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Numerical approximation of solution of an inverse heat conduction problem based on Legendre polynomials
Applied Mathematics and Computation 175 (2006) 1366–1374
www.elsevier.com/locate/amc
Numerical approximation of solution of an inverse heat conduction problem based on Legendre polynomials A. Shidfar *, R. Pourgholi Department of Mathematics, Iran University of Science and Technology, Narmak, Tehran 16, Iran
Abstract In this paper, we consider an inverse heat conduction problem (IHCP). The given heat conduction equation, the boundary condition, and the initial condition are presented in a dimensionless form. A set of temperature measurements at a single sensor location inside the heat conduction body is required. Using a linear transformation, the ill-posed IHCP becomes a Cauchy problem. This problem will be solved by applying Legendre polynomials. Results show that an excellent estimation can be obtained within a couple of minutes CPU time at pentium IV-2.4 GHz PC. Ó 2005 Elsevier Inc. All rights reserved. Keywords: Inverse heat conduction problem; Existence; Uniqueness; Stability; Legendre polynomials
*
Corresponding author. E-mail addresses: [email protected] (A. Shidfar), [email protected] (R. Pourgholi).
0096-3003/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.08.040
A. Shidfar, R. Pourgholi / Appl. Math. Comput. 175 (2006) 1366–1374
1367
1. Introduction Solving IHCPÕs needs additional information about temperature history. This new data is usually given by temperature sensor which is located on the boundary or inside the body. To date, various methods have been developed for the analysis of the inverse problems and inverse heat conduction problems involving the estimation of heat flux by measuring temperature inside the material [1–11]. In this work, by using the Legendre polynomials, a stable solution for an inverse heat conduction problem will be presented.
2. Mathematical formulation Consider a one dimensional inverse heat conduction problem in the domain D ¼ fðx; tÞj0 < x < 1; t > 0g. The dimensionless mathematical formulation of this problem may be expressed as follows: T t ¼ T xx ; ðx; tÞ 2 D; T ðx; 0Þ ¼ f ðxÞ; 0 < x1 < x < 1;
ð1Þ ð2Þ
T ðx1 ; tÞ ¼ gðtÞ; T ð1; tÞ ¼ hðtÞ; t > 0; jT ðx; tÞj < C; 0 < x < 1; t > 0;
ð3Þ ð4Þ
where C is a known positive constant, the location x1 of the sensor is assumed to be measured and to have negligible error. Thickness of the bar with unite length also known and considered errorless, and f(x) is piecewise continuous known function, g(t) and h(t) are infinitely differentiable functions while the temperature T(x, t) and heat flux Tx(0, t) are unknown which remain to be determined. This problem is quite different from the direct problem in that the boundary condition is not specified at x = 0 but instead a measured temperature history is given at one or more internal locations. The problem (1)– (4) can be subdivided into two separate problems, one of which is the following direct problem: T t ¼ T xx ;
0 < x1 < x < 1; t > 0;
T ðx; 0Þ ¼ f ðxÞ;
0 < x1 6 x < 1;
T ðx1 ; tÞ ¼ gðtÞ;
T ð1; tÞ ¼ hðtÞ;
jT ðx; tÞj < C;
0 < x < 1; t > 0.
ð5Þ ð6Þ
t > 0;
ð7Þ ð8Þ
The portion of the body from x = x1 to x = 1 may be analyzed as a direct problem because there are known boundary condition at both boundaries x = x1 and x = 1. Other of which is the following inverse problem:
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A. Shidfar, R. Pourgholi / Appl. Math. Comput. 175 (2006) 1366–1374
T t ¼ T xx ; 0 < x < x1 ; T ðx1 ; tÞ ¼ gðtÞ; t > 0;
ð9Þ ð10Þ
jT ðx; tÞj < C;
ð11Þ
0 < x < 1; t > 0.
3. Main results In this section, we consider existence, uniqueness, and instability of the inverse problem (1)–(4). In additional we show that there is always a unique instable solution to the problem (1)–(4). Now using transformations 1
1
X ¼ ð1 x1 Þ x ð1 x1 Þ x1
ð12Þ
and 2
s ¼ ð1 x1 Þ t
ð13Þ
and setting 2
uðX ; sÞ ¼ T ðð1 x1 ÞX þ x1 ; ð1 x1 Þ sÞ;
ð14Þ
then the problem (5)–(8) becomes us ¼ uXX ;
0 < X < 1; s > 0;
uðX ; 0Þ ¼ F ðX Þ; uð0; sÞ ¼ GðsÞ; juðX ; sÞj < C;
0 < X < 1; uð1; sÞ ¼ H ðsÞ; s > 0; 0 < X < 1; s > 0.
ð15Þ ð16Þ ð17Þ ð18Þ
Now we state the following results. Theorem 1 (Existence). For piecewise continuous functions F(X), G(s), and H(s) on their domains, the function Z 1 uðX ; sÞ ¼ fhðX n; sÞ hðX þ n; sÞgF ðnÞ dn 0 Z s Z s oh oh ðX ; s gÞGðgÞ dg þ 2 ðX 1; s gÞH ðgÞ dg; 2 oX oX 0 0 ð19Þ P1 where hðX ; sÞ ¼ m¼1 KðX þ 2m; sÞ, s > 0, and the fundamental solution K(X, s) is defined by 1 X2 KðX ; sÞ ¼ pffiffiffiffiffiffiffiffi e 4s ; 4ps
s > 0;
is a solution of problem (15)–(18) [8].
ð20Þ
A. Shidfar, R. Pourgholi / Appl. Math. Comput. 175 (2006) 1366–1374
1369
Theorem 2 (Uniqueness). If the assumptions of Theorem 1 are fulfilled, then the solution (19) is the unique solution for the problem (15)–(18). Proof. Let u1 and u2 are two solutions of the problem (15)–(18), now by putting u = u1 u2 in the problem (15)–(18) and using the energy method, we will find u1 = u2. By the above result the following IHCP arises, T t ¼ T xx ;
ð21Þ
0 < x < x1 ; t > 0;
T ðx1 ; tÞ ¼ gðtÞ; jT ðx; tÞj < C 1 ;
T x ðx1 ; tÞ ¼ kðtÞ; t > 0;
ð22Þ ð23Þ
0 < x < x1 ; t > 0.
The problem (21)–(23) is a Cauchy problem with the solution [8] " # 2n 2nþ1 1 X dn gðtÞ ðx x1 Þ dn kðtÞ ðx x1 Þ T ðx; tÞ ¼ þ . dtn dtn ð2nÞ! ð2n þ 1Þ! n¼0
ð24Þ
The series given by Eq. (24) is uniformly and absolutely convergent for any bounded t, provided k(t) and g(t) for some positive constants C2 and C3 satisfy n n d gðtÞ ð2nÞ! ð2nÞ! and the condition, d dtkðtÞ dtn 6 C 3 x1 , where n = 1, 2, . . . and x = x1 n 6 C2 x 1 is the heated surface [9–11]. The solution (24), exists and is unique but not always stable [8, chapter 11, Section 4]. We can construct an example of this behavior by considering the following Cauchy problem oT n o2 T n ðx; tÞ ¼ ðx; tÞ; 0 < x < x1 ; t > 0; ot ox2 2 T n ðx1 ; tÞ ¼ n cos 2n2 t; t > 0; oT n ðx; tÞ ¼ n1 ðsin 2n2 t cos 2n2 tÞ; t > 0; n ¼ 1; 2; 3; . . . ; ox
ð25Þ
with the solution T n ðx; tÞ ¼ n2 expfnðx1 xÞg cos½nðx1 xÞ þ 2n2 t.
ð26Þ oT n ox
Note that, as n tends to infinity, the boundary data Tn(x1, t) and ðx1 ; tÞ tend uniformly to zero, while for 0 < x < x1 the function Tn assumes values n2 exp {n(x1 x)}, x < x1, which tend to infinity as n tends to infinity. h
4. Description of the method In this section, let us consider the problem (21)–(23) in the domain 0 < x < x1, 0 < t < T0, where T0 > 2 is a positive constant. Using the transformation
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s ¼ 1 þ
2 t T0
ð27Þ
and putting
ð1 þ sÞT 0 uðx; sÞ ¼ T x; ; 2 we arrive to the following problem T0 us ¼ uxx ; 0 < x < x1 ; 1 < s < 1; 2 uðx1 ; sÞ ¼ G1 ðsÞ; ux ðx1 ; sÞ ¼ K 1 ðsÞ; 1 < s < 1; juðx; sÞj < C 1 ;
0 < x < x1 ; 1 < s < 1.
Now we seek a solution in the form 1 X 1 X uðx; sÞ ¼ C ij pj ðsÞðx x1 Þi ; i¼0
ð28Þ
ð29Þ ð30Þ ð31Þ
ð32Þ
j¼0
where pj(s) is a Legendre polynomial of degree j. Putting (32) in (29) and equating the coefficient of each power of (x x1)i to zero, we find C 0;jþ1 p0jþ1 ðsÞ ¼ T 0 C 2;j pj ðsÞ. In additional from boundary conditions we obtain Z 2j þ 1 1 C 0;j ¼ G1 ðsÞpj ðsÞ ds 2 1 and
Z 2j þ 1 1 K 1 ðsÞpj ðsÞ ds. 2 1 Substituting from the above results into (32), we find ( ! s 1 X 1 s Y X p0jþk ðsÞ 2 C 0;jþs pj ðsÞðx x1 Þ2s uðx; sÞ ¼ T p ðsÞ ð2SÞ! 0 j¼0 s¼0 k¼1;k6¼0 jþk1 !) C 1;jþs 2sþ1 p ðsÞðx x1 Þ þ . ð2s þ 1Þ! j C 1;j ¼
ð33Þ
Now, suppose that G1(s) and K1(s) belong to Holmgren class H(x1, 1, C1, 0), then Eq. (33) represents a unique solution of problem (29)–(31) [8]. Furthermore, if G1(s) and K1(s) approximated by two polynomials of degree n, then by the orthogonality of Legendre polynomials with respect to the weight function W(s) = 1, and the following properties: Z 1 G1 ðsÞP jþs ðsÞ ¼ 0; j þ s > n ð34Þ 1
A. Shidfar, R. Pourgholi / Appl. Math. Comput. 175 (2006) 1366–1374
and
Z
1371
1
K 1 ðsÞP jþs ðsÞ ¼ 0;
j þ s > n;
ð35Þ
1
we will obtain an approximate solution in the form ( ! s n X n s Y X p0jþk ðsÞ 2 uðx; sÞ ffi T0 p ðsÞ j¼0 s¼0 k¼1;k6¼0 jþk1 ) C 0;jþs C 1;jþs 2s 2sþ1 p ðsÞðx x1 Þ þ p ðsÞðx x1 Þ ; ð2SÞ! j ð2s þ 1Þ! j
ð36Þ
for the problem (29)–(31). The unknown heat flux at x = 0 will be obtained by differentiation of (33) with respect to x and putting x = 0. The uniqueness of the solution follows from the analyticity of the solution (33) of the problem (29)–(31) in the spatial variable x [8]. 5. Test problem Mathematically, IHCPs belong the class of ill-posed problems, i.e. small errors in the measured data can lead to large deviations in the estimated quantities. The physical reason for the ill-posedness of the estimation problem is that variations in the surface conditions of the solid body are damped towards the interior because of the diffusive nature of heat conduction. As a consequence, large-amplitude changes at the surface have to be inferred from small-amplitude changes in the measurements data. Errors and noise in the data can therefore be mistaken as significant variations of the surface state by the estimation procedure. The purpose of this section is to illustrate the applicability of the present method described in Section 4 for solving IHCP. As expected, the IHCP is ill-posed and therefore it is necessary to investigate the stability of the present method by giving a test problem. Example. Let us consider T 1 ðx; tÞ ¼ T ðx; tÞ þ T ðx; tÞ;
ð37Þ
where T(x, t) is the solution of the problem (21)–(23) with the given data T ðx1 ; tÞ ¼ x21 þ 2t; T x ðx1 ; tÞ ¼ 2x1 ;
0 6 t 6 T 0;
0 6 t 6 T 0;
and the perturbation function T*(x, t) is expressed as pffiffiffi p T ðx; tÞ ¼ 2xðcÞ cos 2c2 t cx expðcxÞ; 4
ð38Þ
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0; k ¼ 0 and k is a constant [12]. The perturbation 1; k 6¼ 0 function T*(x, t) satisfies the heat equation (21) and it is clear that when c is sufficiently large, the absolute value of T*(x, t) is very small. Therefore T*(x, t) illustrate a small perturbation to the exact input data. Now, consider the problem (21)–(23) with the given data pffiffiffi p T ðx1 ; tÞ ¼ x21 þ 2t þ 2xðcÞ cos 2c2 t cx1 expðcx1 Þ; 4 ð39Þ 0 < t < T 0; h i pffiffiffi oT p p ðx1 ; tÞ ¼ c 2xðcÞ sin 2c2 t cx1 cos 2c2 t cx1 ox 4 4 expðcx1 Þ; 0 < t < T 0 . ð40Þ where c ¼ k 2 p2 , xðcÞ ¼
From Tables 1, 2, and Figs. 1, 2 corresponding to the perturbed and unperturbed solutions, it is found that the results for the surface temperature and the heat flux are almost identical, but they do differ from the perturbed solution (37). Therefore the scheme is able to recognize the correct solution when a small amount of noise is included in the input data. Table 1 The temperature and the heat flux on the boundary x = 0 for T0 = 5 when x1 = 0.25, k = 0 and degree of polynomial is 16 t
0.025 0.225 0.425 0.625 0.825 1.025
Solution (33) when s = 1 + 0.4t
Perturbed solution (38)
T(0, t)
Tx(0, t)
T1(0, t)
q1(0, t)
0.05 0.45 0.85 1.25 1.65 2.05
0.00 0.00 0.00 0.00 0.00 0.00
0.05 0.45 0.85 1.25 1.65 2.05
0.0 0.0 0.0 0.0 0.0 0.0
Table 2 The temperature and the heat flux on the boundary x = 0 for T0 = 5 when x1 = 0.25, k = 3 and degree of polynomial is 16 t
0.025 0.225 0.425 0.625 0.825 1.025
Solution (33) when s = 1 + 0.4t
Perturbed solution (38)
T(0, t)
Tx(0, t)
T1(0, t)
q1(0, t)
0.050519 0.450237 0.850103 1.250040 1.650020 2.050010
0.001907200 0.000870877 0.000372898 0.000151882 0.000058612 0.000022212
0.6873 1.82897 0.70741 0.0380 2.61446 2.72255
41.7004 150.360 137.6450 62.52750 177.544 50.76430
A. Shidfar, R. Pourgholi / Appl. Math. Comput. 175 (2006) 1366–1374 2.5 2 1.5 1 0.5 -0.5
1373
Numeric Exact Perturb 0.2 0.4 0.6 0.8
1
Fig. 1. The exact T(0, t), the estimate T(0, t) and the perturbed T1(0, t) on the boundary x = 0 when x1 = 0.25 and k = 3.
100 50 -50 -100 -150
Perturb 0.2 0.4 0.6 0.8
1
Exact Numeric
Fig. 2. The exact Tx(0, t), the estimate Tx(0, t) and the perturbed q1(0, t) on the boundary x = 0 when x1 = 0.25 and k = 3.
6. Conclusion In this paper the approximate method based on Legendre polynomials is applied successfully for the stable solution of the inverse problem to determine the unknown boundary temperature and heat flux in regular domain by utilizing temperature readings. From numerical example we conclude that the proposed method is efficient and accurate to approximate the unknown boundary temperature and heat flux in an IHCP. In additional all the computations are performed on the PC.
References [1] J. Blum, W. Marquardt, An optimal solution to inverse heat conduction problems based on frequency domain interpretation and observers, Numer. Heat Transfer, Part B: Fundamentals 32 (1997) 453–478. [2] J.V. Beck, B. Blackwell, Ch.R.St. Clair Jr., Inverse Heat Conduction, John Wiley and Sons, Inc., 1985. [3] A. Shidfar, R. Pourgholi, Application of finite difference method to analysis an ill-posed problem, Appl. Math. Comput. 168 (2) (2005) 1400–1408. [4] A. Shidfar, G.R. Karamali, Numerical solution of inverse heat conduction problem with nonstationary measurements, Appl. Math. Comput. 168 (2005) 540–548. [5] A. Shidfar, A. Shahrezaee, M. Garshasbi, A method for solving an inverse biharmonic problem, J. Math. Anal. Appl. 302 (2005) 457–462. [6] A. Shidfar, M. Djalalvand, M. Garshasbi, A numerical scheme for solving special class of nonlinear diffusion–convection equation, Appl. Math. Comput. 167 (2005) 1080–1089.
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[7] O.M. Alifanov, Inverse Heat Conduction Problems, Springer-Verlag, 1994. [8] J.R. Cannon, The One-dimensional Heat Equation, Addison Wesley, Reading, MA, 1984. [9] A.N. Tikhonov, V.Y. Arsenin, Solutions of Ill-posed Problems, V. H. Winston and Sons, Washington, DC, 1977. [10] D.V. Widder, The Heat Equation, Academic Press, New York, 1975. [11] C.F. Weber, Analysis and Solution of the ill-posed inverse heat conduction problem, Int. J. Heat Mass Transfer 24 (1981) 1783–1792. ¨ zisik, Boundary Value Problems of Heat Conduction, Dover, New York, 1989. [12] M.N. O
www.elsevier.com/locate/amc
Numerical approximation of solution of an inverse heat conduction problem based on Legendre polynomials A. Shidfar *, R. Pourgholi Department of Mathematics, Iran University of Science and Technology, Narmak, Tehran 16, Iran
Abstract In this paper, we consider an inverse heat conduction problem (IHCP). The given heat conduction equation, the boundary condition, and the initial condition are presented in a dimensionless form. A set of temperature measurements at a single sensor location inside the heat conduction body is required. Using a linear transformation, the ill-posed IHCP becomes a Cauchy problem. This problem will be solved by applying Legendre polynomials. Results show that an excellent estimation can be obtained within a couple of minutes CPU time at pentium IV-2.4 GHz PC. Ó 2005 Elsevier Inc. All rights reserved. Keywords: Inverse heat conduction problem; Existence; Uniqueness; Stability; Legendre polynomials
*
Corresponding author. E-mail addresses: [email protected] (A. Shidfar), [email protected] (R. Pourgholi).
0096-3003/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.08.040
A. Shidfar, R. Pourgholi / Appl. Math. Comput. 175 (2006) 1366–1374
1367
1. Introduction Solving IHCPÕs needs additional information about temperature history. This new data is usually given by temperature sensor which is located on the boundary or inside the body. To date, various methods have been developed for the analysis of the inverse problems and inverse heat conduction problems involving the estimation of heat flux by measuring temperature inside the material [1–11]. In this work, by using the Legendre polynomials, a stable solution for an inverse heat conduction problem will be presented.
2. Mathematical formulation Consider a one dimensional inverse heat conduction problem in the domain D ¼ fðx; tÞj0 < x < 1; t > 0g. The dimensionless mathematical formulation of this problem may be expressed as follows: T t ¼ T xx ; ðx; tÞ 2 D; T ðx; 0Þ ¼ f ðxÞ; 0 < x1 < x < 1;
ð1Þ ð2Þ
T ðx1 ; tÞ ¼ gðtÞ; T ð1; tÞ ¼ hðtÞ; t > 0; jT ðx; tÞj < C; 0 < x < 1; t > 0;
ð3Þ ð4Þ
where C is a known positive constant, the location x1 of the sensor is assumed to be measured and to have negligible error. Thickness of the bar with unite length also known and considered errorless, and f(x) is piecewise continuous known function, g(t) and h(t) are infinitely differentiable functions while the temperature T(x, t) and heat flux Tx(0, t) are unknown which remain to be determined. This problem is quite different from the direct problem in that the boundary condition is not specified at x = 0 but instead a measured temperature history is given at one or more internal locations. The problem (1)– (4) can be subdivided into two separate problems, one of which is the following direct problem: T t ¼ T xx ;
0 < x1 < x < 1; t > 0;
T ðx; 0Þ ¼ f ðxÞ;
0 < x1 6 x < 1;
T ðx1 ; tÞ ¼ gðtÞ;
T ð1; tÞ ¼ hðtÞ;
jT ðx; tÞj < C;
0 < x < 1; t > 0.
ð5Þ ð6Þ
t > 0;
ð7Þ ð8Þ
The portion of the body from x = x1 to x = 1 may be analyzed as a direct problem because there are known boundary condition at both boundaries x = x1 and x = 1. Other of which is the following inverse problem:
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T t ¼ T xx ; 0 < x < x1 ; T ðx1 ; tÞ ¼ gðtÞ; t > 0;
ð9Þ ð10Þ
jT ðx; tÞj < C;
ð11Þ
0 < x < 1; t > 0.
3. Main results In this section, we consider existence, uniqueness, and instability of the inverse problem (1)–(4). In additional we show that there is always a unique instable solution to the problem (1)–(4). Now using transformations 1
1
X ¼ ð1 x1 Þ x ð1 x1 Þ x1
ð12Þ
and 2
s ¼ ð1 x1 Þ t
ð13Þ
and setting 2
uðX ; sÞ ¼ T ðð1 x1 ÞX þ x1 ; ð1 x1 Þ sÞ;
ð14Þ
then the problem (5)–(8) becomes us ¼ uXX ;
0 < X < 1; s > 0;
uðX ; 0Þ ¼ F ðX Þ; uð0; sÞ ¼ GðsÞ; juðX ; sÞj < C;
0 < X < 1; uð1; sÞ ¼ H ðsÞ; s > 0; 0 < X < 1; s > 0.
ð15Þ ð16Þ ð17Þ ð18Þ
Now we state the following results. Theorem 1 (Existence). For piecewise continuous functions F(X), G(s), and H(s) on their domains, the function Z 1 uðX ; sÞ ¼ fhðX n; sÞ hðX þ n; sÞgF ðnÞ dn 0 Z s Z s oh oh ðX ; s gÞGðgÞ dg þ 2 ðX 1; s gÞH ðgÞ dg; 2 oX oX 0 0 ð19Þ P1 where hðX ; sÞ ¼ m¼1 KðX þ 2m; sÞ, s > 0, and the fundamental solution K(X, s) is defined by 1 X2 KðX ; sÞ ¼ pffiffiffiffiffiffiffiffi e 4s ; 4ps
s > 0;
is a solution of problem (15)–(18) [8].
ð20Þ
A. Shidfar, R. Pourgholi / Appl. Math. Comput. 175 (2006) 1366–1374
1369
Theorem 2 (Uniqueness). If the assumptions of Theorem 1 are fulfilled, then the solution (19) is the unique solution for the problem (15)–(18). Proof. Let u1 and u2 are two solutions of the problem (15)–(18), now by putting u = u1 u2 in the problem (15)–(18) and using the energy method, we will find u1 = u2. By the above result the following IHCP arises, T t ¼ T xx ;
ð21Þ
0 < x < x1 ; t > 0;
T ðx1 ; tÞ ¼ gðtÞ; jT ðx; tÞj < C 1 ;
T x ðx1 ; tÞ ¼ kðtÞ; t > 0;
ð22Þ ð23Þ
0 < x < x1 ; t > 0.
The problem (21)–(23) is a Cauchy problem with the solution [8] " # 2n 2nþ1 1 X dn gðtÞ ðx x1 Þ dn kðtÞ ðx x1 Þ T ðx; tÞ ¼ þ . dtn dtn ð2nÞ! ð2n þ 1Þ! n¼0
ð24Þ
The series given by Eq. (24) is uniformly and absolutely convergent for any bounded t, provided k(t) and g(t) for some positive constants C2 and C3 satisfy n n d gðtÞ ð2nÞ! ð2nÞ! and the condition, d dtkðtÞ dtn 6 C 3 x1 , where n = 1, 2, . . . and x = x1 n 6 C2 x 1 is the heated surface [9–11]. The solution (24), exists and is unique but not always stable [8, chapter 11, Section 4]. We can construct an example of this behavior by considering the following Cauchy problem oT n o2 T n ðx; tÞ ¼ ðx; tÞ; 0 < x < x1 ; t > 0; ot ox2 2 T n ðx1 ; tÞ ¼ n cos 2n2 t; t > 0; oT n ðx; tÞ ¼ n1 ðsin 2n2 t cos 2n2 tÞ; t > 0; n ¼ 1; 2; 3; . . . ; ox
ð25Þ
with the solution T n ðx; tÞ ¼ n2 expfnðx1 xÞg cos½nðx1 xÞ þ 2n2 t.
ð26Þ oT n ox
Note that, as n tends to infinity, the boundary data Tn(x1, t) and ðx1 ; tÞ tend uniformly to zero, while for 0 < x < x1 the function Tn assumes values n2 exp {n(x1 x)}, x < x1, which tend to infinity as n tends to infinity. h
4. Description of the method In this section, let us consider the problem (21)–(23) in the domain 0 < x < x1, 0 < t < T0, where T0 > 2 is a positive constant. Using the transformation
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s ¼ 1 þ
2 t T0
ð27Þ
and putting
ð1 þ sÞT 0 uðx; sÞ ¼ T x; ; 2 we arrive to the following problem T0 us ¼ uxx ; 0 < x < x1 ; 1 < s < 1; 2 uðx1 ; sÞ ¼ G1 ðsÞ; ux ðx1 ; sÞ ¼ K 1 ðsÞ; 1 < s < 1; juðx; sÞj < C 1 ;
0 < x < x1 ; 1 < s < 1.
Now we seek a solution in the form 1 X 1 X uðx; sÞ ¼ C ij pj ðsÞðx x1 Þi ; i¼0
ð28Þ
ð29Þ ð30Þ ð31Þ
ð32Þ
j¼0
where pj(s) is a Legendre polynomial of degree j. Putting (32) in (29) and equating the coefficient of each power of (x x1)i to zero, we find C 0;jþ1 p0jþ1 ðsÞ ¼ T 0 C 2;j pj ðsÞ. In additional from boundary conditions we obtain Z 2j þ 1 1 C 0;j ¼ G1 ðsÞpj ðsÞ ds 2 1 and
Z 2j þ 1 1 K 1 ðsÞpj ðsÞ ds. 2 1 Substituting from the above results into (32), we find ( ! s 1 X 1 s Y X p0jþk ðsÞ 2 C 0;jþs pj ðsÞðx x1 Þ2s uðx; sÞ ¼ T p ðsÞ ð2SÞ! 0 j¼0 s¼0 k¼1;k6¼0 jþk1 !) C 1;jþs 2sþ1 p ðsÞðx x1 Þ þ . ð2s þ 1Þ! j C 1;j ¼
ð33Þ
Now, suppose that G1(s) and K1(s) belong to Holmgren class H(x1, 1, C1, 0), then Eq. (33) represents a unique solution of problem (29)–(31) [8]. Furthermore, if G1(s) and K1(s) approximated by two polynomials of degree n, then by the orthogonality of Legendre polynomials with respect to the weight function W(s) = 1, and the following properties: Z 1 G1 ðsÞP jþs ðsÞ ¼ 0; j þ s > n ð34Þ 1
A. Shidfar, R. Pourgholi / Appl. Math. Comput. 175 (2006) 1366–1374
and
Z
1371
1
K 1 ðsÞP jþs ðsÞ ¼ 0;
j þ s > n;
ð35Þ
1
we will obtain an approximate solution in the form ( ! s n X n s Y X p0jþk ðsÞ 2 uðx; sÞ ffi T0 p ðsÞ j¼0 s¼0 k¼1;k6¼0 jþk1 ) C 0;jþs C 1;jþs 2s 2sþ1 p ðsÞðx x1 Þ þ p ðsÞðx x1 Þ ; ð2SÞ! j ð2s þ 1Þ! j
ð36Þ
for the problem (29)–(31). The unknown heat flux at x = 0 will be obtained by differentiation of (33) with respect to x and putting x = 0. The uniqueness of the solution follows from the analyticity of the solution (33) of the problem (29)–(31) in the spatial variable x [8]. 5. Test problem Mathematically, IHCPs belong the class of ill-posed problems, i.e. small errors in the measured data can lead to large deviations in the estimated quantities. The physical reason for the ill-posedness of the estimation problem is that variations in the surface conditions of the solid body are damped towards the interior because of the diffusive nature of heat conduction. As a consequence, large-amplitude changes at the surface have to be inferred from small-amplitude changes in the measurements data. Errors and noise in the data can therefore be mistaken as significant variations of the surface state by the estimation procedure. The purpose of this section is to illustrate the applicability of the present method described in Section 4 for solving IHCP. As expected, the IHCP is ill-posed and therefore it is necessary to investigate the stability of the present method by giving a test problem. Example. Let us consider T 1 ðx; tÞ ¼ T ðx; tÞ þ T ðx; tÞ;
ð37Þ
where T(x, t) is the solution of the problem (21)–(23) with the given data T ðx1 ; tÞ ¼ x21 þ 2t; T x ðx1 ; tÞ ¼ 2x1 ;
0 6 t 6 T 0;
0 6 t 6 T 0;
and the perturbation function T*(x, t) is expressed as pffiffiffi p T ðx; tÞ ¼ 2xðcÞ cos 2c2 t cx expðcxÞ; 4
ð38Þ
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A. Shidfar, R. Pourgholi / Appl. Math. Comput. 175 (2006) 1366–1374
0; k ¼ 0 and k is a constant [12]. The perturbation 1; k 6¼ 0 function T*(x, t) satisfies the heat equation (21) and it is clear that when c is sufficiently large, the absolute value of T*(x, t) is very small. Therefore T*(x, t) illustrate a small perturbation to the exact input data. Now, consider the problem (21)–(23) with the given data pffiffiffi p T ðx1 ; tÞ ¼ x21 þ 2t þ 2xðcÞ cos 2c2 t cx1 expðcx1 Þ; 4 ð39Þ 0 < t < T 0; h i pffiffiffi oT p p ðx1 ; tÞ ¼ c 2xðcÞ sin 2c2 t cx1 cos 2c2 t cx1 ox 4 4 expðcx1 Þ; 0 < t < T 0 . ð40Þ where c ¼ k 2 p2 , xðcÞ ¼
From Tables 1, 2, and Figs. 1, 2 corresponding to the perturbed and unperturbed solutions, it is found that the results for the surface temperature and the heat flux are almost identical, but they do differ from the perturbed solution (37). Therefore the scheme is able to recognize the correct solution when a small amount of noise is included in the input data. Table 1 The temperature and the heat flux on the boundary x = 0 for T0 = 5 when x1 = 0.25, k = 0 and degree of polynomial is 16 t
0.025 0.225 0.425 0.625 0.825 1.025
Solution (33) when s = 1 + 0.4t
Perturbed solution (38)
T(0, t)
Tx(0, t)
T1(0, t)
q1(0, t)
0.05 0.45 0.85 1.25 1.65 2.05
0.00 0.00 0.00 0.00 0.00 0.00
0.05 0.45 0.85 1.25 1.65 2.05
0.0 0.0 0.0 0.0 0.0 0.0
Table 2 The temperature and the heat flux on the boundary x = 0 for T0 = 5 when x1 = 0.25, k = 3 and degree of polynomial is 16 t
0.025 0.225 0.425 0.625 0.825 1.025
Solution (33) when s = 1 + 0.4t
Perturbed solution (38)
T(0, t)
Tx(0, t)
T1(0, t)
q1(0, t)
0.050519 0.450237 0.850103 1.250040 1.650020 2.050010
0.001907200 0.000870877 0.000372898 0.000151882 0.000058612 0.000022212
0.6873 1.82897 0.70741 0.0380 2.61446 2.72255
41.7004 150.360 137.6450 62.52750 177.544 50.76430
A. Shidfar, R. Pourgholi / Appl. Math. Comput. 175 (2006) 1366–1374 2.5 2 1.5 1 0.5 -0.5
1373
Numeric Exact Perturb 0.2 0.4 0.6 0.8
1
Fig. 1. The exact T(0, t), the estimate T(0, t) and the perturbed T1(0, t) on the boundary x = 0 when x1 = 0.25 and k = 3.
100 50 -50 -100 -150
Perturb 0.2 0.4 0.6 0.8
1
Exact Numeric
Fig. 2. The exact Tx(0, t), the estimate Tx(0, t) and the perturbed q1(0, t) on the boundary x = 0 when x1 = 0.25 and k = 3.
6. Conclusion In this paper the approximate method based on Legendre polynomials is applied successfully for the stable solution of the inverse problem to determine the unknown boundary temperature and heat flux in regular domain by utilizing temperature readings. From numerical example we conclude that the proposed method is efficient and accurate to approximate the unknown boundary temperature and heat flux in an IHCP. In additional all the computations are performed on the PC.
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[7] O.M. Alifanov, Inverse Heat Conduction Problems, Springer-Verlag, 1994. [8] J.R. Cannon, The One-dimensional Heat Equation, Addison Wesley, Reading, MA, 1984. [9] A.N. Tikhonov, V.Y. Arsenin, Solutions of Ill-posed Problems, V. H. Winston and Sons, Washington, DC, 1977. [10] D.V. Widder, The Heat Equation, Academic Press, New York, 1975. [11] C.F. Weber, Analysis and Solution of the ill-posed inverse heat conduction problem, Int. J. Heat Mass Transfer 24 (1981) 1783–1792. ¨ zisik, Boundary Value Problems of Heat Conduction, Dover, New York, 1989. [12] M.N. O