¼ fxigNi¼0 on the interval [0, 1] is constructed by partitioning the interval

into two subintervals, ð0; 1� dÞ; ð1� d; 1Þ. The resulting piecewise uniform mesh depends on one parameterwhich is called the transition parameter d and is chosen such that

d � min½0:5;C� ln N �; ð5:70Þ
where C is the constant whose value depends upon the method applied. Also it is assumed that N ¼ 2k, wherek P 2 is an integer, which guarantees that there is at least one point in the boundary regions. On each of thesesubintervals a uniform mesh is then placed.


Define

xi ¼ihi for 0 6 i 6 N=2;

ð1� dÞ þ ði� N=2Þhi for N=2þ 1 6 i 6 N ;

�

where N is the number of mesh points.

hi ¼2ð1� dÞ=N for 0 6 i 6 N=2;

2d=N for N=2þ 1 6 i 6 N :

�

The fitted mesh finite difference method for the time semidiscretized problem (4) at ðjþ 1Þth time step on thepiecewise uniform mesh XN

x is defined by

LNx;�ui;jþ1 ¼ hðxiÞ; ð5:71Þ

u0;jþ1 ¼ uN ;jþ1 ¼ 0; ð5:72Þ

where the discrete operator is defined as

LNx;�ui;jþ1 ¼ ��DþD�ui;jþ1 þ aðxiÞDþui;jþ1 þ bðxiÞui;jþ1: ð5:73Þ

6. Convergence and stability analysis

6.1. Stability analysis

A discrete maximum principle can also be established directly, without appealing to the properties of theelements in the system matrix. Analogously to the continuous case, this is usually proved by the method ofcontradiction as we now show.

Theorem 6.1 (Discrete Maximum Principle). Assume that mesh function Wi;jþ1 satisfies W0;jþ1 P 0 andWN ;jþ1 P 0. Then ðLN

x;�ÞWi;jþ1 P 0 for all xi 2 XNx implies that Wi;jþ1 P 0 for all xi 2 XN

x .

Proof. Assume that there exists a positive integer k such that Wk;jþ1 ¼ min06i6NWi;jþ1 and assume Wk;jþ1 < 0.Then we have W0;jþ1 P 0 and WN ;jþ1 P 0, therefore it follows from the hypothesis that k 62 f0;Ng. AlsoWk;jþ1 �Wk�1;jþ1 6 0 and Wkþ1;jþ1 �Wk;jþ1 P 0 and note that ðLN

x;�ÞWk;jþ1 < 0 since Wk;jþ1 < 0 by assumptionand bðxiÞ > h > 0, which is a contradiction to our assumption. Therefore the assumption Wk;jþ1 < 0 is wrong,hence Wk;jþ1 P 0. We have chosen k to be fixed and arbitrary, so Wi;jþ1 P 0 for all xi 2 XN

x . h

As an immediate application of the maximum and minimum principle, we have the following uniform sta-bility estimates:

Lemma 6.2. Let Zi;jþ1 be any mesh function such that. Z0;jþ1 ¼ ZN ;jþ1 ¼ 0, then for all i; 0 6 i 6 N

jZi;jþ1j 6 h�1 max16k6N�1

jLNx;�Zk;jþ1j:

Proof. To prove the estimate we construct two mesh functions

W�i;jþ1 ¼ Mxi;jþ1 � Zi;jþ1;

where M ¼ h�1max16k6N�1jðLNx�ÞZk;jþ1j. Then we clearly have

W�0;jþ1 ¼ h�1 max16k6N�1

jLNx;�Zk;jþ1jxi;jþ1 � Z0;jþ1 ¼ 0; ð6:74aÞ

W�N ;jþ1 ¼ h�1 max16k6N�1

jLNx;�Zk;jþ1jxN ;jþ1 � ZN ;jþ1 ¼ h�1 max

16k6N�1jLN

x;�Zk;jþ1jxN ;jþ1 P 0 ð6:74bÞ


and for 1 6 i 6 N � 1,

ðLNx;�ÞW�i;jþ1 ¼ ðaðxiÞ þ bðxiÞxi;jþ1Þh�1 max

16k6N�1jðLN

x;�ÞZk;jþ1j � LNx;�Zi;jþ1 P 0;

Thus an application of the discrete maximum principle leads to the required estimates. h

6.2. Convergence analysis

The discrete problem is decomposed in an analogous manner as the second decomposition of the solution u�of the continuous problem, in order to estimate the error in the smooth and singular component separately.Thus

uN� ðxi;jþ1Þ ¼ vN

� ðxi;jþ1Þ þ wN� ðxi;jþ1Þ;

where vN� is the solution of the inhomogeneous problem

LNx;�v

N� ðxi;jþ1Þ ¼ f ðxi;jþ1Þ;

vN� ðx0;jþ1Þ ¼ 0 and vN

� ðx1;jþ1Þ ¼ 0

and wN� ðxi;jþ1Þ is the solution of the homogeneous problem

LNx;�w

N� ðxi;jþ1Þ ¼ 0;

wN� ðx0;jþ1Þ ¼ 0 and wN

� ðx1;jþ1Þ ¼ 0:

Then the error at each time step (j + 1) can be written in the form

ðuN� � u�Þðxi;jþ1Þ ¼ ðvN

� � v�Þðxi;jþ1Þ þ ðwN� � w�Þðxi;jþ1Þ:

Now in the next two results, we estimate the error in the smooth and singular component, respectively.

Lemma 6.3. At each time step (j + 1) and mesh point xi 2 XNx , the smooth component of error satisfies

jðvN� � v�Þðxi;jþ1Þj 6 CN�1: ð6:75Þ

Proof. This is obtained using the following standard stability and consistency argument, we consider the localtruncation error
� � � �
LNx;�ðvN

� � v�Þðxi;jþ1Þ ¼ ��o

2

ox2� DþD� v�ðxi;jþ1Þ þ aðxiÞ

d

dx� D� v�ðxi;jþ1Þ:

At each time step, to reduce the order of differentiation in the integral, integration by parts leads us to� � Z
D� � o
oxv�ðxi;jþ1Þ ¼

1

xi;jþ1 � xi�1;jþ1

xi;jþ1

xi�1;jþ1

ðxi�1;jþ1 � sÞv00� ðsÞds:

Taking the modulus on both the sides, we have� �
o
ox� D�

� �v�ðxi;jþ1Þ

�� 6 jv00� ðxi;jþ1Þjxi;jþ1 � xi�1;jþ1

Z xi;jþ1

xi�1;jþ1

ðs� xi�1;jþ1Þds ¼ 1

2jv00� ðxi;jþ1Þjðxi;jþ1 � xi�1;jþ1Þ: ð6:76Þ

Also, integrating by parts twice we have

DþD� � o2

ox2

� �v�ðxi;jþ1Þ ¼

1

xiþ1;jþ1 � xi�1;jþ1

1xiþ1;jþ1�xi;jþ1

R xi;jþ1

xiþ1;jþ1ðxiþ1;jþ1 � sÞ2v000� ðsÞds

� 1xi;jþ1�xi�1;jþ1

R xi�1;jþ1

xi;jþ1ðs� xi�1;jþ1Þ2v000� ðsÞds

24

35:

From this it follows that

o2

ox2� DþD�

� �v�ðxi;jþ1Þ

�� 6 v000� ðxi;jþ1Þ

�� xiþ1;jþ1 � xi�1;jþ1

1xiþ1;jþ1�xi;jþ1

R xiþ1;jþ1

xi;jþ1ðxiþ1;jþ1 � sÞ2 ds

þ 1xi;jþ1�xi�1;jþ1

R xi;jþ1

xi�1;jþ1ðs� xi�1;jþ1Þ2 ds

24

35

61

3ðxiþ1;jþ1 � xi�1;jþ1Þ v000� ðxi;jþ1Þ

�� : ð6:77Þ


Substituting (6.76) and (6.77) into (6.76) yields

jLNx;�ðvN

� � v�Þðxi;jþ1Þj 6 Cðxiþ1;jþ1 � xi�1;jþ1Þð�jv000� ðxi;jþ1Þj þ jv00� ðxi;jþ1ÞjÞ: ð6:78Þ
Now, since xiþ1;jþ1 � xi�1;jþ1 6 2N�1 holds for all choice of piecewise uniform meshes, and the bounds on thesmooth components lead to jLN
x;�ðvN� � v�Þðxi;jþiÞj 6 C N�1 by an application of the preceding Lemma 6.2, we

have jðvN� � v�Þðxi;jþ1Þj 6 jLN

x;�ðvN� � v�Þðxi;jþ1Þj 6 CN�1. h

Lemma 6.4. For all N P 4, at each time step (jþ 1) and mesh point xi 2 XNx the singular component of the error

satisfies the estimates

jðwN� � w�Þðxi;jþ1Þj 6 CN�1 ln N : ð6:79Þ

For total discrete uniform convergence, we have the following parameter uniform convergence result.

Theorem 6.5. The fitted mesh finite difference method with the standard upwind finite difference operator and the

piecewise uniform fitted mesh XNx , condensing at the boundary points, is �-uniform for the continuous problem Eq.

(1.1) provided that d is chosen to satisfy condition (5.70). Moreover, the solution u� of the continuous problem Eq.

(1.1) and the solution uN� of the discrete problem satisfy the following �-uniform error estimates:

sup0<�61

kuN� � u�k1;XN

x6 CðN�1ðln NÞ2 þM�1Þ;

where C is a constant independent of �.

7. Computational results and discussion

To examine the performance of the method, a set of numerical experiments is carried out. Choice of therange of the parameter � reflects our interests in the perturbed case. In order to tabulate the maximum errorand the order of convergence of the method, we use the double mesh principal [15]. These are defined as

EN� ¼ max

06i;j6N ;MjuNðxi; tjÞ � u2N ðxi; tjÞj; EN ¼ max

�EN� ;

where superscript indicates the number of mesh points used in the spatial direction, and tj ¼ jDt and Dt is thetime step followed by the tabulation of computed order of convergence pN

� , and the computed parameter uni-form order of convergence pN on the same lines, defined as

pN� ¼

logðEN� =E2N

� Þlog 2

; pN ¼ logðEN=E2NÞlog 2

:

Example 1. ut þ uux � �uxx ¼ ð1� uÞðu� 0:5Þu, under the homogeneous boundary conditions uð0; tÞ ¼ 0;
uð1; tÞ ¼ 0; 0 6 t 6 T , and initial condition u0ðx; 0Þ ¼ sinðpxÞ 0 6 x 6 1.
Example 2. ut þ uux ¼ �uxx, under the homogeneous boundary conditions uð0; tÞ ¼ 0; uð1; tÞ ¼ 0; 0 6 t 6 T ,and initial condition u0ðx; 0Þ ¼ xð1� x2Þ; 0 6 x 6 1.

8. Conclusion

In this paper, the initial boundary value problem for generalized singularly perturbed time dependent Burger–Huxley equation is considered. There are two factors in the problem that lead to severe difficulties in approxi-mating the solution of the problem. The first one is due to the presence of the perturbation parameter andthe second one is due to the nonlinearity in the problem. Further, the current interest in the problem is due toits great practical value. The equation admits two particular cases each for a = 0 and b = 0, respectively. Inthe first case, the equation reduces to an initial boundary value problem of reaction diffusion type, and in thesecond case, the resultant is a convection diffusion type problem. Now in the limiting case both the cases willhave different behaviors. For the problem of reaction diffusion type, the solution admits the layer at both endsof the outflow boundary region, while for the convection diffusion problem the solution exibits only one layer oneither side of the interval. The first case requires a different analysis to deal with, so it is differently treated in [7].


In order to overcome the difficulties, on a specially generated piecewise uniform mesh, a uniformly conver-gent numerical scheme using the standard finite difference upwind operator is proposed to analyze the layerbehavior. An extensive amount of theoretical analysis is carried out to obtain the stability and the error esti-mates. It was found that the method proposed is unconditionally stable and the convergence obtained is

Table 1The maximum absolute error for Example 1, when Dt ¼ 0:01

e N

8 16 32 64 128 256 512

2�2 0.046959 0.027434 0.014867 0.007745 0.003955 0.001999 0.0010052�4 0.064927 0.039037 0.021442 0.011273 0.005787 0.002932 0.0014762�6 0.069603 0.042693 0.023719 0.012526 0.006450 0.003274 0.0016502�8 0.071201 0.043571 0.024304 0.012873 0.006632 0.003368 0.0016972�10 0.071695 0.043786 0.024446 0.012957 0.006678 0.003391 0.0017092�12 0.071818 0.043839 0.024481 0.012977 0.006689 0.003397 0.0017122�14 0.071849 0.043852 0.024490 0.012983 0.006692 0.003399 0.0017132�16 0.071857 0.043856 0.024492 0.012984 0.006693 0.003399 0.0017132�18 0.071859 0.043856 0.024493 0.012984 0.006693 0.003399 0.0017132�20 0.071859 0.043857 0.024493 0.012984 0.006693 0.003399 0.001713

EN 0.071859 0.043857 0.024493 0.012984 0.006693 0.003399 0.001713


e N

8 16 32 64 128 256 512

2�2 0.045165 0.026306 0.014235 0.007411 0.003783 0.001911 0.0009612�4 0.061563 0.036563 0.020058 0.010533 0.005402 0.002736 0.0013772�6 0.066019 0.039974 0.021987 0.011611 0.005969 0.003028 0.0015252�8 0.067127 0.040840 0.022530 0.011904 0.006126 0.003108 0.0015662�10 0.067403 0.041057 0.022675 0.011982 0.006166 0.003129 0.0015762�12 0.067472 0.041111 0.022711 0.012002 0.006176 0.003134 0.0015792�14 0.067489 0.041124 0.022720 0.012006 0.006179 0.003135 0.0015802�16 0.067494 0.041128 0.022722 0.012008 0.006179 0.003136 0.0015802�18 0.067495 0.041129 0.022723 0.012008 0.006179 0.003136 0.0015802�20 0.067495 0.041129 0.022723 0.012008 0.006179 0.003136 0.001580

EN 0.067495 0.041129 0.022723 0.012008 0.006179 0.003136 0.001580


e N

8 16 32 64 128 256 512

2�2 0.044977 0.026188 0.014169 0.007377 0.003765 0.001902 0.0009562�4 0.061213 0.036307 0.019914 0.010457 0.005362 0.002716 0.0013672�6 0.065642 0.039685 0.021829 0.011520 0.005921 0.003003 0.0015122�8 0.066747 0.040547 0.022338 0.011803 0.006074 0.003082 0.0015532�10 0.067022 0.040763 0.022482 0.011881 0.006114 0.003103 0.0015632�12 0.067091 0.040817 0.022518 0.011900 0.006125 0.003108 0.0015662�14 0.067108 0.040831 0.022527 0.011905 0.006127 0.003109 0.0015662�16 0.067112 0.040834 0.022529 0.011906 0.006128 0.003109 0.0015662�18 0.067113 0.040835 0.022530 0.011907 0.006128 0.003110 0.0015662�20 0.067114 0.040835 0.022530 0.011907 0.006128 0.003110 0.001566

EN 0.067114 0.040835 0.022530 0.011907 0.006128 0.003110 0.001566


parameter uniform. Further, in Section 7, a set of numerical experiments is carried out in support of the pre-dicted theory, which were found in good accordance with the predicted theory.

The maximum absolute error is tabulated for both the examples for different time steps and can be seenfrom Tables 1–3 for example 1 and Tables 4–6 for example 2. It was observed from the tables that the order


e N

8 16 32 64 128 256 512

2�2 0.007691 0.004348 0.002311 0.001195 0.000607 0.000306 0.0001542�4 0.012118 0.006814 0.003621 0.001867 0.000948 0.000477 0.0002402�6 0.014024 0.007701 0.004086 0.002099 0.001065 0.000536 0.0002692�8 0.014374 0.007865 0.004174 0.002148 0.001089 0.000549 0.0002752�10 0.014449 0.007919 0.004194 0.002160 0.001095 0.000552 0.0002772�12 0.014467 0.007931 0.004199 0.002163 0.001097 0.000552 0.0002772�14 0.014471 0.007935 0.004200 0.002163 0.001097 0.000553 0.0002772�16 0.014472 0.007935 0.004201 0.002163 0.001097 0.000553 0.0002772�18 0.014472 0.007936 0.004201 0.002163 0.001097 0.000553 0.0002772�20 0.014472 0.007936 0.004201 0.002163 0.001097 0.000553 0.000277

EN 0.014472 0.007936 0.004201 0.002163 0.001097 0.000553 0.000277

Table 5The maximum absolute error for Example 2, when At Dt ¼ 0:001

e N

8 16 32 64 128 256 512

2�2 0.007368 0.004169 0.002217 0.001147 0.000583 0.000294 0.0001472�4 0.011327 0.006351 0.003369 0.001735 0.000880 0.000443 0.0002222�6 0.012945 0.007084 0.003736 0.001917 0.000971 0.000489 0.0002452�8 0.013244 0.007205 0.003810 0.001953 0.000990 0.000498 0.0002502�10 0.013309 0.007232 0.003828 0.001963 0.000995 0.000501 0.0002512�12 0.013325 0.007239 0.003832 0.001965 0.000996 0.000501 0.0002522�14 0.013328 0.007241 0.003833 0.001966 0.000996 0.000502 0.0002522�16 0.013329 0.007241 0.003833 0.001966 0.000996 0.000502 0.0002522�18 0.013330 0.007241 0.003833 0.001966 0.000996 0.000502 0.0002522�20 0.013330 0.007241 0.003833 0.001966 0.000996 0.000502 0.000252

EN 0.013330 0.007241 0.003833 0.001966 0.000996 0.000502 0.000252

Table 6The maximum absolute error for Example 2, when At Dt ¼ 0:0001

e N

8 16 32 64 128 256 512

2�2 0.007334 0.004150 0.002207 0.001142 0.000580 0.000292 0.0001472�4 0.011247 0.006304 0.003343 0.001722 0.000873 0.000440 0.0002212�6 0.012836 0.007022 0.003701 0.001898 0.000962 0.000484 0.0002432�8 0.013130 0.007141 0.003774 0.001935 0.000980 0.000493 0.0002482�10 0.013194 0.007168 0.003791 0.001943 0.000985 0.000496 0.0002492�12 0.013209 0.007175 0.003795 0.001946 0.000986 0.000496 0.0002492�14 0.013213 0.007176 0.003796 0.001946 0.000986 0.000497 0.0002492�16 0.013214 0.007177 0.003796 0.001946 0.000987 0.000497 0.0002492�18 0.013214 0.007177 0.003797 0.001946 0.000987 0.000497 0.0002492�20 0.013214 0.007177 0.003797 0.001946 0.000987 0.000497 0.000249

EN 0.013214 0.007177 0.003797 0.001946 0.000987 0.000497 0.000249


of convergence is very good. But the convergence of the time discretization is a little low, the reason lying inthe fact that the choice of the temporal step size is governed by the accuracy and stability considerations. Inthe ideal case, accuracy and stability restrictions on the time step should be about the same. If the time step ismuch more restricted by stability than by accuracy, efficiency may be gained by switching to a possibly more

0 0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

T=0.9

Com

pute

d S

olut

ions

0 0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

T=0.1

Com

pute

d S

olut

ions

ε=2

ε=2

ε=2

ε=2

ε=2ε=2

ε=2

ε=2

Fig. 1. Computed solutions of Example 1 for different values of �, with N = 128 and M = 10.

0 0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

=2

Com

pute

d S

olut

ions

0 0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

=2

Com

pute

d S

olut

ions

T=0.1

T=0.6

T=0.3

T=0.9

T=0.1

T=0.3

T=0.6

T=0.9

Fig. 2. Computed solution of Example 1 for different values of T, with N = 128 and M = 10.


expansive but more stable time discretization. Further, the effect of perturbation parameter and the final timeon the layer behavior can be seen from Figs. 1 and 2, and Figs. 3 and 4, for example 1 and example 2, respec-tively. It can be seen from the figures that this scheme faithfully mimics the dynamics of the correspondingdifferential equation.

0 0.5 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

T=0.1

Com

pute

d S

olut

ions

0 0.5 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

T=0.9

Com

pute

d S

olut

ions

ε=2ε=2

ε=2

ε=2

ε=2

ε=2

ε=2

ε=2

Fig. 3. Computed solutions of Example 2 for different values of �, with N = 128 and M = 10.

0 0.5 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

ε=2

Com

pute

d S

olut

ions

0 0.5 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

ε=2

Com

pute

d S

olut

ions

T=0.1

T=0.3

T=0.6

T=0.9

T=0.1

T=0.3

T=0.6

T=0.9

Fig. 4. Computed solution of Example 2 for different values of T, with N = 128 and M = 10.


References

[1] X.Y. Wang, Z.S. Zhu, Y.K. Lu, Solitary wave solutions of the generalized Burger’s–Huxley equation, J. Phys. A: Math. Gen. 23(1990) 271–274.

[2] A.C. Scott, Neurophysics, John Wiley, New York, 1977.[3] J. Satsuma, Explicit solutions of nonlinear equations with density dependent diffusion, J. Phys. Soc. Jpn. 56 (1987) 1947–1950.[4] X.Y. Wang, Nerve propagation and wall in liquid crystals, Phy. Lett. 112A (1985) 402–406.[5] X.Y. Wang, Brochard–Lager wall in liquid crystals, Phys. Rev. A 34 (1986) 5179–5182.[6] X.Y. Wang, Z.S. Zhu, Y.K. Lu, Solitary Waves solutions of the generalized Burger–Huxley equations, J. Phys.: Math. Gen. 23 (1990)

271–274.[7] A. Kaushik, M.D. Sharma, Numerical analysis of a mathematical model describes propagation of electrical pulses in a neuron,

Numer. Meth. Partial Differ. Equation, in press.[8] J.M. Burgers, A mathematical model illustrating the theory of turbulence, Adv. Appl. Mech. 1 (1948) 171–199.[9] M.J. Lighthill, Viscosity effects in sound waves of finite amplitude, in: G.K. Batchelor, R.M. Davies (Eds.), Surveys in Mechanics,

Cambridge University Press, Cambridge, 1956, pp. 250–351.[10] H.G. Roos, M. Stynes, L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations. Convection–Diffusion and

Flow Problems, Springer-Verlag, Berlin, 1996.[11] H.N.A. Ismail, K. Raslan, A.A.A. Rabboh, Adomian decomposition method for Burgers’–Huxley and Burger’s–Fisher equations,

Appl. Math. Comput. 159 (2004) 291–301.[12] P.G. Estevez, Non-classical symmetries and the singular modified the Burger’s and Burgers’–Huxley equation, J. Phys. A 27 (1994)

2113–2127.[13] M. Javidi, A numerical solution of the generalized Burger’s–Huxley equation by pseudospectral method and Darvishi’s

preconditioning, Appl. Math. Comput. 175 (2006) 1619–1628.[14] A.A. Samarskii, Theory of Difference Schemes, third ed., Nauka, Moscow, 1989.[15] E.P. Doolan, J.J.H. Miller, W.H.A. Schilders, Uniform Numerical Methods for Problems with Initial and Boundary Layers, Boole

Press, Dublin, 1980.

A uniformly convergent numerical method on non-uniform mesh for singularly perturbed unsteady...

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