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Research Article

On the Numerical Solution of One-Dimensional Nonlinear

Nonhomogeneous Burgers’ Equation

Maryam Sarboland and Azim Aminataei

Faculty of Mathematics, Department of Applied Mathematics, K. N. Toosi University of Technology, P.O. Box 16315-1618, Tehran, Iran

Correspondence should be addressed to Azim Aminataei; ataei@kntu.ac.ir

Received January ; Accepted March ; Published April

AcademicEditor:A.SalarElahi

Copyright © M. Sarboland and A. Aminataei. is is an open access article distributed under the Creative Commons

Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is

properly cited.

e nonlinear Burgers’ equation is a simple form of Navier-Stocks equation. e nonlinear nature of Burgers’ equation has been

exploited as a useful prototype dierential equation for modeling many phenomena. is paper proposes two meshfree methods

for solving the one-dimensional nonlinear nonhomogeneous Burgers’ equation. ese methods are based on the multiquadric

(MQ) quasi-interpolation operator LW2and direct and indirect radial basis function networks (RBFNs) schemes. In the present

schemes, the Taylors series expansion is used to discretize the temporal derivativeand the quasi-interpolation is used to approximate

the solution function and its spatial derivatives. In order to show the eciency of the present methods, several experiments are

considered. Our numerical solutions are compared with the analytical solutions as well as the results of other numerical schemes.

Furthermore, the stability analysis of the methods is surveyed. It can be easily seen that the proposed methods are ecient, robust,

and reliable for solving Burgers’ equation.

1. Introduction

In this paper, we consider the one-dimensional nonlinear

nonhomogeneous Burgers’ equation:

𝑡+𝑥=𝑥𝑥 +(,),∈=

[,],

0,()

with the initial condition,

,0=(),()

and the boundary conditions,

(,)=1(),

(,)=2(),()

where (),1(),and2() are known functions, is

the positive parameter that related to the Reynolds number

=1/,and(,)is a known nonhomogeneous term.

is equation was rst derived from the hydrodynamics

equations and used in surveying the laser generation of sound

[]. Later on, it was applied to other physical phenomena such

as wind forcing the buildup of water waves, electrohydrody-

namic eld in plasma physics, and design of feedback control

[–].

When (,) = 0,()isthewell-knownBurgers’equa-

tion:

𝑡+𝑥=𝑥𝑥.()

Burgers’ equation in homogeneous form was rst introduced

by Bateman [] who found its steady solutions, descriptive of

certain viscous ows. It was later presented by Burgers as one

of class of equations describing mathematical models of tur-

bulence []. In the context of gas dynamics, it was surveyed by

Hopf [] and Cole []. e homogeneous Burgers’ equation

appears in various areas of applied mathematics and physics

such as the phenomena of turbulence and supersonic ow,

heat conduction, elasticity, and fusion [–].

From an analytical point of view, the nonhomogeneous

form is poorly studied, the complete analytical solution being

closely dependent on the form of the nonhomogeneous term.

For example, Karabutov et al. [] obtained the analytical

solution of the nonhomogeneous Burgers’ equation with

(,) = sin(),>0,Dingetal.[]studiedthe

solution of () for the time-independent nonhomogeneous

term (,) = −, and Rao and Yadav []represented

Hindawi Publishing Corporation

Journal of Applied Mathematics

Volume 2014, Article ID 598432, 15 pages

http://dx.doi.org/10.1155/2014/598432

Journal of Applied Mathematics

the solutions of the nonhomogeneous Burgers’ equation for

the nonhomogeneous term (,) = /(2+1)2that

>0and >0are constant. Recently, Moreau and vall´

ee

have obtained the analytical solution of the nonhomogeneous

Burgers’ equation with an elastic forcing term (,)=−+

(),∈R[].

Up to now, various numerical methods are presented for

the homogeneous Burgers’ equation such as nite dierence,

nite element, boundary element, and collocation methods.

For a survey of these methods refer to [–] and references

cited therein. Among the methods that are mentioned above,

the spatial domain where the partial dierential governing

equations are dened is oen discretized into meshes. In

these methods, the creation of suitable meshes is very

essential for getting accurate results. However, the procedure

of mesh generation consumes a lot of time and labor for some

problems, especially for discontinuous and high gradient

problems, for which these methods will become complicate.

e root of these diculties is the use of mesh in the formu-

lation step. To avoid the mesh generation, recently, a kind of

so-called meshfree or meshless method has extended quickly.

In these methods, the scattered nodes are only used instead

of meshing the domain of the problem.

For the last years, the radial basis functions (RBFs)

method was known as a powerful tool for the scattered data

interpolation problem. e use of RBFs as a meshless process

for the numerical solution of partial dierential equations

(PDEs) is based on the collocation scheme. e meshless

methods based on RBFs were studied for approximating the

solution of PDEs since initial development of Kansa’s work

() []. Kansa’s method was extended to solve various

ordinary and partial dierential equations [–]. In these

works, the solution function is decomposed into RBFs and

its derivatives are then arrived through dierentiation that

caused the reduction in convergence rate. In order to avoid

this problem, Mai-Duy and Tran-Cong proposed an inte-

grated MQ-RBFNs scheme for the approximation of function

and its derivatives []. Numerical experiments and theoreti-

cal analysis demonstrate that for solving PDEs integrated RBF

(IRBF)procedureismoreaccurateincomparisonwithdirect

RBF (DRBF) procedure. Also, IRBF scheme is more stable

than DRBF for a range of PDEs [,].

In both DRBF and IRBF schemes, one must resolve a

linear system of equations at each time step. In the past

decade, the other meshless method was introduced by using

aMQquasi-interpolationwithoutsolvingalinearsystem

of equations. MQ quasi-interpolation is constructed directly

from linear combination of MQ-RBF and the approximated

function. In , Beaston and Powell [] presented three

univariate MQ quasi-interpolations named as LA,LB,and

LC.WuandSchaback[]proposedtheMQquasi-

interpolation LDand indicated that the scheme is shape pre-

serving and convergent. Recently, Jiang et al. []haveintro-

duced a new multilevel univariate MQ quasi-interpolation

approach with high approximation order compared with ini-

tial MQ quasi-interpolation scheme named as LWand LW2.

is approach is based on inverse multiquadric (IMQ) RBF

interpolation and Wu and Schaback’s MQ quasi-interpolation

operator LD.ChenandWuappliedinitialMQquasi-

interpolation scheme for solving one-dimensional nonlinear

homogeneous Burgers’ equation [].

In numerical solution of time dependent PDEs, such as

Burgers’ and Sine-Gordon equations, by using MQ quasi-

interpolation scheme, there is a limitation for discretization of

the temporal derivative. One has to use low order nite dif-

ference approximation for discretization of time derivatives

because one does not solve any system of equations at each

time step; otherwise one must solve a system of equations

[]. Also, large number of nodes must be used for getting

appropriate accuracy ; see [,,].

In this paper, we present two numerical methods by

using MQ quasi-interpolation for the numerical solution of

the nonhomogeneous Burgers’ equation. In both of them,

we use a two-order approximation for discretization of the

time derivative. e main idea behind the discretization is

to use more time derivatives in Taylor series expansion. is

approach was demonstrated by Lax and Wendro in nite

dierence []andusedbyDa

˘

getal.forthehomogeneous

Burgers’ equation []. By using this discretization, we have

to solve a linear system of equations at each time step that the

size of the system is equivalent to the number of the centers in

the spatial domain. Also, because the IRBFN scheme requires

fewer centers in comparison with the DRBFN scheme, we

apply MQ quasi-interpolation scheme in the indirect form in

order not to encounter with large scale matrix.

e Jiang et al. MQ quasi-interpolation operator LW2is

summation of two series that the second series coecients

are combined with rst series coecients. By giving relation

between two series coecients based on function values, we

canconvertittoacompactformbasedononeseriesanduse

it in direct and indirect forms for the numerical solution of

PDEs.

e rest of present paper is organized as follows. A brief

explanation of the MQ quasi-interpolation scheme is given

in Section .Ournumericalmethodsareappliedonthenon-

linear Burgers’ equation in Section .InSection ,thesta-

bility analysis of the methods is discussed. e results of

several numerical experiments are reported in Section .

Finally, some conclusions based on obtained results are

drawn in Section .

2. The MQ Quasi-Interpolation Scheme

In this section, three univariate MQ quasi-interpolation

schemes named as LD,LW,andLW2are described. en,

we describe our approach which converts operator LW2to

thecompactform.Moredetailscanbeseenin[–].

For a given region =[,]andanitesetofdistinct

points,

=0<1<⋅⋅⋅<𝑁=, =max

1⩽𝑖⩽𝑁 𝑖−𝑖−1, ()

if we are supplied with a function :[,]→R,quasi-

interpolation of takes the form:

L= 𝑁

𝑖=0𝑖𝑖(),()

Journal of Applied Mathematics

where each function 𝑖() is a linear combination of

the Hardy MQs basis function [],

𝑖()=2+−𝑖2,()

and low order polynomials and ∈R+is a shape parameter.

is formula requires the derivative values of at the end

points that is not convenient for practical purposes [].

Wu and Schab ack []presentedtheunivariateMQquasi-

interpolation operator LDthat is dened as

LD()=𝑁

𝑖=0𝑖

𝑖(),()

where

0()=1

2+1()−−0

21−0,

1()=2()−1()

22−1−1()−−0

21−0,

𝑖()=𝑖+1 ()−𝑖()

2𝑖+1 −𝑖−𝑖()−𝑖−1 ()

2𝑖−𝑖−1,

2−2,

𝑁−1 ()=𝑁−−𝑁−1 ()

2𝑁−𝑁−1 −𝑁−1 ()−𝑁−2 ()

2𝑁−1 −𝑁−2,

𝑁()=1

2+𝑁−1 ()−𝑁−

2𝑁−𝑁−1 .()

Suppose that {𝑘𝑖}𝑁

𝑖=1 is a smaller set from the given points

{𝑖}𝑁

𝑖=0,whereis a positive integer satisfying <and 0=

0<1<⋅⋅⋅<𝑁+1 =. Using the IMQ-RBF, the second

derivative of ()can be approximated by RBF interpolant

𝑓 as

𝑓 =𝑁

𝑗=1𝑗−𝑘𝑗, ()

where

()=2

2+23/2 ,()

and ∈R+is a shape parameter.

e coecients {𝑗}𝑁

𝑗=1 are uniquely determined by the

interpolation condition

𝑓 𝑘𝑖=𝑁

𝑗=1𝑗𝑘𝑖−𝑘𝑗= 𝑘𝑖, 1.

()

Since ()issolvable[], so

=−1

𝑋⋅

𝑋,()

where

=𝑘1,...,𝑘𝑁, =1,...,𝑁𝑇,

𝑋=𝑘𝑖−𝑘𝑗,

𝑋= 𝑘1,..., 𝑘𝑁𝑇.

()

By using the and the coecient dened in (), a function

()is constructed in the form

()=()−𝑁

𝑗=1𝑗2+−𝑘𝑗2.()

Now, the MQ quasi-interpolation operator LWby using LD

dened by ()and()onthedata{(𝑖,(𝑖))}𝑁

𝑖=1 with the

shape parameter is given by

LW()=𝑁

𝑗=1𝑗2+−𝑘𝑗2+LD().()

e shape parameters and should not be the same constant

in ().

In (), the value of

𝑥𝑘𝑗canbereplacedby

𝑥𝑘𝑗=2𝑘𝑗−𝑘𝑗−1 𝑘𝑗+1

−𝑘𝑗+1 −𝑘𝑗−1 𝑘𝑗

+𝑘𝑗+1 −𝑘𝑗𝑘𝑗−1

×𝑘𝑗−𝑘𝑗−1 𝑘𝑗+1 −𝑘𝑗𝑘𝑗+1 −𝑘𝑗−1 −1,()

when the data’s {(𝑘𝑖,(𝑘𝑖))}𝑁

𝑖=1 are given. So, if

𝑋in ()is

replaced by

𝑋=

𝑥𝑘1,...,

𝑥𝑘𝑁𝑇,()

then the quasi-interpolation operator dened by ()and()

is denoted by LW2e linear reproducing property and the

high convergence rate of LW2were also studied in [].

e operator LW2can be written in the compact form

LW2()=𝑁

𝑖=0𝑖

𝑖(),()

where the basis functions

𝑖() are obtained by

substituting (), (), and ()into(). As such, let

Journal of Applied Mathematics

={

0,1,2,3,4}and ={

2}.So=4,=1,

1=2,and

=2𝑘1−𝑘0𝑘2

−𝑘2−𝑘0𝑘1+𝑘2−𝑘1𝑘0

×𝑘1−𝑘0𝑘2−𝑘1𝑘2−𝑘0−1,()

()=()−2+−𝑘12.()

Substituting ()into()yields

()=()−2𝑘1−𝑘0𝑘2

−𝑘2−𝑘0𝑘1

+𝑘2−𝑘1𝑘0

×𝑘1−𝑘0𝑘2−𝑘1𝑘2−𝑘0−1

×2+−𝑘12.()

Hence, the substitution of ()and()into()leadsto

LW2()=2𝑘1−𝑘0𝑘2

−𝑘2−𝑘0𝑘1

+𝑘2−𝑘1𝑘0

×𝑘1−𝑘0𝑘2−𝑘1𝑘2−𝑘0−1

×2+−𝑘12

+4

𝑖=0 𝑖−2𝑘1−𝑘0𝑘2

−𝑘2−𝑘0𝑘1

+𝑘2−𝑘1𝑘0

×𝑘1−𝑘0𝑘2−𝑘1

×𝑘2−𝑘0−1

×2+𝑖−𝑘12

𝑖(),()

whereas (𝑘0)=(0),(𝑘1)=(2),and(𝑘2)=(4).

erewith, ()canberewrittenas

LW2()=22−04

−4−02+4−20

×2−04−24−0−1

×2+−22+4

𝑖=0𝑖

𝑖()

−4

𝑖=0 22−04−4−02

+4−20

×2−04−24−0−1

×2+𝑖−22

𝑖().()

Hence, the basis function

𝑖()are arrived as follows:

0()=2

2−04−0

×2+−22

−4

𝑖=02+𝑖−22

𝑖()+

0(),

2()=−2

2−04−2

×2+−22

−4

𝑖=02+𝑖−22

𝑖()+

2(),

4()=2

4−24−0

×2+−22

−4

𝑖=02+𝑖−22

𝑖()+

4(),

𝑖()=

𝑖(),=1,3.

()

By writing operator LW2in the compact form (), we

can use it in two indirect and direct forms for the numerical

solution of PDEs.

Journal of Applied Mathematics

3. The Numerical Methods

In this section, the numerical schemes are presented for

solving the nonlinear Burgers’ equation ()byusingtheMQ

quasi-interpolation LW2In our approach, the MQ quasi-

interpolation approximates the solution function and the spa-

tial derivatives of the dierential equation and Taylor’s series

expansion is employed to approximate the temporal deriva-

tive similar to the work that Da˘

getal.didin[]. e MQ

quasi-interpolation method is applied in direct and indirect

forms.

We discretize the problem using the following Taylor’s

series expansion with step size :

𝑛

𝑡=𝑛+1 −𝑛

−

2𝑛

𝑡𝑡 +2, ()

where 𝑛

𝑡=𝑡(,𝑛)and 𝑛=0+.

Dierentiating ()withrespecttotime,𝑛

𝑡𝑡 can be written

as follows: 𝑛

𝑡𝑡 =−𝑛𝑛

𝑥+𝑛

𝑥𝑥 +𝑛𝑡

=−𝑛𝑛

𝑡𝑥−𝑛

𝑥𝑛

𝑡+𝑛

𝑡𝑥𝑥 +𝑛

𝑡,()

where 𝑛

𝑡=

𝑡(,𝑛). Using forward dierence formula for

the time derivative 𝑛

𝑡in (), 𝑛

𝑡𝑡 canberewrittenas

𝑛

𝑡𝑡 =−𝑛𝑛+1

𝑥−𝑛

𝑥−𝑛

𝑥𝑛+1 −𝑛

+𝑛+1

𝑥𝑥 −𝑛

𝑥𝑥+𝑛

𝑡.()

Substituting ()into() and using the expression achieved

in (), the following time discretized form of nonlinear

Burgers’ equation is yielded:

𝑛+1 +

2𝑛𝑛+1

𝑥+

2𝑛

𝑥𝑛+1 −

2𝑛+1

𝑥𝑥

=𝑛+

2𝑛

𝑥𝑥 +𝑛+

2𝑛

𝑡. ()

3.1. e Direct MQ Quasi-Interpolation Scheme. In this

scheme, the unknown function 𝑛is approximated by using

MQ quasi-interpolation scheme, and its spatial derivatives 𝑛

𝑥

and 𝑛

𝑥𝑥 arecalculatedbydierentiatingsuchclosedformof

quasi approximation as follows:

𝑛()=𝑁

𝑖=0𝑛

𝑖

𝑖(),()

𝑛

𝑥()=𝑁

𝑖=0𝑛

𝑖

𝑖

()=𝑁

𝑖=0𝑛

𝑖

𝑖(),()

𝑛

𝑥𝑥 ()=𝑁

𝑖=0𝑛

𝑖2

𝑖

2()=𝑁

𝑖=0𝑛

𝑖𝑖(),()

where

𝑖/=

𝑖and 2

𝑖/2=𝑖.

Now, replacing ()–()into()andapplyingcolloca-

tion method yield

𝑁

𝑘=0𝑛+1

𝑘

1+

2𝑁

𝑗=0𝑛

𝑗

𝑖𝑗

𝑖𝑘

+

2𝑁

𝑗=0𝑛

𝑗

𝑖𝑗

𝑖𝑘 −

2𝑖𝑘

=𝑁

𝑗=0𝑛

𝑗

𝑖𝑗 +

2𝑖𝑗

+𝑛

𝑖+

2𝑛

𝑡𝑖, 1−1,

()

where 𝑛

𝑖=(𝑖,𝑛),𝑛

𝑡(𝑖)=

𝑡(𝑖,𝑛),

𝑖𝑗 =

𝑗(𝑖),

𝑖𝑗 =

𝑗(𝑖),and𝑖𝑗 =𝑗(𝑖), whereas, according to (), we have

𝑛

0=0,𝑛=,𝑛=1𝑛, ()

𝑛

𝑁=𝑁,𝑛=,𝑛=2𝑛. ()

Substituting ()and()into(), wherein () generates a

system of −1linear equations in −1unknown parameters

𝑛+1

𝑖.

Equation ()canbewritteninthematrixform

A1+

2u𝑛

𝑥∗A1+

2u𝑛∗D1−

2

D1u𝑛+1

=

A1+

2

D1−

22

𝑗=1𝑛+1

𝑗

Ψ𝑗∗D1+

Ψ𝑗∗A1

u𝑛

−1

𝑘=0

2

𝑗=1𝑛+𝑘

𝑗

Ψ𝑗−

2Ψ𝑗

−2

𝑗=1𝑛+1

𝑗𝑛

𝑗

Ψ𝑗∗

Ψ𝑗

−

2𝑛+1

1𝑛

2+𝑛+1

2𝑛

1

Ψ2∗

Ψ1+

Ψ1∗

Ψ2

+F𝑛

1+2

2́

F𝑛

1,()

Journal of Applied Mathematics

where symbol ∗stands for component by component multi-

plication,

A1𝑖𝑗 =

𝑖𝑗𝑁−1

𝑖,𝑗=1,D1𝑖𝑗 =

𝑖𝑗𝑁−1

𝑖,𝑗=1,

D1𝑖𝑗 =𝑖𝑗𝑁−1

𝑖,𝑗=1,

Ψ1=

01,

02,...,

0𝑁−1𝑇,

Ψ2=

𝑁1,

𝑁2,...,

𝑁𝑁−1𝑇,

Ψ1=

01,

02,...,

0𝑁−1𝑇,

Ψ1=

𝑁1,

𝑁2,...,

𝑁𝑁−1𝑇,

Ψ1=01,02,...,0𝑁−1𝑇,

Ψ2=𝑁1,𝑁2,...,𝑁𝑁−1𝑇,

F𝑛

1=𝑛1,𝑛2,...,𝑛𝑁−1𝑇,

́

F𝑛

1=𝑛

𝑡1,𝑛

𝑡2,...,𝑛

𝑡𝑁−1𝑇.()

Subsequently, ()canbewrittenas

u𝑛+1 =M−1

1N1u𝑛+M−1

1Ψ, ()

where

M1=A1+

2u𝑛

𝑥∗A1+

2u𝑛∗D1−

2

D1,

N1=A1+

2

D1−

22

𝑗=1𝑛+1

𝑗

Ψ𝑗∗D1+

Ψ𝑗∗A1,

Ψ=−1

𝑘=0

2

𝑗=1𝑛+𝑘

𝑗

Ψ𝑗−

2Ψ𝑗

+−2

𝑗=1𝑛+1

𝑗𝑛

𝑗

Ψ𝑗∗

Ψ𝑗

−1

2𝑛+1

1𝑛

2+𝑛+1

2𝑛

1

Ψ2∗

Ψ1+

Ψ1∗

Ψ2

+F𝑛

1+

2́

F𝑛

1.

()

In order to make reduction in error, the obtained 𝑖from ()

issubstitutedintherighthandsideof()thatcanbewritten

as follows:

u𝑛=A1u𝑛+𝑛

1

Ψ1+𝑛

2

Ψ2,()

and the obtained value is considered as 𝑖. erefore, from

()and(), it yields that

u𝑛+1 =A1M−1

1N1A−1

1u𝑛+A1M−1

1Ψ

−A1M−1

1N1A−1

1𝑛

1

Ψ1+𝑛

2

Ψ2+𝑛+1

1

Ψ1+𝑛+1

2

Ψ2.

()

Hence, the unknown parameters 𝑖are specied from ()

instead of ().

3.2. e Indirect MQ Quasi-Interpolation Scheme. In indirect

scheme, the highest order derivatives (second order in this

paper) of the solution function are rst approximated by (),

and their lower order derivatives and the solution function

are then obtained by symbolic integration. erefore, 𝑛

𝑥𝑥 can

be approximated by MQ quasi-interpolation LW2on data

{𝑗}𝑁−1

𝑗=1 as follows:

𝑛

𝑥𝑥 ()=𝑁−1

𝑗=1 𝑛

𝑥𝑥 𝑗

𝑗().()

Now, integrating () yields

𝑛

𝑥()=𝑁−1

𝑗=1 𝑛

𝑥𝑥 𝑗

𝑗()+1,()

𝑛()=𝑁−1

𝑗=1 𝑛

𝑥𝑥 𝑗

𝑗()+1+2.()

Equations ()–()canberewritteninthecompactformas

follows:

𝑛()=𝑁

𝑗=0𝑛

𝑗𝑗(),

𝑛

𝑥()=𝑁

𝑗=0𝑛

𝑗

𝑗(),

𝑛

𝑥𝑥 ()=𝑁

𝑗=0𝑛

𝑗

𝑗(),

()

where

𝑗()=

𝑗(), 1−1,

0()=, 𝑁()=1,

𝑗()=

𝑗(), 1−1,

0()=1,

𝑁()=0,

𝑗()=

𝑗(), 1−1,

0()=0,

𝑁()=0,

𝑛

𝑗=𝑛

𝑥𝑥 𝑗, 1−1,

𝑛

0=1,

𝑛

𝑁=2.

()

Journal of Applied Mathematics

Similar to direct scheme, replacing ()into()and()and

applying collocation method lead to

𝑁

𝑘=0𝑛+1

𝑘

1+

2𝑁

𝑗=0𝑛

𝑗

𝑖𝑗𝑖𝑘

+

2𝑁

𝑗=0𝑛

𝑗𝑖𝑗

𝑖𝑘 −

2

𝑖𝑘

=𝑁

𝑗=0𝑛

𝑗𝑖𝑗 +

2

𝑖𝑗

+𝑛

𝑖+

2𝑛

𝑡𝑖, 1−1,

𝑁

𝑘=0𝑛+1

𝑘𝑘0=𝑛+1

1,

𝑁

𝑘=0𝑛+1

𝑘𝑘𝑁=𝑛+1

2,

()

where 𝑖𝑗 =

𝑗(𝑖),

𝑖𝑗 =

𝑗(𝑖),

𝑖𝑗 =

𝑗(𝑖),and𝑛+1

𝑖=

𝑖(𝑛+1),=1,2.

Equations () generate a system of +1linear equations

in +1unknown parameters 𝑛+1

𝑖.

Similar to the direct quasi-interpolation scheme, ()can

be written in matrix form

A𝑑+A𝑏+

2u𝑛

𝑥∗A𝑑+

2u𝑛∗D2−

2

D2w𝑛+1

=A𝑑+

2

D2w𝑛+G𝑛+1 +F𝑛

2+2

2́

F𝑛

2,()

where, in this case,

A𝑑(𝑖+1)(𝑗+1) =𝑖𝑗,A𝑏(𝑖+1)(𝑗+1) =0,

D2(𝑖+1)(𝑗+1) =

𝑖𝑗,

D2(𝑖+1)(𝑗+1) =

𝑖𝑗,()

for =1,...,−1;=0,1,...,and

A𝑑(𝑖+1)(𝑗+1) =0, A𝑏(𝑖+1)(𝑗+1) =𝑖𝑗 ,

D2(𝑖+1)(𝑗+1) =0,

D2(𝑖+1)(𝑗+1) =0, ()

for =0,;=0,1,...,and

G𝑛+1 =𝑛+1

1,0,...,0,𝑛+1

2𝑇,

F𝑛

2=0,𝑛1,...,𝑛𝑁−1,0𝑇,

́

F𝑛

2=0,𝑛

𝑡1,...,𝑛

𝑡𝑁−1,0𝑇.

()

Subsequently, ()canbewrittenas

w𝑛+1 =M−1

2N2w𝑛+M−1

2G𝑛+1 +M−1

2

F𝑛,()

where

M2=A2+

2u𝑛

𝑥∗A𝑑+

2u𝑛∗D2−

2

D2,

N2=A𝑑+

2

D2,

F𝑛=F𝑛

2+2

2́

F𝑛

2,

()

and A2=A𝑑+A𝑏.From(), it yields that

u𝑛=A2w𝑛.()

Hence, the combination of ()and()isgivenas

u𝑛+1 =A2M−1

2N2A−1

2u𝑛+A2M−1

2G𝑛+1 +A2M−1

2

F𝑛.()

4. The Stability Analysis

In this section, the stability analysis from direct and indirect

quasi-interpolation schemes is presented by using spectral

radius of the amplication matrix similar to the work that

Siraj-ul-Islam et al. did in []. Let ube the exact and u∗

the numerical solution of (); then the error vector 𝑛+1 =

u𝑛+1 −u∗𝑛+1 in the direct and indirect quasi-interpolation

schemes can be written as

𝑛+1 =u𝑛+1 −u∗𝑛+1 =A1M−1

1N1A−1

1𝑛=E1𝑛,

𝑛+1 =u𝑛+1 −u∗𝑛+1 =A2M−1

2N2A−1

2𝑛=E2𝑛,()

where E1=A1M−1

1N1A−1

1and E2=A2M−1

2N2A−1

2.Forthe

stability of the numerical schemes, we must have 𝑛→0as

→∞;thatis,(E1)1,(E2)1,whichisthenecessary

and sucient condition for the numerical schemes to be

stable, where (E1)and (E2)denote the spectral radius of

the amplication matrices E1and E2,respectively.Equations

()canbewrittenas

M1A−1

1𝑛+1 =N1A−1

1𝑛,

M2A−1

2𝑛+1 =N2A−1

2𝑛.()

Equations ()canbewrittenintothefollowingformsby

using the values of M1,N1,M2,andN2dened in ()and

(): I+

2R1𝑛+1 =I+

2R2𝑛,

I+

2S1𝑛+1 =K+

2S2𝑛,()

where

R1=u𝑛

𝑥∗A1+u𝑛∗D1−

1A−1

1,

R2=

D1−2

𝑗=1𝑛+1

𝑗

Ψ𝑗∗D1+

Ψ𝑗∗A1

A−1

1,

S1=A−1

2u𝑛

𝑥∗A𝑑+u𝑛∗D2−

2,

K=A−1

2A𝑑,S2=A−1

2

2.

()

Journal of Applied Mathematics

e condition of stability will be satised if maximum eigen-

value of the matrix E1=[I+(/2)R1]−1[I+(/2)R2]and

maximum eigenvalue of the matrix E2=[I+(/2)S1]−1[K+

(/2)S2]are less than unity (in direct and indirect MQ quasi-

interpolation schemes, resp.); that is,

1+(/2)𝑅2

1+(/2)𝑅1

1,

𝐾+(/2)𝑆2

1+(/2)𝑆1

1, ()

where 𝑅1,𝑅2,𝑆1,𝑆2,and𝐾denote the eigenvalues of the

matrices R1,R2,S1,S2,andK,respectively.Itisclearfrom

() that the stability of the methods depends on the time

step and eigenvalues of the matrices 𝑅1,𝑅2,𝑆1,𝑆2,and

𝐾. e condition numbers and magnitude of the eigenvalues

of the matrices R1,R2,S1,S2,andKdepend on the shape

parameter and the number of collocation points. Hence, the

condition number and the spectral radius of the matrices

E1and E2are dependent on the shape parameter and the

number of collocation points. Since it is not possible to nd

explicit relationship among the spectral radius of the matrices

and the shape parameter, this dependency is approximated

numerically by keeping the number of collocation points

xed.

5. The Numerical Experiments

Five test experiments are studied to investigate the robustness

andtheaccuracyoftheproposedmethods.esolution

function of Burgers’ equation is approximated by direct

MQ quasi-interpolation (DMQQI) and indirect MQ quasi-

interpolation (IMQQI) schemes and the results are compared

with analytical solutions and the results in [,,,]. e

∞and 2error norms which are dened by

∞=∗𝑛−𝑛∞=max

0⩽𝑗⩽𝑁 ∗𝑛𝑗−𝑛𝑗,

2=∗𝑛−𝑛2=𝑁

𝑗=0∗𝑛𝑗−𝑛𝑗2()

areusedtomeasuretheaccuracy.Also,thestabilityanalysis

of the methods is considered for rst experiment. In all

experiments, the shape parameter is considered twice the

shape parameter and is chosen twice . Also, the centers

and the collocation points have been chosen as the same and

equidistant.

e computations associated with the experiments dis-

cussedabovewereperformedinMapleonaPCwithaCPU

of . GHZ.

Experiment 1. In this experiment, we consider nonlinear

Burgers’ equation ()with(,)=0and the initial and the

boundary conditions:

(,0)=sin (), 01,

(0,)=(1,)=0, 0. ()

e exact series solution of this experiment was given by Cole

[]:

(,)=2∑∞

𝑘=1 𝑘sin ()exp −22

0+∑∞

𝑘=1 𝑘cos ()exp −22,()

where

0=1

0exp −1−cos ()

2 ,

𝑘=21

0cos ()exp −1−cos ()

2 ,

(=1,2,3,...).

()

Numerical results are presented for = 0.1and = 0.01

with =0.001andcomparedwiththeexactsolutionsand

the results of the MQ quasi-interpolation scheme (MQQI; see

[])andadaptiveMQscheme(AMQ;see[]) for the cases

of =0.1and =0.01in Tables and ,respectively.Also,the

numericalsolutionsarecomparedwiththeresultsobtained

by MQQI scheme, AMQ scheme, and Galerkin scheme []

for =0.0001in Tabl e .Inthecases=0.1and =0.01,

the shape parameter is 0.815.Inthecase = 0.0001,the

parameter is 2.78×10−1 and 1.389×10−4 for =36and

=72, respectively. e space-time graph of the estimated

solution for =0.1and =0.01is presented in Figures and

.

Numerical comparison in these cases shows that the

obtained results, particularly in IMQQI scheme, are in good

agreement with the exact solutions and the results of the other

schemes.

Relation between the spectral radius of the matrices E1

and E2and the dierent values of the shape parameter

is shown in Table by keeping the number of collocation

points xed. It is clear from Tab l e that if the values of shape

parameter are greater than the critical value = 0.1(=

0.01), then the solution obtained from the IMQQI (DMQQI)

method breaks down and hence the IMQQI and DMQQI

methods become unstable. erefore, the interval stability

of IMQQI and DMQQI schemes is (0,0.1)and (0.004,0.01),

respectively.

ItcanbeseenfromTab l e that the schemes are very

sensitivetothevaluesoftheshapeparameterand the

interval stability of methods is a small interval.

Experiment 2. In this experiment, we consider the shock

propagation solution of the homogeneous Burgers’ equation

[] as a numerical experiment. is solution is given by

(,)=

1+/∗exp 2/4,

1, ∗=exp 1

8, 01.2. ()

e initial condition of the problem is obtained from ()at

time =1and the boundary conditions in ()canbeobtained

from the exact solution. Propagation of the shock is studied

Journal of Applied Mathematics

T : Comparison of results with the exact solution and the results in []of=0.1with =0.001at =1for dierent values of of

Experiment .

Exact MQQI []IMQQI DMQQI ErrorError

=100 =10 =20 =10 =20 IMQQI DMQQI

. . . . . . . 8.59−07 4.41−05

. . . . . . . 2.18−06 7.77−05

. . . . . . . 4.49−06 1.21−04

. . . . . . . 8.59−06 1.88−04

. . . . . . . 1.55−05 2.85−04

. . . . . . . 2.26−05 4.15−04

. . . . . . . 3.39−05 5.53−04

. . . . . . . 4.96−05 5.98−04

. . . . . . . 4.47−05 4.16−04

T : Comparison of results with the exact solution and the results in []of=0.01with =0.00at=1for dierent values of of

Experiment .

Exact MQQI []IMQQI DMQQI ErrorError

=100 =20 =30 =30 =60 IMQQI DMQQI

. . . . . . . 4.17−07 2.84−04

. . . . . . . 1.43−07 1.12−04

. . . . . . . 3.20−08 3.16−05

. . . . . . . 4.40−10 8.83−06

. . . . . . . 6.09−08 4.67−06

. . . . . . . 6.97−08 4.55−06

. . . . . . . 5.19−07 4.99−06

. . . . . . . 5.05−06 4.54−06

. . . . . . . 7.78−05 1.10−06

T : Comparison of results with the results of [,,]for=0.0001and =0.001at =1for dierent values of of Experiment .

Galerkin

method []

AMQ []MQQI[]IMQQI DMQQI

=10 =72 =36 =72 =72

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

T:espectralradiusand∞and 2error norms versus shape parameter when = 0.001, = 100,and=0.01at =1of

Experiment .

IMQQI DMQQI

(E1)

∞2(E2)

∞2

1.00−20 . 2.4628−04 5.1531−04 4.00−03 . 5.0580−02 8.8134−03

1.00−10 . 2.4628−03 5.1531−04 6.00−03 . 9.7461−03 1.6022−03

1.00−05 . 2.4642−03 5.1413−04 8.00−03 . 4.7521−03 7.5521−04

1.00−02 . 2.1783−03 2.8715−04 1.00−02 . 5.1918−03 6.7218−04

1.00−01 . 2.8855−04 3.3028−05 3.00−02 . 8.3973−01 1.2667−01

1.20−01 . 1.3242+04 1.7896+04 5.00−02 . 9.3609−01 1.7319−01

Journal of Applied Mathematics

1

0.8

0.6

0.4

0.2

0

1

0.8

0.6

0.4

0.2

0

1

0.8

0.6

0.4

0.2

0

x

t

(a)

1

0.8

0.6

0.4

0.2

0

x

t

1

0.8

0.6

0.4

0.2

0

1

0.8

0.6

0.4

0.2

0

x

t

08

0.6

0.4

0.2

0.

0.6

0.4

0.2

0

(b)

F : e space-time graph of the estimated solution of Burgers’ equation by using IMQQI for ∈[0,1],∈[0,1], = 0.1(a), and

=0.01(b) of Experiment .

1

0.8

0.6

0.4

0.2

0

1

0.8

0.6

0.4

0.2

0

1

0.8

0.6

0.4

0.2

0

06

0

.

4

0

.

2

0

0.6

0

.

4

0

.

2

x

t

(a)

1

0.8

0.6

0.4

0.2

0

1

0.8

0.6

0.4

0.2

0

1

0.8

0.6

0.4

0.2

0

x

t

(b)

F : e space-time graph of the estimated solution of Burgers’ equation by using DMQQI for ∈[0,1],∈[0,1],=0.1(a), and

=0.01(b) of Experiment .

with = 0.01for = 0.005and = 0.001.eshape

parameter is denoted by 6.0×10−2,2.4×10−2,and1.2×10−2

for =20,=50,and = 100,respectively.e2

and ∞error norms are calculated in and points

for =0.005and =0.001,respectively,andcomparedwith

the results of []inTa b l e at dierent times. e space-time

graph of the estimated solution for = 0.005is showed in

Figure .

Experiment 3. In this experiment, we study the fusion phe-

nomenon of the two solitary waves of the homogeneous

Burgers’ equation. e fusion phenomenon happens when

two or more solitons will fusion to one soliton at a specic

time. In [], Wang et al. studied the following Burgers’

equation: 𝑡+2𝑥−𝑥𝑥 =0. ()

ey obtained the two-solitary-wave solution

(,)=−1𝑘1(𝑥+𝑘1𝑡) +2𝑘2(𝑥+𝑘2𝑡)

1+𝑘1(𝑥+𝑘1𝑡) +𝑘2(𝑥+𝑘2𝑡) ,()

where 1and 2are constant. Let

→, →

2.()

Hence, () converts to Burgers’ equation form ()wherein

(,) = 0and = 0.25. In this case, two-solitary-wave

fusion happens at a specic time =0.Becausewecanshow

the fusion phenomenon, we consider an interval [−5,5]for

and . For this purpose, we introduce a new time variable

=+5andapproximatethesolution(,) of ()by

using our schemes for ∈ [0,10].en,weobtain(,)

for ∈[−5,5]. e initial condition can be obtained from the

exact solution at =0. e boundary conditions can be also

taken from the exact solution.

Journal of Applied Mathematics

T : e c o mparison o f 2and ∞errors between the numerical results by using our schemes and the results of []with= 0.01,

=0.005,and=0.001of Experiment .

=0.005 ∞2

=1.7 =2.4 =3.1 =1.7 =2.4 =3.1

IMQQI; =20 9.88−03 3.74−03 9.70−04 2.18−03 7.85−04 2.48−04

IMQQI; =50 7.63−05 2.88−05 1.45−05 1.79−05 8.35−06 4.86−06

DMQQl; =50 1.82−04 1.58−04 9.88−05 4.12−05 3.24−05 2.09−05

TCM []; =240 6.48−05 4.32−05 3.13−05 1.69−05 1.21−05 9.20−06

TGM []; =240 1.78−03 1.28−03 1.00−03 3.23−04 2.99−04 2.75−04

=0.001 =1.7 =3.0 =3.5 =1.7 =3.0 =3.5

IMQQI; =100 7.57−03 2.85−03 1.70−03 1.07−03 3.20−04 2.21−04

DMQQI; =100 1.71−02 4.03−03 5.29−03 2.26−03 5.37−04 6.43−04

TCM []; =2400 1.54−03 3.58−04 2.41−04 1.89−04 5.12−05 3.59−05

TGM []; =2400 1.56−03 3.67−04 2.49−04 1.93−04 5.54−05 4.07−05

0.4

0.3

0.2

0.1

0

0

0.2

0.4

0.6

0.8

1

1.2 1

1.5

2

2.5

3

3.5

x

t

(a)

0.4

0.3

0.2

0.1

0

0

0.2

0.4

0.6

0.8

1

1.2 1

1.5

2

2.5

3

3.5

x

t

(b)

F : e space-time graph of the estimated solution of Burgers’ equation by using IMQQI (a) and DMQQI (b) for ∈ [0,1.2]and

∈[1,3.5]of Experiment .

e 2and ∞errors of IMQQI scheme are calculated

in 1000points for 1=1,2=−1, = 0.001,and=

and listed in Ta ble . From our numerical experiment whose

results are not given here, we see that the accuracy of the

DMQQI scheme is very bad in this experiment but we can

clearly see from Ta b l e that the results of the IMQQI scheme

are in good agreement with the exact solutions.

e space-time graph of the estimated solution by using

IMQQI is presented in Figure .

Experiment 4. In this experiment, we consider Burgers’

equation ()withthenonhomogeneousterm(,) =

/(2+1)2that >0and 0are constant. So, Burgers’

equation ()hasthefollowingform:

𝑡+𝑥=𝑥𝑥 +

2+12.()

In , Rao and Yadav [] obtained the solution of Burgers’

equation () with the initial condition (,0) = 0() ∈

2(R,𝑥2/2). ey showed that the solution of ()isgivenby

(,)=0

2+1,

0=+2+, ()

for the initial condition (,0)=,>.

In this paper, we simulate this solution for =5and

=2in ∈[−1,1]. e boundary conditions in ()canbe

obtained from the exact solution (). e 2and ∞error

norms are calculated in 1000points for arbitrary ,=0.01,

and =and listed in Tabl e .Also,thespace-timegraphof

the estimated solution by using IMQQI and DMQQI schemes

is presented in Figure .

Table shows that the accuracy of the DMQQI scheme

is low even if the number of collocation points increases,

Journal of Applied Mathematics

1

0.5

0

−0.5

−1

−4

−2

0

2

4−4

−2

0

2

4

x

t

F : e space-time graph of the estimated solution of Burgers’ equation by using IMQQI for ∈[−5,5]and ∈[−5,5]of Experiment .

4

2

0

−2

−4

−1

−0.5

0

0.5

10

2

4

6

8

10

x

t

(a)

4

2

0

−2

−4

−1

−0.5

0

0.5

10

2

4

6

8

10

x

t

(b)

F : e space-time graph of the estimated solution of Burgers’ equation by using IMQQI (a) and DMQQI (b) for ∈ [−1,1]and

∈[0,10]of Experiment .

T : e 2and ∞errors of the IMQQI scheme with =0.001at dierent times of Experiment .

∞2

=−4 =0 =5 =−4 =0 =5

IMQQI; =20 3.907−04 3.242−03 1.592−02 5.660−04 3.181−03 1.728−02

IMQQI; =40 7.678−06 2.661−05 3.379−04 1.174−05 1.670−05 2.119−04

T : e c o mparison o f 2and ∞errors between the numerical results of our schemes with =0.01of Experiment .

∞2

=1 =5 =10 =1 =5 =10

IMQQI; =10 1.171−06 2.816−09 1.876−10 8.394−08 2.020−09 1.345−10

DMQQI; =20 1.533−01 4.146−02 2.159−02 7.441−02 2.102−02 1.101−02

DMQQI; =40 1.234−01 3.446−02 2.034−02 6.612−02 2.102−02 1.037−02

Journal of Applied Mathematics

2

1.5

1

0.5

0

1

2

30

1

2

3

x

t

(a)

2

1.5

1

0.5

0

1

2

30

1

2

3

x

t

(b)

F : e space-time graph of the estimated solution of Burgers’ equation by using IMQQI (a) and DMQQI (b) for =0.1,∈[0,]

and ∈[0,3]of Experiment .

T : Comparison of numerical results with the exact solutions for =1with =20and =0.001at =3of Experiment .

Exact IMQQI DMQQI Error Error

=10 =20 =20 =30 IMQQI DMQQI

. . . . . . 1.025−05 6.383−04

. . . . . . 6.026−06 4.374−04

. . . . . . 1.619−05 3.516−04

. . . . . . 3.306−05 3.083−04

. . . . . . 2.109−04 3.599−04

. . . . . . 2.169−04 1.750−02

whereas the IMQQI scheme provides the good results with

a small number of points.

Experiment 5. We nally closed our analysis by considering

the following Burgers’ equation with the nonhomogeneous

term: 𝑡+𝑥=𝑥𝑥 +sin (),>0, ()

with the initial condition

(,0)=0, ∈[0,],()

which was discussed in [], as a nonlinear model for describ-

ing hypersound generation in prescribed light eld. e

solution of the dierential equation ()canbewrittenasa

series of Mathieu’s functions []:

(,)

=2

ln ∞

𝑘=02𝑘 exp −2𝑘

42𝑘 −

2,,

()

where

2𝑘 =∫2𝜋

00/2,

∫2𝜋

02

2𝑘 /2, ()

and =/2.enotationsusedherecorrespondtothose

from the book by Strutt []. e boundary conditions are

(0,)=(,)=0, 0. ()

e numerical results compared with the exact solutions for

=1with =20and =0.1with =100in Tables and ,

respectively. e numerical calculations are performed with

=0.001and =. e space-time graph of the estimated

solutions is also presented for =0.1in Figure .

6. Conclusion

In this paper, two numerical schemes based on high accuracy

MQ quasi-interpolation scheme and RBFs approximation

schemes (IRBF and DRBF approximation schemes) have

been presented for solving the nonlinear nonhomogeneous

Burgers’ equation. e accuracy of the methods can be

increasedbyselectingtheappropriateshapeparameter.e

choice of the shape parameter is still a pendent question.

e numerical results which are given in the previous sec-

tion indicate that the performance of the methods specially

IMQQI is in excellent agreement with the exact solutions.

Tables –show that the IMQQI scheme is more accurate

than DMQQI scheme as expected, and the interval stability

of the IMQQI method is greater than the DMQQI method.

Also, the IMQQI scheme required less nodes in comparison

with the DMQQI scheme. We can even get good results with

Journal of Applied Mathematics

T : Comparison of numerical results with the exact solutions for =0.1with =100and =0.001at =3of Experiment .

Exact IMQQI DMQQI Error Error

=20 =30 =30 =60 IMQQI DMQQI

. . . . . . 2.679−03 2.696−03

. . . . . . 6.056−03 6.006−03

. . . . . . 7.894−03 9.878−03

. . . . . . 4.911−03 1.490−02

. . . . . . 1.833−03 2.195−02

. . . . . . 5.433−03 3.238−02

less number of points specially in IMQQI method but the

results are bad at the ends of interval that we can improve it

by using the knot method []. Hence, we will not encounter

with large scale matrix. Besides, we use equidistant points in

our numerical experiments but our schemes can be used for

the scattered points.

Conflict of Interests

e authors declare that there is no conict of interests

regarding the publication of this paper.

Acknowledgments

e authors are grateful to the anonymous reviewers and

theeditorDr.A.SalarElahifortheirhelpfulcommentsand

suggestions which indeed improved the quality of this paper.

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