Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

50, 4, pp. 1011-1023, Warsaw 2012
50th Anniversary of JTAM

APPLICATION OF THE METHOD OF FUNDAMENTAL SOLUTIONS WITH

THE LAPLACE TRANSFORMATION FOR THE INVERSE TRANSIENT

HEAT SOURCE PROBLEM

Magdalena Mierzwiczak, Jan Adam Kołodziej

Poznan University of Technology, Institute of Applied Mechanics, Poznan, Poland

e-mail: magdalena.mierzwiczak@wp.pl; jan.kolodziej@put.poznan.pl

The paper deals with the inverse determination of heat source in an unsteady heat con-
duction problem. The governing equation for the unsteady Fourier heat conduction in 2D
region with unknown internal heat source is known as the inverse boundary-initial-value
problem.The identification of strength of the heat source is achieved by using the boundary
condition, initial condition and a known value of temperature in chosen points placed inside
the domain. For the solution of the inverse problem of determination of the heat source,
the Laplace transformationwith themethod of fundamental solution and radial basis func-
tions is proposed. Due to ill conditioning of the inverse transient heat conduction problem,
the Tikhonov regularizationmethod based on SVD and L-curve criterion was used. As the
test problems, the 2D inverse boundary-initial-value problems (2D IBIVP) in region Ω with
known analytical solutions are considered.

Key words: method of fundamental solutions, radial basis functions, inverse transient heat
source problem

1. Introduction

Theproblemof inverseheat sourcespresentsan interesting challenge inmanyareasof engineering
where the strength of heat sources is not exactly recognized. It belongs to a broad class of inverse
heat conductionproblemswhichareusually ill-posedbecause low randomerrors inmeasurements
can lead to big errors in identifications. Direct heat conduction problems on the other hand are
well-posed. So far, many different methods have been applied to solve the inverse heat source
problem, and in some papers, the problem of uniqueness was considered, e.g. Ling et al. (2006).
Generally speaking, the papers published in this field can be divided into two groups. The first
group consists of papers in which point or line heat sources are considered, e.g. Karami and
Hematiyan (2000). The second group is concerned with the sources as a continuous function in
a considered region, e.g. Jin and Marin (2007). The majority of authors consider the transient
heat conduction problem, e.g. Yan et al. (2008), but some authors analyze the steady state
heat conduction problem, e.g. Kołodziej et al. (2010). Among numerical methods, the conjugate
gradient method coupled with the finite difference method, e.g. Le Niliot and Lefevre (2001),
finite differencemethod (FDM) for discretization of the time termand the finite elementmethod
(FEM) for discretization of the space term, e.g. Yang (1998), finite difference with the discrete
mollification method, e.g. Yi and Murio (2004), finite difference coupled with other special
methods, e.g. Yan et al. (2010), and in the last decade, boundarymethods were used.Generally
speaking, the boundary methods can be divided into two groups: boundary element methods
(BEM), e.g. Karami and Hematiyan (2000), and the method of fundamental solutions (MFS)
(Kołodziej et al., 2010).
The MFS is a relatively new meshless method for solution of certain boundary value pro-

blems, and was first proposed in the 60ties by Kupradze and Aleksidze. In recent years, it has



1012 M.Mierzwiczak, J.A. Kołodziej

become increasingly popular because of its simplicity of implementation to problems of compli-
cated geometries. The solution is approximated by linear combinations of fundamental solutions
in terms of singularities which are placed on a fictitious boundary lying outside the considered
domain. Due to this, an approximated solution is given as a continuous function with continu-
ous derivatives which is very convenient in the case of inverse problems. These are the basic
advantages of this method in comparison with BEM.

In papers by Alves et al. (2008), Jin and Marin (2007), and Kołodziej et al. (2010), the
authors employed the method of fundamental solutions to recover the heat source in steady
state heat conduction problems. In papers by Alves et al. (2008), and Jin and Marin (2007),
the problem is solved using the a priori information that the source is harmonic or satisfies the
Helmholtz equation, and for the reconstruction of source theCauchydatawasused.Onthe other
hand, in the paper by Kołodziej et al. (2010), the identification of source in the steady state
heat conduction problem was considered without restriction for the form of heat source, but
some information about temperature in some inner point in the considered region was used. In
the paper byYan et al. (2008), a 1D transient heat conduction problemwas considered inwhich
heat source is taken to be time-dependent only. The identification of a heat source dependent
only on space was considered in the paper by Yan et al. (2009) for 1D and 2D transient heat
conduction problems. In papers by Yan et al. (2008, 2009), the authors used MFS with the
fundamental solution to the diffusion equation. The identification of heat source dependent on
space and time was considered in paper by Mierzwiczak and Kołodziej (2010) using the MFS
with discretization of time derivative by θ-method.

For the inverse heat transfer problem other than the inverse source problem, this method
was applied for boundary heat flux (Hon and Wei, 2004), Cauchy problem (Marin, 2005), or
identification of heat transfer conductivity (Mierzwiczak and Kołodziej, 2011).

Thegoal of thispaper is toapply theboundarycollocation technique (Kołodziej andZieliński,
2009) to the identification of 2D heat sources in a transient case. In our work, the type of heat
sources considered is not limited by any restrictions with the exception that it is a continuous
function of space and time. Contrary to the paper byYan et al. (2008) where sources depended
only on time, and the paper by Yan et al. (2009) where the sources were exclusively space
dependent, in this paper the source is a function of space and time.Using theLaplace transform,
the transient heat conduction problem can be expressed by inhomogeneous modifiedHelmholtz
equationswith adequate conditions in theLaplace transfer domain (the s-domain). Theproblem
is then solved by solving the non-homogeneous equation for each s-parameter. The radial basis
function (RBF) is used for interpolation of the right- hand side of the governing equation and
obtaining a particular solution. Several numerical examples for the inverse source problem are
presented to demonstrate the efficacy of the proposedmethod.

2. Mathematical formulation of the problem

Consider a general two-dimensional homogeneous isotropic region Ω with boundary ∂Ω. We
want to find the non-dimensional function q(ξ,η,τ) of heat sources and the non-dimensional
temperature field T(ξ,η,τ), which is governed by

∂T

∂τ
(ξ,η,τ) =∇2T(ξ,η,τ)+q(ξ,η,τ) for (ξ,η)∈ Ω

T(ξ,η,0)= T0(ξ,η) for (ξ,η)∈ Ω

a1T(ξ,η,τ)+a2
∂T(ξ,η,τ)

∂n
= a3 for (ξ,η)∈ ∂Ω

(2.1)



Application of the method of fundamental solutions... 1013

where ∇2 = ∂2/∂ξ2 + ∂2/∂η2 is the Laplace operator, n outward normal to the boundary,
a1, a2, a3 are the specified functions of position, τ is non-dimensional time, T0(ξ,η) is the
specified initial non-dimensional temperature distribution.
TheMFS can be used to solve the initial-boundary value problem formulated by Eq. (2.1)1,

initial condition (2.1)2 and boundary condition (2.1)3. Because the source function q(ξ,η,τ)
is unknown, the problem expressed by equations (2.1) is an inverse problem and requires an
additional condition. This extra condition is provided by the known temperature in a few points
placed inside the domain

T(ξi,ηi,τ)= Ti for {(ξi,ηi)}Mi=1 ∈ Ω (2.2)

3. MFS with Laplace transformation

Taking the Laplace transform of equation (2.1)1, we obtain a new differential equation in the
Laplace transfer domain (the s-domain)

∇2T(ξ,η,s)−sT(ξ,η,s) =−T0(ξ,η)−q(ξ,η,s) for (ξ,η)∈ Ω (3.1)

where T(ξ,η,s) is the Laplace transform of T(ξ,η,τ) with respect to dimensionless time τ

T(ξ,η,s) =

∞∫

0

e−sτT(ξ,η,τ) dτ (3.2)

The Laplace transform of the boundary condition in Eq. (2.1)3 is

a1T(ξ,η,s)+a2
∂T(ξ,η,s)

∂n
=
a3
s

(3.3)

and of additional condition (2.2)

T(ξi,ηi,s)=
Ti
s

{(ξi,ηi)}Mi=1 ∈ Ω (3.4)

Now observe that (3.1) is a sequence of the inhomogeneous modified Helmholtz equation

∇2T(ξ,η,s)−sT(ξ,η,s) = f(ξ,η,s) (3.5)

The right hand side in the modified Helmholtz equation f(ξ,η,s) = f(T0(ξ,η),q(ξ,η,s)) is
unknown, because the heat source function q(ξ,η,s) is unknown. The solution to Eq. (3.5) is

the sum T = T
(h)
+T

(p)
of the homogeneous solution T

(h)
, and the particular solution T

(p)

∇2T(h)−sT(h) =0 ∇2T(p)−sT(p) = f(ξ,η,s) (3.6)

The approximate solution to Eq. (3.6)1 is a superposition of fundamental solutions to the mo-
dified Helmholtz equation with the unknown coefficients {Wj}NSj=1

T
(h)
(ξ,η,s) =

NS∑

j=1

WjK0(r̃j
√
s) (3.7)

where r̃j =
√
(ξ− ξ̃j)2+(η− η̃j)2, K0 is the modified Bessel function of the second kind and

zero order and {(ξ̃j, η̃j)}NSj=1 are coordinates of source points which are located outside the
region Ω. In order to find the particular solution for non-homogeneous equation (3.6)2, the



1014 M.Mierzwiczak, J.A. Kołodziej

right hand side of the modified Helmholtz equation f(ξ,η,s) should be interpolated by means
of the RBFs andmononomials

f(ξ,η,s)=
M∑

m=1

αmϕ̂(rm)+
K∑

k=1

βkϕ̃k(ξ,η) (3.8)

where rm =
√
(ξ− ξm)2+(η−ηm)2 is the argument of RBF, ϕ(rm) = r2m ln(rm) is a thin

plate spline function, {ϕ̃k(ξ,η)}Kk=1 are mononomials given in Table 1 (K is the number of
polynomials) and {(ξm,ηm)}Mm=1 are the interpolation points placed inside the region Ω (Fig. 1).
The unknown coefficients {αm}Mm=1, {βk}Kk=1 are connected to the unknown source function by
a system of linear equations

M∑

m=1

αmϕ̂(rmi)+
K∑

k=1

βkϕ̃k(ξi,ηi)= f(ξi,ηi,s) 1¬ i ¬ M

M∑

m=1

αmϕ̃k(ξm,ηm)= 0 1¬ k ¬ K
(3.9)

where rmi =
√
(ξi− ξm)2+(ηi−ηm)2 and f(ξi,ηi,s) the function of the right hand side is

unknown.

Table 1. Form of mononomials and their particular solutions

k 1 2 3 4 5 6

ϕ̃k(ξ,η) 1 ξ η ξη ξ
2 η2

ψ̃k(ξ,η,s) −
1

s
−
ξ

s
−
η

s
−
ξη

s
−
ξ2

s
−
2

s2
−
η2

s
−
2

s2

Fig. 1. Location ◦ collocation, △ sources and • interpolation points

The particular solution to the non-homogeneous modified Helmholtz equation has form

T
(p)
(ξ,η,s)=

M∑

m=1

αmψ̂(rm,s)+
K∑

k=1

βkψ̃k(ξ,η,s) (3.10)

where functions ψ̂(rm,s) and ψ̃k(ξ,η,s) are solutions to equations

∇2ψ̂(rm,s)−λ2ψ̂(rm,s)= ϕ̂(rm) 1¬ m ¬ M
∇2ψ̃k(ξ,η,s)−λ2ψ̃k(ξ,η,s) = ϕ̃k(ξ,η) 1¬ k ¬ K

(3.11)



Application of the method of fundamental solutions... 1015

The functions ψ̃k(ξ,η,s) are given in Table 1, whereas the function

ψ̂(rm,s)=





−
4

s2
[K0(rm

√
s)+ ln(rm)+1]−

r2m ln(rm)

s
for rm > 0

4

s2

[
γeuler +ln

(√
s
2

)
−1
]

for rm =0

(3.12)

where γeuler =0.57721 is the Euler constant.

Benefiting from homogeneous solution (3.7), and particular solution (3.10), the solution to
differential equation (3.5) can now be given in the following form

T(ξ,η,s) = T
(h)
+T

(p)
=
NS∑

j=1

WjK0(r̃j
√
s)+

M∑

m=1

αmψ̂(rm,s)+
K∑

k=1

βkψ̃k(ξ,η,s) (3.13)

Theunknown coefficients Wj, αm, βk are determined fromboundary conditions (3.3) and added
condition (3.4) – the known value of temperature in M-points placed inside the domain Ω
necessary to solve the inverse value problem.

For each value of s, the collocation condition leads to a system of linear equations:

— for {(ξi,ηi)}Mi=1 ∈ Ω

NS∑

j=1

WjK0(r̃ji
√
s)+

M∑

m=1

αmψ̂(rmi,s)+
K∑

k=1

βkψ̃k(ξi,ηi,s)=
Ti
s

(3.14)

— for {(ξi,ηi)}M+NBi=M+1 ∈ ∂Ω

a1i

[NS∑

j=1

WjK0(r̃ji
√
s)+

M∑

m=1

αmψ̂(rmi,s)+
K∑

k=1

βkψ̃k(ξi,ηi,s)

]
+

+a2i

[NS∑

j=1

Wj
∂K0(r̃ji

√
s)

∂n
+
M∑

m=1

αm
∂ψ̂(rmi,s)

∂n
+
K∑

k=1

βk
∂ψ̃k(ξi,ηi,s)

∂n

]
=
a3i
s

(3.15)

— for {(ξm,ηm)}Mm=1 ∈ Ω

M∑

m=1

αmϕ̃i(ξm,ηm)= 0 i =1,2, . . . ,K (3.16)

where N1= NB+M+K create analgebraic systemof linear equationswith N2= NS+M+K
unknown coefficients Wj, αm, βk, which can be written in matrix form as

Aω= r (3.17)

where A is the N1×N2 influencematrix, ω is a columnvector of unknowncoefficients Wj,αm,
βk, and r is a column vector containing the prescribed boundary, additional and interpolation
conditions.

For each parameter s, the temperature field and the source function in the transformed
domain (s-domain) in the NF field points at which we want the solution, can be calculated
from equation (3.13) and from formula

q(ξ,η,s) =−
M∑

m=1

αmϕ̂(rm)−
K∑

k=1

βkϕ̃k(ξ,η)−T0(ξ,η) (3.18)



1016 M.Mierzwiczak, J.A. Kołodziej

For each NF inner point, the transformed solution of temperature and of sources must be
inverted back to the time domain. We utilize a relatively accurate method which states that a
function of dimensionless time p(τ) may be approximated by the following formula

p(τ)= A+Bτ +
N∑

j=1

Cj exp(−Djτ) (3.19)

where A, B, Cj, Dj, j =1, . . . ,N coefficients are evaluated by taking the Laplace transform of
Eq. (3.19) andmultiplying the result by the Laplace parameter s

sp(s)= A+
B

s
+
N∑

j=1

Cj

1+
Dj
s

(3.20)

We select a set of N +2 linear algebraic equations in N +2 coefficients A, B, and Cj, Dj,
j =1,2, . . . ,N. These equations are obtained by input of the value of transformed temperature
or transformed strength of source for the field point to equation (3.20) for different values of the
parameter s. Solving this system of linear equations for each inner point, we obtain A, B, Cj,
Dj coefficients which allow us to calculate the value of temperature or strength of the source in
s chosen point at different time increments using (3.19).

4. Numerical experiments

In order to validate the proposed numerical method, 2D numerical examples given in Table 2
are carried out. The accuracy of themethod is verified by calculating themaximal and the root
mean squared relative error

δMAX =
maxi{|Ti,ANAL−Ti,NUM|}

maxi{Ti,ANAL}

δRMSE =

√∑NN
i=1(Ti,ANAL−Ti,NUM)2√∑NN

i=1 T
2
i,ANAL

(4.1)

for the source function and the temperature field in NN = 200 inner control points. The
temperature Ti,ANAL is the exact temperature in the i-th point estimated from the analytical
solution (Table 2), and Ti,NUM is the approximated temperature in the i-th point calculated
using the numerical algorithm.

Table 2.The source function and the analytical solution of test examples

Example T(ξ,η,τ) = q(ξ,η,τ) =

1st [1−exp(−4τ)][cos(2ξ)+cos(2η)] 4[cos(2ξ)+cos(2η)]
2nd τ sin[π(ξ +η)/5] (2π2τ/25+1)sin[π(ξ+Y )/5]

3rd τ[(ξ−6)3+(η−6)3]/6 (ξ3+η3)/6−3(ξ2+η2)+18(ξ +η−4)
− τ(ξ+η−12)

4th τ(ξ+η)/2 (ξ +η)/2

All these examples are formulated in the unit square Ωa = [0,1]× [0,1] (Fig. 1a) with the
exception of the fourth example, which is formulated in a polygonal area Ωb, shown in Fig. 1b.



Application of the method of fundamental solutions... 1017

The form of the source function as well as the exact solutions are known. In the numerical
solution, the boundary condition as well as the values of temperature in the chosen points of the
region result from the exact solution. Inside the area Ωa, M = 25 and the area Ωb, M = 22
uniformly distributed interpolation points {(ξm,µm)}Mm=1 are chosen (see Fig. 1). The number
of collocation points on the boundary ∂Ωa or on the boundary ∂Ωb is NB =44. The number
of source points equals the number of collocation points NS = NB, and they are placed on a
contour geometrically similar to the contour of the boundary (see Fig. 1). The distance between
the contour of the boundary and the contour of sources D equals 0.2.
If the exact data are considered, and if the number of equations resulting from collocation

conditions (3.3) and (3.4) is greater than the number of unknowns N1 > N2, we obtain an
overdetermined systemof equationswhich can be solved in the least squared sense. If N1= N2,
the Gauss elimination method is used.
The results for the undisturbed data for all examples in the form of the root mean squared

relative error of identification of the temperature field and the source function are shown in
Fig. 2a and 2b, respectively.

Fig. 2. Values of the root mean squared relative error of identification of (a) temperature field and
(b) sources function for M =25, NB =44, ∆T =0%

For the undisturbed data, the proposed algorithm is quite accurate. The analysis of the
results shows that the unknown source function is identified much more accurately than the
temperature field.
The worst results of identification were obtained for the first test example. The important

advantage of the proposedmethod is that it allows one to determine the value of temperature of
the source function at any point in the concerned area. This is possible because the temperature
field and the source functions are approximated using the continuous functions.
The test calculations carried out for data from the known exact solutions are a certain

idealization of reality. While solving the inverse heat conduction problem, the input data will
come from the measurement, which is as we know, not exactly accurate, but disrupted by
some error. In order to consider the measuring error, the exact data (from the analytical so-
lution) are loaded with some random disturbance value TMEAS = TANAL(1+RN∆T), where
∆T =1% is the disturbance coefficient, and RN ∈ [−1,1] is a random number. The results for
the disturbeddata ∆T =1% for the all examples in formof the rootmean squared relative error
of identification of the temperature field and the source function are shown in Fig. 3a and 3b,
respectively.
For the disturbance data, the collocation matrix A is highly ill-conditioned, and some regu-

larization techniques are needed. In this paper, the Tikhonov regularization method based on
the singular value decomposition (SVD) is used. The Tikhonov regularization solution ωλ for
problem (3.14)-(3.16) is defined as the solution to the following least-square problem

min
ωλ
{‖Aωλ−r‖+λ2‖ωλ‖}



1018 M.Mierzwiczak, J.A. Kołodziej

Fig. 3. Values of the root mean squared relative error of identification of (a) temperature field and
(b) source function for M =25, NB =44, ∆T =1% (one regularization)

where ‖·‖ denotes the 2-norm and λ is a regularization parameter. The regularization solution
can be then expressed as

ωλ =
L∑

i=1

fi
wTi r

σi
vi

where

A=WΣVT =
L∑

i=1

wiσiv
T
i

and the filter factors are fi = σ
2
i/(σ

2
i +λ

2). In our computations, the L-curve criterion given by
Hansen andO’Leary (1993) for choosing the regularization parameter λ was used. The suitable
regularization parameter λ is the one that corresponds to the regularized solution near the
corner of the L-curve: {(log‖Aωλ−r‖, log‖ωλ‖),λ > 0}. The coefficients Wj, αm, βk obtained
for each s-parameter for the optimal regularization parameter enable determination of the value
of temperature field (3.13), and the source function given by equation (3.18) in the s-domain.
Then, using formula (3.19), we can determine the value of the temperature field or strength

of sources at any point in the considered area.
The results of identification of the temperaturefield and the source function for the disturbed

data ∆T =1%and for theTikhonov regularization for all the test examples are shown inFig. 4a
and 4b, respectively.

Fig. 4. Values of the root mean squared relative error of identification of (a) temperature field and
(b) source function for M =25, NB =44, ∆T =1% (with Tikhonov regularization)

Figure 5 presents examples of L-curveswith the designated regularization parameter located
in the corner of this curve for all the test examples.
Figure 6 shows the impact of the disturbance coefficient ∆T on the value of the maximum

and root mean squared relative error of the identification of the temperature fields and heat
sources for the third test example, when τ =0.8.



Application of the method of fundamental solutions... 1019

Fig. 5. L-curve with the regularization parameter λ located near the corner for all test examples and
for ∆T =1%

Fig. 6. Maximal and root mean squared relative error of the identification of the temperature field and
the source function for 3rd test example as a function of the number of the disturbance

coefficient ∆T [%]

As it was expected, with increasing values of the disturbance coefficient ∆T , the relative
error of identification increases.

In order to demonstrate the impact of the values of disturbance coefficient ∆T on the
quality of the results, two test examples were considered. Figure 7 presents the results of the
test comparison for the second example and Fig. 8 for the fourth example. Both figures clearly
show an increase in the error identification, taking into account themeasurement error. For the
disturbance coefficient of data ∆T = {1%,2%,3%}, the results are comparable.
TheMFS used in the computing algorithm depends on the number of interpolation points,

collocation and source points and the distance between the contour of the boundary and the
contour of sources D. Therefore, we decided to present the effect of these parameters on the
quality of results of identification of the temperature field and the heat source function for the
third test example at the time τ =0.8.

The dependence of the maximum error and root mean square error on the number of col-
location points is shown in Fig. 10 and on the interpolation points in Fig. 9. In the case of
the interpolation points, a regular distribution of these points in the region Ω and the random
distribution was considered.



1020 M.Mierzwiczak, J.A. Kołodziej

Fig. 7. Root mean squared relative error of the identification of (a) temperature field and (b) source
function for the 2nd test different value of the disturbance coefficient ∆T = {1%,2%,3%}

Fig. 8. Root mean squared relative error of the identification of (a) temperature field and (b) source
function for the 4th test different value of the disturbance coefficient ∆T = {1%,2%,3%}

Fig. 9. Maximal and root mean squared relative error of the identification of the temperature field and
source function for the 3rd test example (∆T =1%,τ =0.8) as a function of the number of

(a) uniformly distributed and (b) randomly distributed interpolation points M

Fig. 10. Maximal and root mean squared relative error of the identification of the temperature field and
source function for the 3rd test example (∆T =1%,τ =0.8) as a function of the number of collocation

points NB



Application of the method of fundamental solutions... 1021

One of the important parameters of the MFS is the location of source points outside the
area Ω. In the presented work, the source points, equal in the number the collocation points,
lay on the contour similar to the boundary at a distance D.

Fig. 11. Maximal and root mean squared relative error of the identification of the temperature field and
the source function for the 3rd test example (∆T =1%,τ =0.8) as a function of distance between the

contour of the boundary and the contour of sources D

Changing the value of the parameter D in the range from 10−3 to 1.55 for the inverse tran-
sient heat conduction problemof the identification of source functiondoes not affect significantly
the quality of the results (see Fig. 11) for the identification of source function and temperature
field.

Presented in this workMFSwith Laplace Transformation (MFS&LT) for the identification
of the temperature field and heat source function was compared to the MFS combined with
the finite differential (MFS&FD). A detailed description of this method (MFS&FD) to the
transient inverse source problem can be found inMierzwiczak andKołodziej (2010).

The comparison of the methods was conducted for the second and third test examples (Ta-
ble 2) for both the exact (∆T =0%) and perturbed data (∆T =1%).

The results of the comparison of MFS with the Laplace Transform (MFS&LT) with MFS
with the finite of differential (MFS&FD) in the form of root mean square relative error of the
identification of the temperature field and heat source function are shown in Fig. 12 andFig. 13
respectively, for the exact (∆T =0%) and disturbed case (∆T =1%).

Fig. 12. Comparison ofMFS&LTwith MFS&FDwith respect to values of the root mean squared
relative error of the identification of (a) temperature function and (b) source function for M =25,

NB =44, ∆T=0%

For the non-disturbed data, more accurate results of identification of both the temperature
field and the source function were obtained with MFS&LT. However, for the perturbed data
(∆T =1%) better results of the identification of the temperature field are obtained with MFS
combinedwith the finite differentialmethod. It should be noted that forMFS&LT in the initial
moments of time, the error is big and for MFS&FD error has a similar value for the whole
computation time.



1022 M.Mierzwiczak, J.A. Kołodziej

Fig. 13. Comparison ofMFS&LTwith MFS&FDwith respect to values of the root mean squared
relative error of the identification of (a) temperature function and (b) source function for M =25,

NB =44, ∆T =1% (with Tikhonov regularization)

The error of identification of source function in the case of MFS&FD alarmingly increases
over time. Such behavior has no place in the application ofMFS&LT, hence thismethod should
be regarded as appropriate in solving this type of heat conduction problem.

5. Conclusions

The paper presents a version of the MFS with Laplace transformation for solving the inverse
transient heat source problem in a 2D domain. The advantage of the presented method is its
meshless character and, consequently, the temperature is a continuous function at each time
contrary to traditional mesh methods such as FDM, FEM, and BEM. Due to ill-conditioning
of the inverse transient heat conduction problem, the Tikhonov regularization method based
on SVDwas used. In order to determine the optimum value of the regularization parameter λ,
the L-curve criterion was used. Based on our numerical experiments, it was shown that the
proposed method is easy to implement and accurate for undisturbed data. For the distorted
data, the proposed algorithm is not accurate, and its computation is time-consuming due to the
use of regularization based on SVD.The unsatisfactory accuracy of the results for the disturbed
data indicate a need for furtherwork on the application ofMFS for solving the inverse transient
heat conduction problems.

Acknowledgments

The second authormade this work within a frame of research project 4917/B/T02/2010/39 granted

by PolishMinistry of Science and Higher Education.

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Zastosowanie metody rozwiązań podstawowych i transformacji Laplace’a do odwrotnego

niestacjonarnego problemu źródeł ciepła

Streszczenie

Artykuł dotyczy odwrotnego problemu określenia mocy źródeł ciepła dla nieustalonego zagadnie-
nia przewodzenia ciepła. Równanie różniczkowe nieustalonego przewodzenia ciepła Fouriera w obsza-
rze dwuwymiarowym z nieznanymi wewnętrznymi źródłami ciepła jest znane jako odwrotny problem
brzegowo-początkowy. Przy określeniu mocy źródeł ciepła korzysta się z warunku brzegowego, warun-
ku początkowego oraz znanej wartości temperatury w wybranych punktach rozmieszczonych wewnątrz
rozważanego obszaru. Do rozwiązania odwrotnego problemu źródeł ciepła została zaproponowana trans-
formacja Laplace’a połączona zmetodą rozwiązań podstawowych i promieniowymi funkcjami bazowymi.
Ze względu na złe uwarunkowanie problemu, w pracy zastosowanometodę regularyzacji Tichonowa dla
rozkładu SVD oraz kryterium L-krzywej. Jako przykłady testowe rozważono dwuwymiarowe odwrotne
problemy brzegowo-początkowe (2D IBIVP) ze znanym rozwiązaniem analitycznymi.

Manuscript received May 25, 2011; accepted for print January 26, 2012