Solving the Axisymmetric Inverse Heat Conduction Problem by a Wavelet Dual Least Squares Method

Volume 2009, Article ID 260941,13pages doi:10.1155/2009/260941

Research Article

Solving the Axisymmetric Inverse Heat Conduction Problem by a Wavelet Dual Least Squares Method

Wei Cheng

and Chu-Li Fu

1College of Science, Henan University of Technology, Zhengzhou 450001, China

2School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China

Correspondence should be addressed to Chu-Li Fu,[email protected] Received 17 August 2008; Revised 23 January 2009; Accepted 10 March 2009 Recommended by Ugur Abdulla

We consider an axisymmetric inverse heat conduction problem of determining the surface temperature from a fixed location inside a cylinder. This problem is ill-posed; the solutionif it existsdoes not depend continuously on the data. A special project method—dual least squares method generated by the family of Shannon wavelet is applied to formulate regularized solution.

Meanwhile, an order optimal error estimate between the approximate solution and exact solution is proved.

Copyrightq2009 W. Cheng and C.-L. Fu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Inverse heat conduction problemsIHCPhave become an interesting subject recently, and many regularization methods have been developed for the analysis of IHCP 1–13. These methods include Tikhonov method 1, 2, mollification method 3, 4, optimal filtering method 5, lines method 6, wavelet and wavelet-Galerkin method 7–11, modified Tikhonov method 12 and “optimal approximations” 13, and so forth. However, most analytical and numerical methods were only used to dealing with IHCP in semiunbounded region. Some works of numerical methods were presented for IHCP in bounded domain14–

19.

Chen et al. 14 applied the hybrid numerical algorithm of Laplace transform technique to the IHCP in a rectangular plate. Busby and Trujillo 15 used the dynamic programming method to investigate the IHCP in a slab. Alifanov and Kerov 16 and Louahlia-Gualous et al. 17 researched IHCP in a cylinder. However to the authors’

knowledge, most of them did not give any stability theory and convergence proofs.

In this paper, we will treat with a special IHCP whose physical model consists of an infinitely long cylinder of radius R. It is considered axisymmetric and a thermocouple

measurement equipment of temperatureis installed inside the cylinderat the radiusr1, 0 < r₁ < R. The correspondingly mathematical model of our problem can be described by the following axisymmetric heat conduction problem:

∂u

∂t Δu ∂²u

∂r² 1 r

∂u

∂r, 0< r≤R, t >0, ur,0 0, 0≤r≤R,

ur1, t gt, t≥0, ur, t bounded inr0, t >0,

1.1

where the functions ur,· and g·belong to L²0,∞ for every fixedr ∈ 0, R, r is the radial coordinate,gtdenotes the temperature history at one fixed radius r₁0 < r₁ < R of cylinder. We want to recoverur,·forr1 < r ≤ R. This problem is ill-posed problem; a small perturbation in the data may cause dramatically large errors in the solutionur,· The details can be seen inSection 2.

To the authors’ knowledge, up to now, there is no regularization theory with error estimate for problem1.1in the intervalr1< r ≤R. The major objective of this paper is to do the theoretic stability and convergence estimates for problem1.1.

Xiong and Fu 11 and Regi ´nska 20 solved the sideways heat equation in semi- unbounded region by applying the wavelet dual least squares method, which is based on the family of Meyer wavelet. In this paper, we will apply a wavelet dual least squares method generated by the family of Shannon wavelet to problem1.1in bounded domain for determining surface temperature. According to the optimality results of general regularization theory, we conclude that our error estimate on surface temperature is order optimal.

2. Formulation of Solution of Problem 1.1

As we consider problem1.1inL²Rwith respect to variablet, we extendur ,·, g· : ur1,·, f·:uR ,·, and other functions of variabletappearing in the paper to be zero for t <0. Throughout the paper, we assume that for the exactgthe solutionuexists and satisfies an apriori bound

f·

p:uR,·

p≤E, p≥0, 2.1

wheref·pis defined by f·

p: _∞

−∞

1ξ²p

fξ²dξ

1/2

. 2.2

Sinceg is measured by the thermocouple, there will be measurement errors, and we would actually have as data some functiong_δ∈L²R, for which

gδ·−g·≤δ, 2.3

where the constantδ >0 represents a bound on the measurement error, and · denotes the L²Rnorm and

hξ 1

√2π _∞

−∞e^−iξthtdt 2.4

is the Fourier transform of functionht. The problem1.1can be formulated, in frequency space, as follows:

iξur, ξ ∂²ur, ξ

∂r² 1 r

∂ur, ξ

∂r , r ∈0, R, ξ∈R, 2.5

ur1, ξ gξ, ξ∈R, 2.6

u0, ξ<∞, ξ∈R. 2.7

Then we have the following lemma.

Lemma 2.1. Problem2.5–2.7has the solution given by

ur, ξ I0

iξr I₀

iξr₁ gξ, r∈0, R, ξ∈R, 2.8 whereI0zdenotes modified spherical Bessel function which given by [21]

I₀z ^∞

k0

1 k!²

z 2

2k

. 2.9

Proof. Due to21, we can solve2.5, in the frequency domain, to obtain

ur, ξ AξI0

iξr

BξK0

iξr

ξ∈R, 2.10

whereK₀zdenotes also modified spherical Bessel function which is given by

K0z −I0z

lnz

2 C ^∞

k1

1 k!²

11

2 · · ·1 k

z 2

2k

. 2.11

Combining limz→0K0z ∞with condition2.7, we obtainBξ 0, that is,

ur, ξ AξI0

iξr

, r∈0, R, ξ∈R. 2.12

According to21, there holds I₀

iξrber

|ξ|r

iσbei

|ξ|r _∞

k0

|ξ|r/24k

k!²2k!

^1/2

, 2.13

whereσsgnξ, both berxand beixdenote the Kelvin functions. Since 0≤r≤R,|ξ| ≥0, we have

∞ k0

|ξ|r/24k

k!²2k! 1

|ξ|r/28

2!²4!

|ξ|r/212

3!²6! · · · ≥1. 2.14

Therefore, for 0≤r≤R, ξ∈R,

I0

iξr _∞

k0

|ξ|r/24k

k!²2k!

^1/2

≥1. 2.15

Solving the systems2.6and2.12using2.15we get

Aξ I₀⁻¹ iξr₁

gξ. 2.16

Substitution ofAξin2.16into2.12, we obtain2.8.

Applying an inverse Fourier transform to2.8, problem1.1has the solution

ur, t 1

√2π _∞

−∞e^iξtI₀ iξr I0

iξr1

gξdξ, r, t∈0, R×R. 2.17

In order to obtain ill-posedness of problem1.1forr, t ∈ r1, R×R, we need the following lemma.

Lemma 2.2. If function |I0

iξr| satisfies 2.15, then there exist positive constants ck, k 1,2,3,4,such that, forr ∈r1, R

c1exp

|ξ|/2 r−r1

≤

I₀ iξr I0

iξr1

≤c2exp

|ξ|/2r−r1

, ξ∈R, 2.18

c₃exp

|ξ|/2r−R

≤

I0

iξr I₀

iξR

≤c₄exp

|ξ|/2r−R

, ξ∈R. 2.19

Proof. First, due to21and2.15, we have, forr ∈r1, Rand|ξ| → ∞, I₀

iξr ber²

|ξ|r

bei²

|ξ|r1/2

exp

|ξ|/2r

2πr

|ξ|

1O

1

|ξ| , 2.20

then there exist positive constantsck, k1,2,3,4,such that, for|ξ|large enough, say|ξ| ≥ξ0

c1

exp

|ξ|/2r

2πr

|ξ| ≤I0

iξr≤c2

exp

|ξ|/2r

2πr

|ξ| , r∈r1, R,

c₃exp

|ξ|/2r1

2πr₁

|ξ| ≤I0

iξr₁≤c₄exp

|ξ|/2r1

2πr₁

|ξ| .

2.21

From these we know that there exist positive constantsc₅andc₆such that, forr ∈r1, Rand

|ξ| ≥ξ0,

c5exp

|ξ|/2r−r1

≤

I₀ iξr I0

iξr1

≤c6exp

|ξ|/2r−r1

. 2.22

Then, since function|I0

iξr/I0

iξr1|is continuous in the closed regionr1, R×−ξ0, ξ0. Threrfore, there exist constantsc7andc8such that, forr∈r1, Rand|ξ| ≤ξ0,

c₇exp

|ξ|/2

r−r₁

≤

I0

iξr I₀

iξr₁

≤c₈exp

|ξ|/2

r−r₁

. 2.23

Finally, combining inequalities2.22with2.23, we can see that there exist others constants c₁ and c₂ such that, for r ∈ r1, R, inequalities 2.18 are valid. Similarly, we obtain inequalities2.19.

In order to formulate problem1.1forr₁ < r≤Rin terms of an operator equation in the spaceXL²R, we define an operatorKr :ur,· →g·, that is,

∀ur,·∈X, K_rur, t gt, r₁< r ≤R. 2.24

From2.8, we have

Krur, ξ I₀ iξr₁ I0

iξr ur, ξ gξ. 2.25

DenoteKrur, ξ : Krur, ξ, and we can see that Kr : L²R → L²Ris a multiplication operator:

K_rur, ξ I0

iξr1

I₀

iξr ur, ξ. 2.26

From2.26, we can prove the following lemma.

Lemma 2.3. LetK_r^∗ be the adjoint toKr, thenK^∗_r corresponds to the following problem where the left-hand side∂u/∂tof problem1.1is replaced by−∂U/∂t, says

−∂U

∂t ΔU ∂²U

∂r² 1 r

∂U

∂r , 0< r ≤R, t≥0, Ur,0 0, 0≤r ≤R,

Ur1, t gt, t≥0, Ur, t bounded inr0, t >0,

2.27

K^∗_r I₀ iξr₁ I0

iξr. 2.28

Proof. Via the the following relations, combining with2.26,

K_ru, υ

K_ru, υ

u,K_r^∗υ

u, K^∗_rυ

u,K^∗_rυ

, 2.29

we can get the adjoint operatorK^∗_r ofKr in frequency domain

K^∗_r K_r^∗ I0

iξr1

I₀

iξr. 2.30

On the other hand, the problem 2.27 can be formulated, in frequency space, as follows:

−iξUr, ξ ∂²Ur, ξ

∂r² 1 r

∂Ur, ξ

∂r , r∈0, R, ξ∈R, Ur 1, ξ gξ, ξ∈R,

U0, ξ<∞, ξ∈R.

2.31

Taking the conjugate operator for problem 2.5–2.7, we realize that Ur, ξ ur, ξ.

Therefore, byLemma 2.1, we conclude that

Ur, ξ ur, ξ I₀ iξr I0

iξr1

gξ, 2.32

that is,

gξ I₀ iξr₁ I0

iξr

Ur, ξ K^∗_rUr, ξ :K^∗_rU. 2.33

Hence the conclusion ofLemma 2.3is proved.

The Parseval formula for the Fourier transform together with inequality2.18, there holds

ur,·²ur,·²

_∞

−∞ur,·²dξ

_∞

−∞

gξe^r−r1√

|ξ|/2²

I₀ iξr I₀

iξr₁

2

dξ

≥c²_{1 ∞}

−∞

gξe^r−r1√

|ξ|/2²dξ.

2.34

This implies thatgξ, which is Fourier transform of exact data gt, must decay rapidly at high frequencies sincer₁< r. But such a decay is not likely to occur in the Fourier transform of the measured noisy datagδtatr r1. So, small perturbation ofgtin high frequency components can blow up and completely destroy the solutionur, tgiven by2.17forr ∈ r1, R.

3. Wavelet Dual Least Squares Method

A general projection method for the operator equationKug,K:X L²R →XL²Ris generated by two subspace families{Vj}and{Yj}ofXand the approximate solutionu_j∈V_j is defined to be the solution of the following problem:

Kuj, yg, y, ∀y∈Y_j, 3.1

where·,·denotes the inner product inX. IfVj⊂RK^∗and subspacesYjare chosen in such a way that

K^∗Yj Vj. 3.2

Then we have a special case of projection method known as the dual least squares method. If {ψλ}_λ∈Ijis an orthogonal basis ofV_jandy_λis the solution of the equation

K^∗yλkλψλ, yλ1, 3.3

then the approximate solution is explicitly given by the expression

uj

λ∈Ij

g, y_λ 1

k_λψλ. 3.4

3.2. Shannon Wavelets

In22, the Shannon scaling function isφ sinπt/πtand its Fourier transform is

φξ

1, |ξ| ≤π,

0, otherwise. 3.5

The corresponding wavelet functionψis given by its Fourier transform

ψξ

e^−iξ/2, π≤ |ξ| ≤2π,

0, otherwise. 3.6

Let us list some notation:φj,kt:2^j/2φ2^jt−k,ψj,kt:2^j/2ψ2^jt−k,j, k∈Z,Ψ_−1,k :φ0,k

andΨl,k:ψ_l,kforl≥0, the index set I

{j, k}:j, k∈Z

⊂Z², I_J

{j, k}:j −1,0, . . . , J−1;k∈Z

⊂Z². 3.7

BecauseV_J V_J−1⊕W_J−1 V_J−2⊕W_J−2⊕W_J−1 · · · V₀⊕W₁⊕ · · · ⊕W_J−1, hence we can define the subspacesVJ

VJspan{Ψλ}_λ∈IJ. 3.8

Define an orthogonal projectionPJ :L²R →VJ: P_Jϕ

λ∈IJ

ϕ,Ψλ

Ψλ, ∀ϕ∈L²R, 3.9

then from3.4we easily conclude u_J P_Ju. From the point of view of an application to the problem1.1, the important property of Shannon wavelets is the compactness of their support in the frequency space. Indeed, since

ψ_j,kξ 2^−j/2e^−i2−jkξψ 2^−jξ

, φ_j,kξ 2^−j/2e^−i2−jkξφ 2^−jξ

, 3.10

it follows that for anyk∈Z supp

ψj,k

ξ:π2^j ≤ |ξ| ≤π2^j1

, supp φj,k

ξ:|ξ| ≤π2^j

. 3.11

From3.9,PJ can be seen as a low-pass filter. The frequencies with greater thanπ2^J1are filtered away.

Theorem 3.1. Ifur, tis the solution of problem1.1satisfying the conditionuR,·p≤E, then for any fixedr∈r1, R

ur,·−PJur,·≤c⁻¹₃ 2^J1−p

e^r−R

√1/2π2^JE. 3.12

Proof. From3.9, we have

ur,·

λ

ur,·,Ψλ

Ψλ,

PJur,·

λ∈IJ

ur,·,Ψλ

Ψλ. 3.13

Due to Parseval relation and2.8,2.19, and2.1, there holds

ur,·−PJur,·ur,·−PJur,·

λ∈I

u, Ψλ Ψλ−

λ∈IJ

u, Ψλ Ψλ

λ∈Ij≥J1

u, Ψλ Ψλ

λ∈Ij≥J1

I0

i·r I₀

i·r1

! g·,Ψλ

"

Ψλ

λ∈Ij≥J1

I0

i·r I₀

i·R

! f·,Ψλ

"

Ψλ

≤ sup

π2^J≤|ξ|≤π2^J1

|ξ|^−p

I₀ iξr I₀

iξR

λ∈Ij≥J1

#1 ·²p/2f·,Ψλ

$Ψλ

≤ sup

π2^J≤|ξ|≤π2^J1

c₄ξ^−pe^r−R√

|ξ|/2E≤c₄ 2^J1−p

e^r−R

√1/2π2^JE.

3.14

Hence the conclusion of Theorem3.1is proved.

4. Error Estimates via Dual Least Squares Method Approximation

Before giving error estimates, we present firstly subspacesY_j. According toK^∗Y_j V_j, the subspacesYjare spanned byρλ, λ∈IJ, where

K^∗ρ_λ Ψλ, k_λρλ⁻¹, y_λ ρ_λ

ρλ k_λρ_λ. 4.1

ρλcan be determined by solving the following parabolic equationseeLemma 2.3:

−∂U

∂t ΔU ∂²U

∂r² 1 r

∂U

∂r , 0< r ≤R, t≥0, Ur,0 0, 0≤r≤R,

Ur1, t Ψj,kt, t≥0, Ur, t bounded inr0, t >0.

4.2

Since suppψj,kis compact, the solution exists for anyt∈0,∞. Similarly the solution of the adjoint equation is unique. Therefore for a givenΨλ,ρ_λcan be uniquely determined according to4.2, furthermore

ρ_λ I0

iξr I₀

iξr₁

Ψλξ⇐⇒y_λ I0

iξr I₀

iξr₁k_λΨλξ, λ{j, k}. 4.3

The approximate solution for noisy datag_δis explicitly given by

PJu^δr, t u^δ_J

λ∈IJ

u^δ,Ψλ

Ψλ

λ∈IJ

gδ, y_λ 1

kλΨλ. 4.4

Now we will devote to estimating the errorPJu^δ−P_Ju.

Theorem 4.1. If gδ is noisy data satisfying the condition g·−gδ· ≤ δ, then for any fixed r∈r1, R

PJu^δ−PJu ≤c4e^r−r1

√1/2π2^Jδ. 4.5

Proof. From4.3, we havey_λ I0

iξr/I0

iξr₁kλΨλ. Note thatP_Ju^δgiven by4.4,P_Ju given by3.4and2.18, forr₁< r ≤R, there holds

P_Ju^δr,·−P_Jur,·

λ∈IJ

g_δ−g, y_λ 1 k_λΨλ

λ∈IJ

g_δ−g, y_λ 1 kλ

Ψλ

λ∈IJ

g_δ−g, I0

i·r I₀

i·r1

k_λΨλ

"

1 kλ

Ψλ

≤ sup

π2^J−1≤|ξ|≤π2^J

I0

iξr I₀

iξr₁ ·

λ∈IJ

g_δ−g, Ψλ Ψλ

≤ sup

π2^J−1≤|ξ|≤π2^J

I0

iξr I0

iξr1

· P_Jg_δ−g

≤ sup

π2^J−1≤|ξ|≤π2^J

I₀ iξr I0

iξr1

·δ

≤c2 sup

π2^J−1≤|ξ|≤π2^J

e^r−r1√

|ξ|/2δ

≤c₂e^r−r1√

1/2π2^Jδ.

4.6

Hence the conclusion ofTheorem 4.1is proved.

The following is the main result of this paper.

Theorem 4.2. Letube the exact solution of 1.1and letP_Ju^δbe given by4.4. Ifg−g_δ ≤δand JJδis such that

J log₂ 2

π 1

R−r₁ ln E

δ

lnE δ

−2p ²

, 4.7

then for any fixedr∈r1, R

ur,·−PJu^δr,·

≤E^{1−R−r/R−r1}δ^R−r/R−r1

lnE δ

−2p1−R−r/R−r1

Co1

forδ−→0,

4.8

whereC c4R−r₁^2pc₂.

Proof. CombiningTheorem 4.1withTheorem 3.1, and noting the choice rule4.7ofJ, we can obtain

ur,·−P_Ju^δr,·

≤c₄ 2^J1−p

e^r−R√

1/2π2^JEc₂e^r−r1√

1/2π2^Jδ

≤c4E R−r1

_2p ln

E δ

lnE δ

−2p !^−2p E δ

lnE

δ

−2p!_r−R/R−r1

c2δ E δ

lnE

δ

−2p!_{r−r1/R−r1}

≤E^{1−R−r/R−r1}δ^R−r/R−r1

lnE δ

−2p1−R−r/R−r1% c4

R−r1lnE/δ_2p ln

E/δlnE/δ^−2p2p c₂

&

. 4.9

Note that

lnE/δ ln

E/δ

lnE/δ_−2p lnE/δ

lnE/δ−2pln

lnE/δ −→1 forδ−→0, 4.10

thus, there holds, forδ → 0

ur,·−P_Ju^δr,·

≤E^{1−R−r/R−r1}δ^R−r/R−r1

lnE δ

−2p1−R−r/R−r1 c₄

R−r₁2pc₂o1 .

4.11

Hence the conclusion ofTheorem 4.2is proved.

Remark 4.3. iWhenp0 andr₁ < r < R, estimate4.8is a H ¨older stability estimate given by

ur,·−PJu^δr,·≤c4c2E^{1−R−r/R−r1}δ^R−r/R−r1. 4.12 iiWhenp >0, r1< r < R, estimate4.8is a logarithmical H ¨older stability estimate.

iiiWhenp >0, rR, estimate4.3becomes uR,·−u^δR,·≤E

lnE

δ

−2p

c₄R−r₁^2pc₂o1

−→0 forδ−→0. 4.13

This is a logarithmical stability estimate.

Remark 4.4. In general, the a-priori boundEis unknown in practice, in this case, with

Jlog₂ 2

π 1

R−r1ln 1

δ

ln1 δ

−2p 2

, 4.14

then

ur,·−P_Ju^δr,·≤δ^R−r/R−r1

ln1 δ

−2p1−R−r/R−r1

Co1

forδ−→0, 4.15

whereCc₄R−r₁^2pEc₂.

Acknowledgments

The work is supported by the National Natural Science Foundation of ChinaNo. 10671085, the Hight-level Personnel fund of Henan University of Technology 2007BS028, and the

Fundamental Research Fund for Natural Science of Education Department of Henan Province of ChinaNo. 2009B110007.

References

1 A. Carasso, “Determining surface temperatures from interior observations,” SIAM Journal on Applied Mathematics, vol. 42, no. 3, pp. 558–574, 1982.

2 C.-L. Fu, “Simplified Tikhonov and Fourier regularization methods on a general sideways parabolic equation,” Journal of Computational and Applied Mathematics, vol. 167, no. 2, pp. 449–463, 2004.

3 D. A. Murio, The Mollification Method and the Numerical Solution of Ill-Posed Problems, A Wiley- Interscience Publication, John Wiley & Sons, New York, NY, USA, 1993.

4 D. N. H`ao and H.-J. Reinhardt, “On a sideways parabolic equation,” Inverse Problems, vol. 13, no. 2, pp. 297–309, 1997.

5 T. I. Seidman and L. Eld´en, “An ‘optimal filtering’ method for the sideways heat equation,” Inverse Problems, vol. 6, no. 4, pp. 681–696, 1990.

6 L. Eld´en, “Solving the sideways heat equation by a method of lines,” Journal of Heat Transfer, vol. 119, pp. 406–412, 1997.

7 L. Eld´en, F. Berntsson, and T. Regi ´nska, “Wavelet and Fourier methods for solving the sideways heat equation,” SIAM Journal on Scientific Computing, vol. 21, no. 6, pp. 2187–2205, 2000.

8 T. Regi ´nska and L. Eld´en, “Solving the sideways heat equation by a wavelet-Galerkin method,”

Inverse Problems, vol. 13, no. 4, pp. 1093–1106, 1997.

9 T. Regi ´nska and L. Eld´en, “Stability and convergence of the wavelet-Galerkin method for the sideways heat equation,” Journal of Inverse and Ill-Posed Problems, vol. 8, no. 1, pp. 31–49, 2000.

10 C.-L. Fu and C. Y. Qiu, “Wavelet and error estimation of surface heat flux,” Journal of Computational and Applied Mathematics, vol. 150, no. 1, pp. 143–155, 2003.

11 X.-T. Xiong and C.-L. Fu, “Determining surface temperature and heat flux by a wavelet dual least squares method,” Journal of Computational and Applied Mathematics, vol. 201, no. 1, pp. 198–207, 2007.

12 W. Cheng, C.-L. Fu, and Z. Qian, “A modified Tikhonov regularization method for a spherically symmetric three-dimensional inverse heat conduction problem,” Mathematics and Computers in Simulation, vol. 75, no. 3-4, pp. 97–112, 2007.

13 U. Tautenhahn, “Optimal stable approximations for the sideways heat equation,” Journal of Inverse and Ill-Posed Problems, vol. 5, no. 3, pp. 287–307, 1997.

14 H.-T. Chen, S.-Y. Lin, and L.-C. Fang, “Estimation of surface temperature in two-dimensionnal inverse heat conduction problems,” International Journal of Heat and Mass Transfer, vol. 44, no. 8, pp. 1455–1463, 2001.

15 H. R. Busby and D. M. Trujillo, “Numerical soluition to a two-dimensionnal inverse heat conduction problem,” International Journal for Numerical Methods in Engineering, vol. 21, no. 2, pp. 349–359, 1985.

16 O. M. Alifanov and N. V. Kerov, “Determination of external thermal load parameters by solving the two-dimensional inverse heat-conduction problem,” Journal of Engineering Physics, vol. 41, no. 4, pp.

1049–1053, 1981.

17 H. Louahlia-Gualous, P. K. Panday, and E. A. Artyukhin, “The inverse determination of the local heat transfer coefficients for nucleate boiling on horizontal cylinder,” Journal of Heat Transfer, vol. 125, no.

1, pp. 1087–1095, 2003.

18 Y. C. Hon and T. Wei, “A fundamental solution method for inverse heat conduction problem,”

Engineering Analysis with Boundary Elements, vol. 28, no. 5, pp. 489–495, 2004.

19 A. Shidfar and R. Pourgholi, “Numerical approximation of solution of an inverse heat conduction problem based on Legendre polynomials,” Applied Mathematics and Computation, vol. 175, no. 2, pp.

1366–1374, 2006.

20 T. Regi ´nska, “Application of wavelet shrinkage to solving the sideways heat equation,” BIT Numerical Mathematics, vol. 41, no. 5, pp. 1101–1110, 2001.

21 M. Abramowitz and I. A. Stegun, Eds., Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, Dover, New York, NY, USA, 1972.

22 J. R. Wang, “The multi-resolution method applied to the sideways heat equation,” Journal of Mathematical Analysis and Applications, vol. 309, no. 2, pp. 661–673, 2005.