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A Meshless Method Based on the Fundamental Solution and Radial Basis Function for Solving an Inverse Heat Conduction Problem

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Research Article
A Meshless Method Based on the FundamentalSolution and Radial Basis Function for Solving an Inverse Heat Conduction Problem
Muhammad Arghand and Majid Amirfakhrian
Department of Mathematics, Central ehran Branch, Islamic Azad University, ehran, Iran
Correspondence should be addressed to Majid Amirakhrian; amirakhrian@iauctb.ac.irReceived November ; Accepted March Academic Editor: Alkesh PunjabiCopyright © M. Arghand and M. Amirakhrian. Tis is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the srcinal work isproperly cited.We propose a new meshless method to solve a backward inverse heat conduction problem. Te numerical scheme, based on theundamental solution o the heat equation and radial basis unctions (RBFs), is used to obtain a numerical solution. Since thecoeﬃcients matrix is ill-conditioned, the ikhonov regularization (R) method is employed to solve the resulted system o linearequations. Also, the generalized cross-validation (GCV) criterion is applied to choose a regularization parameter. A test problemdemonstrates the stability, accuracy, and eﬃciency o the proposed method.
1. Introduction
ransient heat conduction phenomena are generally described by the parabolic heat conduction equation, andi the initial temperature distribution and the boundary conditions are specied, then this, in general, leads to awell-posed problem which may easily be solved numerically by using various numerical methods. However, in many practicalsituationswhendealingwithaheatconductingbody itisnotalwayspossibletospeciytheboundaryconditionsorthe initial temperature. Hence, we are aced with an inverseheat conduction problem. Inverse heat conduction problems(IHCPs) occur in many branches o engineering and science.Mechanical and chemical engineers, mathematicians, andspecialistsinothersciencesbranchesareinterestedininverseproblems. From another point o view, since the existence,uniqueness, and stability o the solutions o these problemsare not usually conrmed, they are generally identiedas ill-posed [–]. According to the act that unknown
solutions o inverse problems are determined throughindirect observable data which contain measurement errors,such problems are naturally unstable. In other words, themain diﬃculty in the treatment o inverse problems is theunstability o their solution in the presence o noise inthe measured data. Hence, several numerical methods havebeen proposed or solving the various kinds o inverseproblems. In addition to, ill-posedness o these kinds o prob-lems,ill-conditioningotheresultingdiscretizedmatrixromthe traditional methods like the nite diﬀerences method(FDM) [], the nite element method (FEM), and so orth[, ], is the main problem making all numerical algorithms
or determining the solution o these kinds o problems.Accordingly, within recent years, meshless methods, as themethod o undamental solution (MFS), radial basis unc-tions (RBFs) method, and some other methods, have beenapplied by many scientists in the eld o applied sciencesand engineering [–]. Kupradze and Aleksidze [] rst
introducedMFSwhichdenesthesolutionotheproblemasalinearcombinationoundamentalsolutions.Honetal.[–] applied the MFS to solve some inverse heat conductionproblems. In s, Kansa applied RBFs method to solve thediﬀerent types o partial diﬀerential equations [, ]. Afer
that, Kansa and many scientists regarded RBFs method tosolve diﬀerent types o mathematical problems rom partialor ordinary diﬀerential equations to integral equations[–]. Following their works, during recent years, many
researchers have made some changes in RBFs and MFSmethods and have developed advance methods to solve
Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2015, Article ID 256726, 8 pageshttp://dx.doi.org/10.1155/2015/256726
Advances in Mathematical Physicssome o these kinds problems [, ]. Consequently, in this
work, we will present a meshless numerical scheme, basedon combining the radial basis unction and the undamentalsolution o the heat equation, in order to approximate thesolution o a backward inverse heat conduction problem(BIHCP),theprobleminwhichanunknowninitialconditionor/and temperature distribution in previous time will bedetermined. Tis kind o problem may emerge in many practical application areas such as archeology and mantleplumes[].Ontheotherhand,sincethesystemothelinearequations obtained rom discretizing the problem in thepresentedmethodisill-conditioned,ikhonovregularization(R) method is applied in order to solve it. Te generalizedcross-validation (GCV) criterion has been assigned to adoptan optimum amount o the regularization parameter. Testructure o the rest o this work is organized as ollows: InSection , we represent the mathematical ormulation o theproblem. Te method o undamental solutions, radial basisunctions method, and method o undamental solution-radial basis unctions (MFSRBF) are described in Section .Section embraces ikhonov regularization method with arule or choosing an appropriate regularization parameter.In Section , we present the obtained numerical results o solving a test problem. Section ends in a brie conclusionand some suggestions.
2. Mathematical Formulation of the Problem
In this section, we consider the ollowing one-dimensionalinverse heat conduction problem:
−
2
= 0, (,) ∈ Ω = (0,) × 0,
max
,
()with the ollowing initial and boundary conditions:
(,0) = (), ∈ 0,
0
,(), ∈
0
,,(0,) = (), ∈ 0,
max
,(,) = (), ∈ 0,
max
,
()where
()
and
()
are considered as known unctions and
max
is a given positive constant, while
()
and
()
areregardedasunknownunctions.So,inordertoestimate
()
and
()
, we consider additional temperature measurementsand heat ux given at a point
0
,
0
∈ (0,1)
, as overspeciedconditions:
0
, = (), ∈ 0,
max
,
0
, = ℎ(), ∈ 0,
max
.
()o solve the above problem, at rst, we divide the problem()–() into two separate problems. Te problem
is asollows:
−
2
= 0, (,) ∈ Ω
= 0,
0
× 0,
max
,(,0) = (), ∈ 0,
0
,(0,) = (), ∈ 0,
max
,
0
, = (), ∈ 0,
max
,
0
, = ℎ(), ∈ 0,
max
,
()and the problem
is considered as ollows:
−
2
= 0, (,) ∈ Ω
=
0
, × 0,
max
,(,0) = (), ∈
0
,,
0
, = (), ∈ 0,
max
,
0
, = ℎ(), ∈ 0,
max
,(,) = (), ∈ 0,
max
.
()Obviously, the problems A and B are considered as IHCP,where
()
,
(,)
and
()
,
(,)
are unknown unctionsin the problems A and B, respectively.
3. Method of Fundamental Solutions andMethod of Radial Basis Functions
In this section, we introduce the numerical scheme orsolving the problem ()–() using the undamental solutions
and radial basis unctions.
..MethodofFundamentalSolutions.
Teundamentalsolu-tion o () is presented as below:
(,) = 12√
−
2
/(4
2
)
(),
()where
()
is Heaviside unit unction. Assuming that
>
max
is a constant, it can be demonstrated that the time shifunction
(,) = (, + )
()is also a nonsingular solution o () in the domain
Ω
.In order to solve an IHCP by MFS, as the problem
,since the basis unction
satises the heat equation ()automatically, we assume that
(
,
)}
=1
is a given seto scattered points on the boundary
Ω
. An approximatesolution is dened as a linear combination o
as ollows:
Φ(,) =
=1
−
, −
,
()where
(,)
isgivenby ()and
’sareunknowncoeﬃcientswhich can be determined by solving the ollowing matrixequation:
A
Λ =
b
,
()where
Λ = [
1
,...,
]
and
b
is a
× 1
known vector.Also, as the undamental unctions are the solution o theheat equation, only the initial and boundary conditions arepracticed to make the system o linear equations; that is,
A
is
Advances in Mathematical Physics
: Some well-known radial basis unctions.Innitely smooth RBFs
()
Gaussian (GA)
(−
2
)
Multiquadrics (MQ)
√
2
+
2
Inverse multiquadrics (IMQ)
√
2
+
2
−1
Inverse quadric (IQ)
(
2
+
2
)
−1
a
×
square matrix which is dened using the initial andthe boundary conditions as ollows:
A
=
,
= Φ(,), (,) ∈ Ω
, , = 1,...,.
()For more details, see [, ].
.. Method of Radial Basis Functions.
In this section, weconsider RBF method or interpolation o scattered data.Supposethat
x
∗
and
x
areaxedpointandanarbitrarypointin
R
, respectively. A radial unction
∗
is dened via
∗
=
∗
()
, where
= ‖
x
−
x
∗
‖
2
. Tat is, the radial unction
∗
dependsonlyonthedistancebetween
x
and
x
∗
.Tisproperty implies that the RBFs
∗
are radially symmetric about
x
∗
.Somewell-knowninnitelysmoothRBFsaregiveninable.As it is observed, these unctionsdepend on a ree parameter
,knownastheshapeparameter,whichhasanimportantrolein approximation theory using RBFs.Let
x
}
=1
be a given set o distinct points in domain
Ω
in
R
. Te main idea o using the RBFs is interpolation witha linear combination o RBFs o the same types as ollows:
Ψ(
x
) =
=1
(
x
),
()where
(
x
) = (‖
x
−
x
‖)
and
’s are unknown coeﬃcientsor
= 1,2,...,
. Assume that we want to interpolate thegiven values
= (
x
)
,
= 1,2,...,
. Te unknowncoeﬃcients
’sareobtained,sothat
Ψ(
x
) =
,
= 1,2,...,
, which results in the ollowing system o linear equations:
A
Λ =
b
,
()where
Λ = [
1
,...,
]
,
b
= [
1
,...,
]
, and
A
= [
]
or
, = 1,2,...,
with entries
=
(
x
)
. Generally,the matrix
A
has been shown to be positive denite (andthereore nonsingular) or distinct interpolation points orinnitely smooth RBFs by Schoenberg’s theorem []. Also,using Micchelli’s theorem [], it is shown that
A
is aninvertible matrix or a distinct set o scattered nodes in thecase o MQ-RBF.
4. Method of Fundamental Solutions-RadialBasis Functions
Tis section is specied to introduce the numerical schemeor solving the problem ()–() using the undamental solu-
tions and radial basis unctions. At rst, we are going togure out how radial basis unctions and the undamentalsolutions are applied to approximate the solution o theproblem
, . Similarly, this method will be used or solvingthe problem
on domain
Ω
. Because the one-dimensionalheat conduction equation depends on both parameters o
and
,
scattered nodes are considered in the domain
Ω
=[0,
0
] × [0,
max
+ ]
. We assume that
Ξ
=
,
=1
, Ξ
=
,
=1
()aretwosetsoscatteredpointsinthedomain
Ω
,where
=
+
and
Ξ
and
Ξ
are the interior and the boundary points in domains
Ω
= [0,
0
] × [0,
max
+ ]
and
Ω
=[0,
0
] × [0,
max
]
, respectively. Also, we assume that
Ξ
= Γ
1
∪ Γ
2
∪ Γ
3
,
()where
Γ
1
=
,
:
= 0, 0 ⩽
⩽
0
, = 1,...,,Γ
2
= {
,
:
=
0
, 0 <
⩽
max
, = 1,...,,Γ
3
=
,
:
=
0
, 0 <
⩽
max
, = 1,...,,
()so that
= + +
.We suppose that the solution o the problem in
Ω
canbe expressed as ollows:
̃(,) =
=1
(,) +
=
+1
(,),
()where
(,) = ( −
, −
+ )
and
(,)
is theundamentalsolutionotheheatequationwhichisdescribedin Section .. Also,
= (‖(,) − (
,
)‖
2
)
and
is aradial unction which is dened in Section .. o determinethecoeﬃcientsin(),weimposetheapproximatesolution
̃
tosatisythegivenpartialdiﬀerentialequationwiththeotherconditions at any point
(,) ∈ Ξ
∪ Ξ
, so we achieve theollowing system o linear equations:
A
Λ =
b
,
()where
Λ = [
1
,...,
]
,
b
= [0,(
),(
),ℎ(
)]
,andis
×1
zeromatrix.Also,the
×
matrix
A
canbesubdividedinto two submatrices as ollows:
A
=
A
A
,
()where
A
= [
(1)
]
is the
×
obtained submatrix o applying the interior points whose entries are dened asollows:
(1)
= 0, = 1,...,
, = 1,...,
,
()because as already mentioned above, the undamental unc-tions
satisy the heat equation and also
(1)
= −
2
2
2
(,), = 1,...,
, , =
+ 1,...,, (,) ∈ Ξ
.
()
Advances in Mathematical PhysicsUsing the boundary points o domain
Ω
, namely,
Ξ
, thesubmatrix
A
= [
(2)
]
results, where
(2)
=
(,), = 1,...,, , = 1,...,
, (,) ∈ Γ
1
,
(2)
=
(,), = 1,...,, , =
+ 1,...,, (,) ∈ Γ
1
,
(2)
=
(,), = + 1,..., + , , = 1,...,
, (,) ∈ Γ
2
,
(2)
=
(,), = + 1,..., + , , =
+ 1,...,, (,) ∈ Γ
2
,
(2)
=
(,),= + + 1,..., + + , ,=1,...,
, (,)∈Γ
3
,
(2)
=
(,), = + + 1,..., + + ,, =
+ 1,...,,(,) ∈ Γ
3
.
()It is essential to keep in mind that inherent errors inmeasurement data are inevitable. On the other hand, as itwas mentioned, a radial basis unction depends on the shapeparameter
, which has an eﬀect on the condition number o
A
. Tereore, the obtained system o linear equations () isill-conditioned. So, it cannot be solved, directly. Since, smallperturbation in initial data may produce a large amount o perturbation in the solution, we use the rand unction inmatlab in the numerical example presented in Section andwe produce noisy data as the ollows:
̃
=
(1 + ⋅
rand
()), = 1,...,,
()where
is the exact data, rand
()
is a random numberuniormlydistributedin
[−1,1]
,andthemagnitude
displaysthe noise level o the measurement data.
5. Regularization Method
Solving the system o linear equations () usually does notlead to accurate results by most numerical methods, becausecondition number o matrix
A
is large. It means that theill-conditioning o matrix
A
makes the numerical solutionunstable. Now, in methods as the proposed method in thisstudy, which is based on RBFs, the condition number o
A
depends on some actors such as the shape parameter
, aswell. On the other hand, or xed values o the shape param-eter
, the condition number increases with the number o scattered nodes
. In practice, the shape parameter mustbe adjusted with the number o interpolating points. Also,the accuracy o radial basis unctions relies on the shapeparameter. So, in case a suitable amount o it is chosen,the accuracy o the approximate solution will be increased.Despite various research works which are done, ndingthe optimal choice o the shape parameter is still an openproblem [–]. Accordingly, some regularization methods
arepresentedtosolvesuchill-conditionedsystems.ikhonov regularization (R) method is mostly used by researchers[]. In this method, the regularized solution
Λ
or thesystem o linear equations () is explained as the solution o the ollowing minimization problem:min
Λ
A
Λ − ̃
b
2
+
2
‖Λ‖
2
, > 0,
()where
‖ ⋅ ‖
denotes the Euclidean norm and
is called theregularization parameter. Some methods such as
-curve[], cross-validation (CV), and generalized cross-validation(GCV) [] are carried out to determine the regularizationparameter
or the R method. In this work, we apply (GCV) to obtain regularization parameter. In this method,regularization parameter
minimizes the ollowing (GCV)unction:
() =
A
Λ
− ̃
b
2
trace
I
−
AA
2
, > 0,
()where
A
= (
A
tr
A
+
2
I
)
−1
A
tr
.Te regularized solution () is shown by
Λ
∗
= [
∗
1
,...,
∗
]
,inwhich
∗
isaminimizero
.Tentheapproximatesolution or the problem is written as ollows:
̃
∗
(,) =
=1
∗
(,) +
=
+1
∗
(,),() =
=1
∗
0 −
, −
+
=
+1
∗
0 −
, −
.
()
6. Numerical Experiment
Inthissection,weinvestigatetheperormanceandtheability o the present method by giving a test problem. Tereore,in order to illustrate the eﬃciency and the accuracy o theproposed method along with the R method, initially, wedene the root mean square (RMS) error, the relative rootmean square (RES) error, and the maximum absolute error
∞
()
as ollows:RMS
() = 1
=1
,
− ̃
∗
,
2
,
RES
() = ∑
=1
,
− ̃
∗
,
2
∑
=1
,
2
,
∞
() =
max
1⩽⩽
,
− ̃
∗
,
,
()
Advances in Mathematical Physics
: Te obtained values o RMS
(),
RES
(),
RMS
()
, and RES
()
using MFSRBF or various values o
,
0
= 0.5
, and
= 0.01
by solving problem
A
.
RMS
()
RES
()
∞
()
RMS
()
RES
()
∞
()
cond
()
. . . . . . .
1.6111 + 092.5
. . . . . .
3.5475 + 105
. . . . . .
5.3298 + 1110
. . . . . .
2.3557 + 1320
. . . . . .
7.9497 + 1330
. . . . . .
1.1148 + 1150
. . . . . .
1.4673 + 11
where
is the total number o testing points in the domain
Ω
,
(
,
)
and
̃
∗
(
,
)
are the exact and the approxi-matedvaluesatthesepoints,respectively.RMS,RES,and
∞
errors o the unctions
()
and
()
are similarly dened,as well. In our computation, the Matlab code developed by Hansen [] is used or solving the discrete ill-conditionedsystem o linear equations ().
Example
. For simplicity, we assume that
0
= 0.5
and
= =
max
= 1
. By these assumptions, the exact solution o theproblem ()–() is given as ollows []:
(,) =
−
sin
() +
2
+ 2,() = 2,() =
sin
() +
2
.
()Since, or a large xed number o scattered points
,the matrix
A
will be more ill-conditioned and also, smaller values o the shape parameter
generate more accurateapproximations, we suppose that
= 0.1
and the numbero interpolating points is
= 20
. Te obtained values o RMS
()
, RES
()
,
∞
()
, RMS
()
, RES
()
,
∞
()
, RMS
()
,RES
()
, and
∞
()
, as well as condition number o
A
or
= 0.01
and
= 208
and various values o
, are giveninablesand.Numericalresultsindicatethatthismethod
is not depended on parameter
. By the assumptions
= 20
and
= 4.1
and with various levels o noise added into thedata, ables and illustrate the relative root mean square
error o
()
and
()
at three points
0
= 0.1
, ., and
0
= 0.9
o the interval
[0,1]
, respectively. Figures and
eatureoutacomparisonbetweentheexactsolutionsandtheapproximate solutions or
0
= 0.5
and
= 3
and variouslevels o noise added into the data. It is observed that as thenoise level increases, the approximated unctions will haveacceptable accuracy. Figures and display the relationship
betweentheaccuracyandtheparameter
withnoiseolevel
= 0.01
added into the data. Tey elucidate not only that thenumerical results are stable with respect to parameter
, butalso that they retain at the same level o accuracy or a widerange o values
. Tereore, the accuracy o the numericalsolutions is not relatively dependent on the parameter
.In addition, the study o the presented numerical results, via MFS in [], indicates that with noise o level
= 0
,namely, with the noiseless data and or
0
= 0.5
o theinterval
[0,1]
, RMS
() = 1.7 × 10
−3
with
= 1.9
and
: Te resulted values o RES
()
using MFSRBF or variouslevels o noise
, diﬀerent values o
0
, and
= 4.1
.
0
= 0.1 = 0.01 = 0.001 = 0.00010.1 3.491 − 02 2.949 − 03 5.095 − 04 4.129 − 040.5 2.703 − 02 3.306 − 03 7.707 − 04 3.069 − 040.9 6.242 − 02 9.011 − 03 3.292 − 03 1.126 − 03
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.511.522.5Exact
p(t)
and its approximation for,
x
0
= 0.5,T = 3
p ( t )
a n d
p
∗
( t )
= 0.01 = 0.03 = 0.05t
F : Te exact
and its approximation
∗
obtained with
=20
,
= 3
,
= 0.01
, and
0
= 0.5
and or various levels o noiseadded into the data using MQ-RBF with
= 0.1
.
RMS
() = 1.89 × 10
−2
or
= 3
. Also, when the data areconsidered with noise, RMS
() = 2.84×10
−2
and RMS
() =3.53 × 10
−2
, while by the proposed method in this work andwith the same assumptions, we achieve RMS
() = 3.0479 ×10
−5
and RMS
() = 3.5691×10
−6
or
= 0
. Also, with noiseo level
= 10
−4
, we obtain RMS
() = 1.1021 × 10
−4
andRMS
() = 1.9204 × 10
−4
. Accordingly, the MFSRBF, whichis based on the undamental solutions o the heat equationand radial basis unctions, is more accurate in comparison toMFS.able indicates the values o the obtained RES
()
or various levels o noise, diﬀerent values o
0
and
= 4.1
by MFSRBF.

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