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# On the Numerical Solution of One-Dimensional Nonlinear Nonhomogeneous Burgers’ Equation

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The nonlinear Burgers’ equation is a simple form of Navier-Stocks equation. The nonlinear nature of Burgers’ equation has been exploited as a useful prototype differential equation for modeling many phenomena. This paper proposes two meshfree methods for solving the one-dimensional nonlinear nonhomogeneous Burgers’ equation. These methods are based on the multiquadric (MQ) quasi-interpolation operator and direct and indirect radial basis function networks (RBFNs) schemes. In the present schemes, the Taylors series expansion is used to discretize the temporal derivative and the quasi-interpolation is used to approximate the solution function and its spatial derivatives. In order to show the efficiency 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 efficient, robust, and reliable for solving Burgers’ equation.
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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 
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 dierential 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 eciency 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 ecient, 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 dierence,
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 dierential governing
equations are dened is oen 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 diculties 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 dierential 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 dierential equations []. In these
works, the solution function is decomposed into RBFs and
its derivatives are then arrived through dierentiation 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
dierence []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 coecients
are combined with rst series coecients. By giving relation
between two series coecients 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 =[,]andanitesetofdistinct
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=0𝑖𝑖(),()
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 dened as
LD()=𝑁
𝑖=0𝑖
𝑖(),()
where
0()=1
2+1()−−0
21−0,
1()=2()−1()
22−11()−−0
21−0,
𝑖()=𝑖+1 ()−𝑖()
2𝑖+1 −𝑖𝑖()−𝑖−1 ()
2𝑖−𝑖−1,
2−2,
𝑁−1 ()=𝑁−−𝑁−1 ()
2𝑁−𝑁−1 𝑁−1 ()−𝑁−2 ()
2𝑁−1 −𝑁−2,
𝑁()=1
2+𝑁−1 ()−𝑁−
2𝑁−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 coecients {𝑗}𝑁
𝑗=1 are uniquely determined by the
interpolation condition
𝑓󸀠󸀠 𝑘𝑖=𝑁
𝑗=1𝑗𝑘𝑖−𝑘𝑗=󸀠󸀠 𝑘𝑖, 1.
()
Since ()issolvable[], so
=−1
𝑋⋅󸀠󸀠
𝑋,()
where
=𝑘1,...,𝑘𝑁, =1,...,𝑁𝑇,
𝑋=𝑘𝑖−𝑘𝑗,
󸀠󸀠
𝑋=󸀠󸀠 𝑘1,...,󸀠󸀠 𝑘𝑁𝑇.
()
By using the and the coecient dened in (), a function
()is constructed in the form
()=()𝑁
𝑗=1𝑗2+−𝑘𝑗2.()
Now, the MQ quasi-interpolation operator LWby using LD
dened 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=2𝑘𝑗−𝑘𝑗−1 𝑘𝑗+1
−𝑘𝑗+1 −𝑘𝑗−1 𝑘𝑗
+𝑘𝑗+1 −𝑘𝑗𝑘𝑗−1 
×𝑘𝑗−𝑘𝑗−1 𝑘𝑗+1 −𝑘𝑗𝑘𝑗+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 dened by ()and()
is denoted by LW2e 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=0𝑖
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
=2𝑘1−𝑘0𝑘2
−𝑘2−𝑘0𝑘1+𝑘2−𝑘1𝑘0
×𝑘1−𝑘0𝑘2−𝑘1𝑘2−𝑘0−1,()
()=()−2+−𝑘12.()
Substituting ()into()yields
()=()2𝑘1−𝑘0𝑘2
−𝑘2−𝑘0𝑘1
+𝑘2−𝑘1𝑘0
×𝑘1−𝑘0𝑘2−𝑘1𝑘2−𝑘0−1
×2+−𝑘12.()
LW2()=2𝑘1−𝑘0𝑘2
−𝑘2−𝑘0𝑘1
+𝑘2−𝑘1𝑘0
×𝑘1−𝑘0𝑘2−𝑘1𝑘2−𝑘0−1
×2+−𝑘12
+4
𝑖=0 𝑖−2𝑘1−𝑘0𝑘2
−𝑘2−𝑘0𝑘1
+𝑘2−𝑘1𝑘0
×𝑘1−𝑘0𝑘2−𝑘1
×𝑘2−𝑘0−1
×2+𝑖−𝑘12
𝑖(),()
whereas (𝑘0)=(0),(𝑘1)=(2),and(𝑘2)=(4).
erewith, ()canberewrittenas
LW2()=22−04
−4−02+4−20
×2−04−24−0−1
×2+−22+4
𝑖=0𝑖
𝑖()
4
𝑖=0 22−04−4−02
+4−20
×2−04−24−0−1
×2+𝑖−22
𝑖().()
Hence, the basis function
𝑖()are arrived as follows:
0()=2
2−04−0
×2+−22
4
𝑖=02+𝑖−22
𝑖()+
0(),
2()=−2
2−04−2
×2+−22
4
𝑖=02+𝑖−22
𝑖()+
2(),
4()=2
4−24−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 dierential 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+.
Dierentiating ()withrespecttotime,𝑛
𝑡𝑡 can be written
as follows: 𝑛
𝑡𝑡 =𝑛𝑛
𝑥+𝑛
𝑥𝑥 +𝑛𝑡
=−𝑛𝑛
𝑡𝑥−𝑛
𝑥𝑛
𝑡+𝑛
𝑡𝑥𝑥 +𝑛
𝑡,()
where 𝑛
𝑡=
𝑡(,𝑛). Using forward dierence formula for
the time derivative 𝑛
𝑡in (), 𝑛
𝑡𝑡 canberewrittenas
𝑛
𝑡𝑡 =−𝑛𝑛+1
𝑥−𝑛
𝑥−𝑛
𝑥𝑛+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
𝑡𝑖, 11,
()
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=1𝑛, ()
𝑛
𝑁=𝑁,𝑛=,𝑛=2𝑛. ()
Substituting ()and()into(), wherein () generates a
system of −1linear equations in −1unknown parameters
𝑛+1
𝑖.
Equation ()canbewritteninthematrixform
A1+
2u𝑛
𝑥A1+
2u𝑛D1−
2
D1u𝑛+1
=
A1+
2
D1
22
𝑗=1𝑛+1
𝑗
Ψ𝑗D1+
Ψ𝑗A1
u𝑛
1
𝑘=0
2
𝑗=1𝑛+𝑘
𝑗
Ψ𝑗−
2Ψ𝑗
−2
𝑗=1𝑛+1
𝑗𝑛
𝑗
Ψ𝑗
Ψ𝑗

2𝑛+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=
01,
02,...,
0𝑁−1𝑇,
Ψ2=
𝑁1,
𝑁2,...,
𝑁𝑁−1𝑇,
Ψ1=
01,
02,...,
0𝑁−1𝑇,
Ψ1=
𝑁1,
𝑁2,...,
𝑁𝑁−1𝑇,
Ψ1=01,02,...,0𝑁−1𝑇,
Ψ2=𝑁1,𝑁2,...,𝑁𝑁−1𝑇,
F𝑛
1=𝑛1,𝑛2,...,𝑛𝑁−1𝑇,
́
F𝑛
1=𝑛
𝑡1,𝑛
𝑡2,...,𝑛
𝑡𝑁−1𝑇.()
Subsequently, ()canbewrittenas
u𝑛+1 =M−1
1N1u𝑛+M−1
1Ψ, ()
where
M1=A1+
2u𝑛
𝑥A1+
2u𝑛D1−
2
D1,
N1=A1+
2
D1
22
𝑗=1𝑛+1
𝑗
Ψ𝑗D1+
Ψ𝑗A1,
Ψ=−1
𝑘=0
2
𝑗=1𝑛+𝑘
𝑗
Ψ𝑗−
2Ψ𝑗
+2
𝑗=1𝑛+1
𝑗𝑛
𝑗
Ψ𝑗
Ψ𝑗
1
2𝑛+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
1𝑛
1
Ψ1+𝑛
2
Ψ2+𝑛+1
1
Ψ1+𝑛+1
2
Ψ2.
()
Hence, the unknown parameters 𝑖are specied from ()
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(), 11,
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
𝑁
𝑘=0𝑛+1
𝑘
1+
2𝑁
𝑗=0𝑛
𝑗
𝑖𝑗𝑖𝑘
+
2𝑁
𝑗=0𝑛
𝑗𝑖𝑗
𝑖𝑘 −
2
𝑖𝑘
=𝑁
𝑗=0𝑛
𝑗𝑖𝑗 +
2
𝑖𝑗
+𝑛
𝑖+
2𝑛
𝑡𝑖, 11,
𝑁
𝑘=0𝑛+1
𝑘𝑘0=𝑛+1
1,
𝑁
𝑘=0𝑛+1
𝑘𝑘𝑁=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
𝑥A𝑑+
2u𝑛D2−
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,𝑛1,...,𝑛𝑁−1,0𝑇,
́
F𝑛
2=0,𝑛
𝑡1,...,𝑛
𝑡𝑁−1,0𝑇.
()
Subsequently, ()canbewrittenas
w𝑛+1 =M−1
2N2w𝑛+M−1
2G𝑛+1 +M−1
2
F𝑛,()
where
M2=A2+
2u𝑛
𝑥A𝑑+
2u𝑛D2−
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 amplication 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 u𝑛+1 in the direct and indirect quasi-interpolation
schemes can be written as
𝑛+1 =u𝑛+1 u𝑛+1 =A1M−1
1N1A−1
1𝑛=E1𝑛,
𝑛+1 =u𝑛+1 u𝑛+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 sucient condition for the numerical schemes to be
stable, where (E1)and (E2)denote the spectral radius of
the amplication 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,andN2dened in ()and
(): I+
2R1𝑛+1 =I+
2R2𝑛,
I+
2S1𝑛+1 =K+
2S2𝑛,()
where
R1=u𝑛
𝑥A1+u𝑛D1−
1A−1
1,
R2=
D12
𝑗=1𝑛+1
𝑗
Ψ𝑗D1+
Ψ𝑗A1
A−1
1,
S1=A−1
2u𝑛
𝑥A𝑑+u𝑛D2−
2,
K=A−1
2A𝑑,S2=A−1
2
2.
()
Journal of Applied Mathematics
e condition of stability will be satised 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 dened by
=𝑛−𝑛=max
0⩽𝑗⩽𝑁 𝑛𝑗−𝑛𝑗,
2=𝑛−𝑛2=𝑁
𝑗=0𝑛𝑗−𝑛𝑗2()
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-
cussedabovewereperformedinMapleonaPCwithaCPU
of . GHZ.
Experiment 1. In this experiment, we consider nonlinear
Burgers’ equation ()with(,)=0and the initial and the
boundary conditions:
(,0)=sin (), 01,
(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
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 dierent 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 dierent 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 =0.00at=1for dierent 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 dierent values of of Experiment .
Galerkin
method []
AMQ []MQQI[]IMQQI DMQQI
=10 =72 =36 =72 =72
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
T:espectralradiusandand 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
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 dierent 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 specic
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 specic 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+12.()
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 dierent 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 dierential equation ()canbewrittenasa
series of Mathieu’s functions []:
(,)
=2
ln
𝑘=02𝑘 exp −2𝑘 
42𝑘 −
2,,
()
where
2𝑘 =∫2𝜋
00/2,
∫2𝜋
02
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 conict of interests
regarding the publication of this paper.
Acknowledgments
e authors are grateful to the anonymous reviewers and
suggestions which indeed improved the quality of this paper.
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Even if numerical simulation of the Burgers’ equation is well documented in the literature, a detailed literature survey indicates that gaps still exist for comparative discussion regarding the physical and mathematical significance of the Burgers’ equation. Recently, an increasing interest has been developed within the scientific community, for studying non-linear convective–diffusive partial differential equations partly due to the tremendous improvement in computational capacity. Burgers’ equation whose exact solution is well known, is one of the famous non-linear partial differential equations which is suitable for the analysis of various important areas. A brief historical review of not only the mathematical, but also the physical significance of the solution of Burgers’ equation is presented, emphasising current research strategies, and the challenges that remain regarding the accuracy, stability and convergence of various schemes are discussed. One of the objectives of this paper is to discuss the recent developments in mathematical modelling of Burgers’ equation and thus open doors for improvement. No claim is made that the content of the paper is new. However, it is a sincere effort to outline the physical and mathematical importance of Burgers’ equation in the most simplified ways. We throw some light on the plethora of challenges which need to be overcome in the research areas and give motivation for the next breakthrough to take place in a numerical simulation of ordinary / partial differential equations.
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This paper develops an efficient numerical meshless method to solve the nonlinear generalized Burgers–Huxley equation (NGB-HE). The proposed method approximates the unknown solution in the two stages. First, the θ-weighted finite difference technique is adopted to discretize the temporal dimension. Second, a combination of the multiquadric quasi-interpolation and pseudospectral (denoted by MQQI-PS) is constructed to approximate the spatial derivatives. In addition, a cross-validation technique is used to find the shape parameter value. Finally, numerical results are illustrated to show the accuracy and efficiency of the MQQI-PS method. © 2022, The Author(s), under exclusive licence to Islamic Azad University.