A method of fundamental solutions for transient heat conduction

ARTICLE IN PRESS

Engineering Analysis with Boundary Elements 32 (2008) 697–703 www.elsevier.com/locate/enganabound

A method of fundamental solutions for transient heat conduction B. Tomas Johanssona,, Daniel Lesnicb a

School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK b Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK Received 30 August 2007; accepted 23 November 2007 Available online 12 February 2008

Abstract In this paper we investigate an application of the method of fundamental solutions (MFS) to transient heat conduction. In almost all of the previously proposed MFS for time-dependent heat conduction the fictitious sources are located outside the time-interval of interest. In our case, however, these sources are instead placed outside the space domain of interest in the same manner as is done for stationary heat conduction. A denseness result for this method is discussed and the method is numerically tested showing that accurate numerical results can be obtained. Furthermore, a test example with boundary singularities shows that it is advisable to remove such singularities before applying the MFS. r 2008 Elsevier Ltd. All rights reserved. MSC: 35K05; 35A35; 65N35 Keywords: Heat conduction; Method of fundamental solutions

1. Introduction The method of fundamental solutions (MFS) is a simple but powerful technique that has been used to obtain highly accurate numerical approximations of solutions to linear elliptic partial differential equations (PDEs) with simple codes and small computational effort, see the excellent review of Fairweather and Karageorghis [1]. The advantages and disadvantages of the MFS with respect to the location of the fictitious sources are described at length in Heise [2] and Burgess and Maharejin [3]. Like the boundary element method (BEM), the MFS is applicable when a fundamental solution of the governing PDE is explicitly known. Justified by denseness results, see Bogomolny [4] and Alves and Chen [5], the solution is approximated by a linear combination of fundamental solutions with sources (singularities) located outside the solution domain. Since the denseness results are mainly valid for elliptic PDEs, e.g. the biharmonic, Helmholtz, Lame`, Laplace and the Stokes equations, the MFS has only Corresponding author.

E-mail address: [email protected] (B.T. Johansson). 0955-7997/$ - see front matter r 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.enganabound.2007.11.012

recently been developed for parabolic PDEs such as the heat equation, see Golberg and Chen [6], Young et al. [7] and Chantasiriwan [8]. Rather surprisingly, the MFS has been developed for the unsteady heat equation more in the context of inverse problems; Dong et al. [9], Hon and Wei [10,11], and Mera [12], instead of in the context of direct problems. Therefore, in this study we take a retrospective approach to the MFS for time-dependent heat conduction. In the above cited papers for the heat equation the fictitious sources are placed outside the time domain of interest, whilst in [1,6] the Laplace transform is first employed and the MFS is applied to the resulting elliptic Helmholtz type equation. Notably, Chantasiriwan [8] employed fictitious source points outside both the space and time domains concluding, however, that a two-timelevel finite difference scheme combined with the MFS for the modified Helmholtz equation performs better than the former MFS applied directly to the time-dependent heat equation. As opposed to the stationary case, no denseness results are mentioned in [6–12] making the use of the MFS questionable. However, in a paper by Kupradze [13] a denseness result is proved if the source points are located outside the space domain, as is done in the stationary case.

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The aim of this paper is then to propose and investigate an MFS for the time-dependent heat equation justified by the denseness result of Kupradze [13]. A brief outline of the paper is as follows. In Section 2 we formulate the problem under investigation. We introduce the fundamental solution and prove that linear combinations of fundamental solutions with source points located in time and outside the space domain can approximate the boundary data and initial condition, see Theorem 2.2. In Section 3 we describe an MFS for an approximation to the solution of the heat equation with Dirichlet boundary conditions. The method proposed can also be applied to other boundary conditions such as the Neumann condition and also mixed ones. Numerical results are discussed in Section 4 where we also investigate how singularities in the solution influence the MFS approximation. These investigations show that accurate approximations can be obtained with relatively few source points and that it is advisable to remove any singularities before applying the MFS.

so-called compatibility conditions which in our case are

2. Formulation of the problem

2.3. The fundamental solution and denseness properties

2.1. Assumptions and functional spaces

The fundamental solution to the one-dimensional heat equation (2.1) is given by

We denote by ða; bÞ and ½a; b an open, respectively, closed interval in R for aob. Let L40 and T40 be real numbers. We consider the heat equation described below in the domain ð0; LÞ  ð0; TÞ. However, if not explicitly stated, the results presented also hold if the interval ð0; LÞ is replaced by a bounded domain O in Rn of class C 2 with a connected complement. The space C k ðða; bÞÞ, where k is a non-negative integer, consists of all functions having continuous derivatives up to order k on ða; bÞ. This is a Banach space under the norm kf kC k ðða;bÞÞ ¼ sup0pj‘jpk;x2ða;bÞ jD‘ f ðxÞj. Let L2 ðða; bÞÞ be the space of square integrable real-valued functions on ða; bÞ with the usual norm. 2.2. The transient heat equation Let T40 be a fixed number. We consider the following problem: Find the temperature u which satisfies the heat conduction equation qu q2 u ðx; tÞ  2 ðx; tÞ ¼ 0 qt qx

for ðx; tÞ 2 ð0; LÞ  ð0; TÞ,

(2.1)

subject to the Dirichlet boundary conditions uð0; tÞ ¼ h0 ðtÞ;

uðL; tÞ ¼ hL ðtÞ for t 2 ð0; TÞ,

(2.2)

and the initial condition uðx; 0Þ ¼ j0 ðxÞ for x 2 ð0; LÞ.

(2.3)

We are mainly interested in a classical solution, i.e. a solution being twice differentiable with respect to x and once with respect to t in ½0; L  ½0; T. To guarantee the existence of such a solution it is necessary to impose the

j0 ðx0 Þ ¼ hx0 ð0Þ and

dhx0 d2 j0 ð0Þ ¼ ðx0 Þ; x0 ¼ 0; L. dt dx2 (2.4)

The following result can be found for example in Friedman [14]. Theorem 2.1. Let hj 2 C 1 ð½0; TÞ for j ¼ 0; L and j0 2 C 2 ð½0; LÞ satisfy (2.4). Then there exists a unique solution u 2 C 2;1 ð½0; L  ½0; TÞ to (2.1)–(2.3) and this solution depends continuously on the data. This implies in particular that the problem (2.1)–(2.3) is well-posed. It is also possible to prove, for example, existence and uniqueness of the weak solution to this problem in anisotropic Sobolev spaces, see e.g. Lions and Magenes [18].

2 Hðt  tÞ F ðx; t; y; tÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eðxyÞ =4ðttÞ , 4pðt  tÞ

(2.5)

where H is the Heaviside function which is introduced to emphasize that the fundamental solution is zero for tpt. There are also expressions for the fundamental solution to the heat equation in higher dimensions and in anisotropic media, see Chang et al. [16]. It is straightforward to verify that, as a function of x and t, F ðx; t; y; tÞ satisfies (2.1) for any ðx; tÞaðy; tÞ. Let ftm gm¼1;2;... be a denumerable, everywhere dense set of points in the interval ðT; TÞ (tm a0) and let y0 o0 and y1 4L. Put vðjÞ m ðx; tÞ ¼ F ðx; t; yj ; tm Þ for j ¼ 0; 1, m ¼ 1; 2; . . . ; and form ua ðx; tÞ ¼

1  X

 ð0Þ ð1Þ ð1Þ cð0Þ m vm ðx; tÞ þ cm vm ðx; tÞ ,

(2.6)

m¼1 ð1Þ where cð0Þ m ¼ cm ¼ 0 for all indices m except a finite number. Note that ua ðx; tÞ is identically zero for tpminm;j:jcðjÞ ja0 tm . m We now prove a denseness result which forms the basis for the MFS proposed in the next section.

Theorem 2.2. The restrictions vðjÞ m ðx0 ; tÞ constitute a linearly independent and dense set in L2 ððT; TÞÞ, for x0 ¼ 0; L. Furthermore, the restrictions vðjÞ m ðx; 0Þ, where tm o0, form a linearly independent and dense set in L2 ðð0; LÞÞ. ðjÞ Proof. That vm ðx0 ; tÞ constitutes a linearly independent and dense set in L2 ððT; TÞÞ, for x0 ¼ 0; L, was proved in Kupradze [13]. We therefore only prove the second part of the theorem. Assume on the contrary that there is an

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integer N40 such that uN ðx; 0Þ ¼

N  X

 ð0Þ ð1Þ ð1Þ cð0Þ m vm ðx; 0Þ þ cm vm ðx; 0Þ ¼ 0

m¼1

and that at least one of the coefficients, denoted by cðjÞ m0 , is non-zero. Note that uN , given by (2.6) with the series truncated after N terms, satisfies the heat equation (2.1) in ð0; LÞ  ðT; 0Þ and both uN ðx; TÞ ¼ 0 and uN ðx; 0Þ ¼ 0. We can uniquely represent the solution uN as a doublelayer potential, see Cannon [17, p. 65],  Z t  qF ðx; t; y; tÞ uN ðx; tÞ ¼ gðy; tÞ dt, (2.7) qnðyÞ T y¼0;L

699

of the previous section, we propose a method for approximating the solution of (2.1)–(2.3). Let us describe this approximation. We first define 2k  1 T for k ¼ ðM  1Þ; . . . ; M. 2M Furthermore, let y0 ¼ h and y1 ¼ L þ h where h40 is a real number. Due to Theorem 2.2 we search for an approximation of the solution to (2.1)–(2.3) of the form

tk ¼

M X

uM ðx; tÞ ¼

cð0Þ m F ðx; t; y0 ; tm Þ

m¼Mþ1 M X

cð1Þ m F ðx; t; y1 ; tm Þ.

for ðx; tÞ 2 ½0; L  ½T; 0, where n is the outward unit normal to the space boundary, nð0Þ ¼ 1 and nðLÞ ¼ 1, if and only if gð0; tÞ and gðL; tÞ are piecewise continuous solutions of Z t qF ðL; t; 0; tÞgðL; tÞ dt uN ð0; tÞ ¼ gð0; tÞ  2 T qx

Now, let tk ¼ ðk=MÞT, k ¼ 0; . . . ; M; and let x‘ ¼ ‘L= ðN þ 1Þ, ‘ ¼ 1; . . . ; N. Imposing the conditions (2.2)–(2.3) we obtain the following system of linear equations to solve ð1Þ for the coefficients cð0Þ m and cm :

and

uM ð0; tk Þ ¼ h0 ðtk Þ;

Z uN ðL; tÞ ¼ gðL; tÞ þ 2

t

qF ðL; t; 0; tÞgð0; tÞ dt. T qx

Imposing the condition uN ðx; 0Þ ¼ 0 for every x 2 ½0; L, we conclude that g is identically zero. Using the analyticity of the solution, it follows that uN is zero in ½y0 ; y1   ½T; 0. Letting the point ðx; tÞ approach the point ðyj ; tm0 Þ such that the ratio ðx  yj Þ=4ðt  tm0 Þ remains bounded, it ðjÞ ðjÞ follows that cm v can be made as large as we wish while 0 m0 the other summands remain bounded, which is a contradiction since uN is identically zero in ½y0 ; y1   ½T; 0. Let us then prove denseness. Assume that there is a function z 2 L2 ð0; LÞ, not identically zero, such that Z vkðjÞ ðx; 0ÞzðxÞ dx ¼ 0, ð0;LÞ

for k ¼ 1; 2; . . . ; and j ¼ 0; 1. Now, let w be the solution (in an appropriate weak sense) to (2.1) supplied with wðx; 0Þ ¼ z, and homogeneous Dirichlet conditions, i.e. h0 ¼ hL ¼ 0. Using Green’s formula, see e.g. Lions and Magenes [15], we conclude that Z T qw ðx0 ; tÞ dt ¼ 0, vðjÞ k ðx0 ; tÞ qx 0

þ

(3.1)

m¼Mþ1

uM ðL; tk Þ ¼ hL ðtk Þ and

uM ðx‘ ; 0Þ ¼ j0 ðx‘ Þ,

(3.2)

where k ¼ 0; . . . ; M and ‘ ¼ 1; . . . ; N. The system of equations (3.2) has 2M þ 2 þ N equations with 4M unknowns. In order to obtain a unique solution we take N ¼ 2M  2. We can write this system as Ac ¼ g,

(3.3)

with the obvious notation. At this stage, it is worth mentioning that although the direct problem (2.1)–(2.3) is well-posed, the resulting MFS matrix A is ill-conditioned, see Ramachandran [19]. Hence, a straightforward inversion of the system of equations (3.3) will produce unstable results. Then, in order to stabilize the solution, instead of solving (3.3), we use Tikhonov’s regularization method and solve ðAT A þ lIÞ c ¼ AT g ,

(3.4)

for x0 ¼ 0; L. But from the first part, vkðjÞ ðx0 ; tÞ is dense in L2 ðð0; TÞÞ, hence we conclude that also the normal derivative of w is zero on the lateral sides. This, according to Saut and Scheurer [18], implies that w ¼ 0 in ½0; L  ½0; T, and the theorem is proved. &

where the regularization parameter l40 is found by trial and error. Note that there are efficient rules for choosing l, for example the L-curve rule, see Hansen [20], and one such investigation will be presented for Example 1. Once the vector c has been found accurately, the heat flux at the boundaries x ¼ 0; L can be determined explicitly by differentiating Eq. (3.1) with respect to x. One can investigate the uniqueness of (3.2) for a sufficiently large number N, see further Katsurada [21].

3. A method of fundamental solutions

4. Numerical results

Based on the denseness results for linear combinations of the fundamental solution with source points located in time and outside the space domain, see Theorem 2.2

We apply the MFS described in the previous section to three different examples and in all of them we choose L ¼ 1 and T ¼ 1. We also choose typically M ¼ 12 and

ARTICLE IN PRESS B.T. Johansson, D. Lesnic / Engineering Analysis with Boundary Elements 32 (2008) 697–703

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N ¼ 2M  2 ¼ 22. In the last example we discuss how to apply this MFS when the solution has singularities. In all the numerical results we plot the MFS approximations at 20 points uniformly distributed on the interval ½0; 1.

It is straightforward to check that the analytical solution of the problem (2.1)–(2.3) with the data (4.1) is given by uðx; tÞ ¼

n¼0

Example 1. In this example the data are generated from the analytical solution uðx; tÞ ¼ x2 þ 2t, i.e. h0 ðtÞ ¼ 2t;

h1 ðtÞ ¼ 2t þ 1

and

1 X

2 2 8 sinðð2n þ 1ÞpxÞeð2nþ1Þ p t . ðð2n þ 1ÞpÞ3

We choose y0 ¼ 1:5, y1 ¼ 2:5, i.e. h ¼ 1:5. In Fig. 3 the analytical value uð; 0:2Þ (—–), calculated with 30 terms in the series (4.2), is plotted together with the MFS approximation (). It is seen that the approximation is accurate throughout the interval but slightly less accurate around x ¼ 12 which is the point of maximum curvature of the plotted curves. Increasing the number of source points to M ¼ 16 does improve slightly the approximation, but increasing the number of source points further does not improve the approximation much more. Again if h is too small or too large the approximation will become less accurate. In Fig. 4 the analytical values ux ð0; Þ and ux ð1; Þ (—–) of the heat flux are plotted together with the MFS approximations () and excellent agreement can be observed.

j0 ðxÞ ¼ x2 .

The fictitious source points are uniformly located on fyj g  ð1; 1Þ for j ¼ 0; 1 and y0 ¼ 1 and y1 ¼ 2, i.e. h ¼ 1, where h is as in the previous section. If the source points on the outer boundaries are chosen too close to the lateral sides (ho0:3) the approximation is not so accurate and the same conclusion holds if these fictitious source points are chosen at a too large distance (h44) from the lateral sides. Increasing the number of source points further does not improve the approximation significantly. In Fig. 1 the analytical values uð; 0:5Þ and uð0:5; Þ (—–) are plotted together with the MFS approximations. In order to investigate the sensitivity of the solution to the choice of the regularization parameter l in (3.4), in Fig. 1 we have included the numerical results for l ¼ 0:5 (þ), l ¼ 108 () and l ¼ 1012 (). The choice of the regularization parameter can be based on the L-curve criterion which plots the residual kA c  g k versus the norm of the solution, i.e. k c k, for various values of l. The corner of the L-curve shown in Fig. 2 corresponds approximately to l ¼ 108 which is taken as the compromise between the a priori expectation (norm k c k) and the a posteriori knowledge (norm kA c  g k). The maximum relative error between the analytical and numerical solutions in Figs. 1(a) and (b) for l ¼ 108 is of order 104 . As was pointed out at the end of the previous section the heat flux at the boundaries x ¼ 0; L can be determined explicitly by differentiating Eq. (3.1) with respect to x. Although not graphically illustrated, it is reported that the heat flux was accurately reconstructed with the maximum relative error between the analytical and numerical solutions being of order 103 .

Example 3. In this final example we investigate how singularities influence the MFS approximation. We choose

3.7

c

3.65

3.6

3.55 0.5

1

Example 2. Here, we choose in (2.2)–(2.3) and

1.5 Ac

j0 ðxÞ ¼ xð1  xÞ.

(4.1)

g

Fig. 2. The L-curve for Example 1.

2

2.5

1.8

2 u (0.5, t)

1.6 u (x, 0.5)

h0 ðtÞ ¼ h1 ðtÞ ¼ 0

1.4 1.2

1.5 1 0.5

1 0.8

0 0

0.2

0.4

0.6 x

0.8

(4.2)

1

0

0.2

0.4

0.6 t

Fig. 1. The temperature u inside the solution domain for Example 1.

0.8

1

ARTICLE IN PRESS B.T. Johansson, D. Lesnic / Engineering Analysis with Boundary Elements 32 (2008) 697–703

consider treatment of singularities in the solution. This example has been previously considered for treatment of singularities in the BEM, see Lesnic et al. [22]. We also wish to point out that this chosen situation is not only of academic interest, but it also models a concrete situation in pressure-head test measurements used to determine the hydraulic properties within rocks, see Al-Dhahir and Tan [23]. The analytical solution of problem (2.1)–(2.3) with the data (4.3) is given by     1  X 2n þ x 2ðn þ 1Þ  x pffiffi pffiffi uðx; tÞ ¼ erfc  erfc . 2 t 2 t n¼0

in (2.2)–(2.3) h0 ðtÞ ¼ 1;

h1 ðtÞ ¼ 0

and

j0 ðxÞ ¼ 0.

701

(4.3)

Then the compatibility conditions (2.4) are violated. This causes a singularity of the first kind for the solution at the point ðx; tÞ ¼ ð0; 0Þ. Note that singularities in the derivative might need a special treatment as well. Here, we only

0.04 0.035

(4.4)

u (x, 0.2)

0.03

We also define

0.025

 using ðx; tÞ ¼ erfc

0.02

 x pffiffi . 2 t

0.015

First, the MFS method with h ¼ 1 is directly applied to approximate the solution u of problem (2.1)–(2.3) with the data (4.3). In Fig. 5(a) the analytical value of u ð; 0:2Þ ¼ uð; 0:2Þ  using ð; 0:2Þ (—–) is plotted together with the MFS approximation (). As it can be noted, the approximation is not accurate and is especially inaccurate near the point ð0; 0Þ, where the function u has a singularity. To improve the approximation we remove the singularity by applying the MFS to u ¼ u  using . The function u

0.01 0.005 0 0

0.2

0.4

0.6

0.8

1

x

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

ux (1, t)

ux (0, t)

Fig. 3. The analytical value uð; 0:2Þ (—–) and the MFS approximation () for Example 2.

0

0.2

0.4

0.6

0.8

1

0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 -0.9 -1 0

0.2

0.4

t

0.6

0.8

1

0.8

1

t

Fig. 4. The heat flux ux at x ¼ 0 (a) and x ¼ 1 (b) for Example 2.

0.02

0

0

u (x, 0.2)

u (x, 0.2)

0.02

0

0.2

0.4

0.6

x

0.8

1

0

0.2

0.4

0.6

x

Fig. 5. The modified temperature u ð; 0:2Þ ¼ uð; 0:2Þ  using ð; 0:2Þ without (a) and with (b) removing the singularity for Example 3.

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1.5

0.1

1

0

0.5

u x (0, t )

u x (0, t )

702

0

0

0.2

0.4

0.6

0.8

0

1

0.2

0.4

t

0.6

0.8

1

t

Fig. 6. The modified heat flux ux ð0; tÞ ¼ ux ð0; tÞ  using;x ð0; tÞ without (a) and with (b) removing the singularity for Example 3.

0.1

0.1

0

u *x (1, t )

u *x (1, t )

0

0

0.2

0.4

0.6

0.8

0

1

0.2

0.4

t

0.6

0.8

1

t

Fig. 7. The modified heat flux ux ð1; tÞ ¼ ux ð1; tÞ  using;x ð1; tÞ without (a) and with (b) removing the singularity for Example 3.

solves (2.1)–(2.3) with

0.04

 h0 ðtÞ ¼ 0;

h1 ðtÞ ¼ erfc

1 pffiffi 2 t



0.035

and j0 ðxÞ ¼ 0.

(4.5)

0.03

5. Conclusions In this paper an MFS for transient heat conduction problems has been investigated. Unlike previous studies in which the fictitious sources (singularities) were located outside the time-interval of interest, here these sources are

u (x, 0.2)

0.025

One can check that u is continuous on ½0; 1  ½0; 1, hence the singularity is removed. We then apply the MFS to approximate u . In Fig. 5(b) the analytical value u ð; 0:2Þ (—–) is plotted together with the MFS approximation (). As compared with Fig. 5(a) the approximation has improved substantially. In Figs. 6(a) and (b) the analytical values ux ð0; tÞ of the modified heat flux without and with the singularity removed, respectively, are presented. Similarly, in Figs. 7(a) and (b) the analytical values ux ð1; tÞ of the modified heat flux without and with the singularity removed, respectively, are shown. From these figures it can be seen that in order to obtain accurate numerical results for the modified heat flux, especially near t ¼ 0, the removal of the singularities in the MFS is essential.

0.02 0.015 0.01 0.005 0

0

0.2

0.4

0.6

0.8

1

x Fig. 8. The analytical value uð; 0:2Þ (—–) and the usual MFS approximation () for Example 2.

placed outside the boundary of the space solution domain, as suggested by Kupradze [13]. For this distribution of sources, the linear independence and density of the MFS for the initial and Dirichlet boundary conditions are proved whilst for the former approach this is not possible. In fact, for Example 2, from the Appendix it can be seen that locating the singularities outside the time-interval of

ARTICLE IN PRESS B.T. Johansson, D. Lesnic / Engineering Analysis with Boundary Elements 32 (2008) 697–703

703

0

1 0.8 ux (1, t)

ux (0, t)

0.6 0.4 0.2 0 0.2 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

t

t

Fig. 9. The heat flux ux at x ¼ 0 (a) and x ¼ 1 (b) obtained using the usual MFS approximation for Example 2.

interest produces less accurate numerical approximations. Similar theoretical results can be proved for Neumann or mixed boundary conditions. Although the original direct problem (2.1)–(2.3) is well-posed, the resulting MFS matrix is ill-conditioned and the ill-conditioning depends on the distance h from the fictitious source points to the boundary. Regularization was found necessary, as given by Eq. (3.4). Further, one can also include noise in the Dirichlet boundary data (2.2) and regularize, as described in Lesnic et al. [24]. Numerical results have been presented and discussed showing the accuracy of the developed MFS. Appendix A For the sake of completeness, we calculate approximations to Example 2 by alternatively placing source points in space at a fixed instant of time t0 o0, which is the usual approach in the available MFS literature, i.e. the approximation is given as uM ðx; tÞ ¼

M X

cm F ðx; t; ym ; t0 Þ.

(A.1)

m¼0

Here, ym ¼ mL=M, m ¼ 0; 1; . . . ; M, and we choose M ¼ 12 and t0 ¼ 0:5. The collocation points for x and t are as in Example 2. The numerically obtained results shown in Figs. 8 and 9 for the interior solution uð; 0:2Þ and for the heat flux are less accurate than the numerical results obtained in Figs. 3 and 4, respectively. References [1] Fairweather G, Karageorghis A. The method of fundamental solutions for elliptic boundary value problems. Adv Comput Math 1998;9:69–95. [2] Heise U. Numerical properties of integral equations in which the given boundary values and the sought solutions are defined on different curves. Comput Struct 1978;8:199–205. [3] Burgess G, Maharejin E. A comparison of the boundary element and superposition methods. Comput Struct 1984;19:697–705. [4] Bogomolny A. Fundamental solutions method for elliptic boundary value problems. SIAM J Numer Anal 1985;22:644–69. [5] Alves CJS, Chen CS. A new method of fundamental solutions applied to nonhomogeneous elliptic problems. Adv Comput Math 2005;23: 125–42.

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