Comp. Appl. Math. DOI 10.1007/s40314-013-0102-y
Restoration of the heat transfer coefficient from boundary measurements using the Sinc method Reza Zolfaghari · Abdollah Shidfar
Received: 18 October 2013 / Accepted: 12 November 2013 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2013
Abstract In this paper, the restoration of the time-dependent heat transfer coefficient in the inverse heat conduction problems from some standard or non-standard boundary measurements is investigated. Depending on additional information, the inverse problems can be transformed into an equivalent set of integral equations with convolution kernels. By using an explicit procedure on the basis of Sinc-function properties, the resulting integral equations are replaced by a system of nonlinear algebraic equations, whose solution yields accurate approximations for missing terms involving the boundary and interior temperatures, the heat fluxes and the heat transfer coefficient. Some examples are considered to illustrate the ability of the proposed method. Keywords Heat conduction problem · Heat transfer coefficient · Inverse problem · Sinc functions Mathematics Subject Classification
35K05 · 80A23 · 45G15
1 Introduction The heat transfer coefficient characterizes the contribution that an interface makes to the overall thermal resistance to the system and is defined in terms of the heat flux across the surface for a unit temperature gradient (Onyango et al. 2009). It is well known that the heat transfer coefficient is an important value to determine in heat transfer.
Communicated by Cristina Turner. R. Zolfaghari (B) Department of Computer Science, Salman Farsi University of Kazerun, 7319673544 Kazerun, Iran e-mail:
[email protected];
[email protected] A. Shidfar School of Mathematics, Iran University of Science and Technology, Tehran, Iran e-mail:
[email protected]
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Over the last years, non-intrusive experimental techniques have been used to determine the convective heat transfer coefficients both in industry and academia rely on analytical solutions for the surface temperature. For instance, techniques based on characteristic colour changes of liquid crystal films at a given temperature (Hippensteele et al. 1983), or on laser-induced fluorescence (Bizzak and Chyu 1995), both rely on the analytical temperature solution for a semi-infinite medium to determine the heat transfer coefficient at a point once the temperature history is obtained from the experiment. Recently, there has been growing interest in developing computational techniques for the reconstruction of the heat transfer coefficient in inverse heat conduction problems (Onyango et al. 2008, 2009; Kostin and Prilepko 1996; Xiong et al. 2010). In this paper, we consider the inverse problem of determination a pair of functions {T, σ }, where T represents the temperature and σ is the heat transfer coefficient, in the following heat conduction equation ∂T ∂2T (x, t) + S(x, t), (x, t) ∈ (0, 1) × (0, tf ), (x, t) = ∂t ∂x2
(1)
subject to the initial condition T (x, 0) = p(x), x ∈ (0, 1),
(2)
boundary conditions −
∂T (0, t) + σ (t)T (0, t) = g0 (t), t ∈ (0, tf ], ∂x ∂T (1, t) + σ (t)T (1, t) = g1 (t), t ∈ (0, tf ], ∂x
(3) (4)
and the specification of either one of the following two boundary temperature measurements λ0 T (0, t) + λ1 T (1, t) = χ(t), t ∈ (0, tf ],
(5)
T 2 (0, t) + T 2 (1, t) = χ(t), t ∈ (0, tf ],
(6)
or
where tf > 0 is an arbitrary fixed time of interest, S is the source term, λ0 and λ1 are given constants, and p, g0 , g1 and χ are known functions. In Onyango et al. (2008), the boundary element method is used for the problem (1)–(4) with the standard boundary measurements (5). In Onyango et al. (2009), the authors considered the additional information given by the non-standard boundary measurement (6) and numerical results obtained using the boundary element method are presented and discussed. Sinc methods have been recognized as powerful tools for solving a wide class of problems arising from scientific and engineering applications including, inverse problems (Shidfar et al. 2009; Shidfar and Zolfaghari 2011), population growth (Al-Khaled 2005), heat transfer (Narasimhan et al. 1998), fluid mechanics (Winnter et al. 2000) and medical imaging (Stenger and ÓReilly 1998). There are several advantages to using approximations based on Sinc numerical methods. Unlike most numerical techniques, it is now well-established that they are characterized by exponentially decaying errors (Keinert 1991; Rashidinia and Zarebnia 2007), and also, they are highly efficient and adaptable in handling problems with singularities (Stenger 1995). Finally, due to their rapid convergence, Sinc numerical methods do not suffer from the common instability problems associated with other numerical methods (Sababheh and Al-Khaled 2003). The books (Lund and Bowers 1992; Cannon 1984) provide excellent
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Restoration of the heat transfer coefficient from boundary measurements
overviews of methods based on Sinc functions for solving ordinary and partial differential equations and integral equations. In this paper, we make use of the Sinc method to determine the functions T and σ which satisfy the problem (1)–(4) with both cases additional information (5) and (6). The method consists of reducing the solution of the inverse problem to a system of integral equations. The properties of Sinc function are then utilized for replacing the resulting integral equations by a system of nonlinear algebraic equations. In the following statement, we call the problem (1)–(5) to Problem I, and the problem (1)–(4) and (6) to Problem II. This paper is organized as follows. In Sect. 2, we briefly review the concept and some properties of the Sinc function and describe the collocation procedure by means of Sinc method for approximating convolution integrals. In Sect. 3, we reduce the inverse problems into the systems of integral equations. Section 4 contains the construction of the Sinc-collocation method to replace the integral equations obtained in Sect. 3 by an explicit system of nonlinear algebraic equations. Finally, some numerical results are presented in Sect. 5.
2 Collocating convolutions In this section, we review some preliminary concepts of Sinc approximation necessary for collocating convolution integrals. The translated Sinc functions with evenly spaced nodes are given by x − kh S(k, h)(x) = sinc , h where k is an integer, h > 0 is a step size and sin(π x) πx , sinc(x) = 1,
x = 0, x = 0.
For purposes of Sinc approximation, consider the case of a finite interval (a, b). Define φ by w = φ = log[(z − a)/(b − z)]; this function φ provides a conformal transformation of the “eye-shaped” region D = {z ∈ C : |arg[(z − a)/(b − z)]| < d}, onto the strip Dd defined by Dd = {z ∈ C : |I m(z)| < d}. The same function φ also provided a one-to-one transformation of (a, b) onto the real line R. The Sinc points are defined for h > 0 and k = 0, ±1, ±2, . . ., by z k = φ −1 (kh) = (a + bekh )/(1 + ekh ). There are three important spaces of functions, H 1 (D ), Lα (D ) and Mα (D ) associated with Sinc approximation on the interval (a, b). Let H 1 (D ) denote the family of all functions f that are analytic in D , such that | f (z)||dz| < ∞. ∂D
Corresponding to number α, let Lα (D ) be the set of all analytic functions f , for which there exists a constant c1 , such that | f (z)| ≤ c1
|ρ(z)|α , z ∈ D, (1 + |ρ(z)|)2α
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where ρ(z) = eφ(z) . The family Mα (D ) consists of all functions f that are analytic in D and continuous in D such that g ∈ Lα (D ), where g(z) = f (z) −
f (a) + ρ(z) f (b) . 1 + ρ(z)
Now, we describe the Sinc-collocation procedure for approximating convolution μ of functions f and g, defined by the integral x μ(x) =
f (x − t)g(t)dt, x ∈ (a, b).
(7)
a
The method of the present section provides an explicit procedure for accurate approximation of μ when either of f or g has singularities at one of both endpoints of its interval of definition, or in the case that μ has singularities at one or both of the endpoints of (a, b) (Stenger 1993). We assume that g ∈ H 1 (D ), and that f is analytic in a domain D f , with φ f denoting a conformal mapping of D f onto Dd , and φ f : (0, c) → R, c being an arbitrary number on the interval [2(b − a), ∞]. Corresponding to a positive integer N we set m = 2N + 1, and 1 we determine h via the formula h = ( απNd ) 2 . Let (−1) δ jk
1 = + 2
j−k 0
sin(π x) dx, πx (−1)
(−1)
then we define a matrix whose ( j, k)th entry is given by δ jk as I (−1) = [δ jk ], and the square matrix Am is obtained by 1 1 Am = h I (−1) diag ,..., . φ (z −N ) φ (z N ) Throughout this paper, the Laplace transformation means the function F defined by c F(s) =
t
f (t)e− s dt,
0
where c defined as above, and we shall assume that the Laplace transformation exists for some c ∈ [2(b − a), ∞], for all s on the right half of the complex plane, i.e., + = {z ∈ C : Re(z) > 0}. Now, by above assumptions, we describe the approximation procedure for μ in (7). If the nonsingular matrix X m and complex numbers s j are determined such that −1 , Am = X m diag[s−N , . . . , s N ]X m
then, square matrix F(Am ) may be defined via the equation −1 . F(Am ) = X m diag[F(s−N ), . . . , F(s N )]X m
Now, define column vectors G m and Pm by G m = [g(z −N ), . . . , g(z N )]T , Pm = [μ−N , . . . , μ N ]T = F(Am )G m ,
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(8)
Restoration of the heat transfer coefficient from boundary measurements
then, the component μ j of vector Pm approximates the value μ(x) at the Sinc point x = z j . Thus, the approximation of μ on (a, b) takes the form μ(x) ≈
N
μ j ω j (x) = {F(Am )G m }T W (x), x ∈ (a, b),
j=−N
where W (x) = [ω−N (x), . . . , ω N (x)]T , and {ω j } is a Sinc basis as follows: λ j (x) = S( j, h)oφ(x),
j = −N , . . . , N ,
ω j (x) = λ j (x),
j = −N + 1, . . . , N − 1, ⎧ ⎫ N ⎬
⎨ λ (x) 1 j − , ω−N (x) = 1 + e−N h ⎩ 1 + ρ(x) 1 + e jh ⎭ j=−N +1 ⎧ ⎫ N −1
⎨ ρ(x) j h e λ j (x) ⎬ . ω N (x) = 1 + e−N h − ⎩ 1 + ρ(x) 1 + e jh ⎭ j=−N
Note that the functions ω j defined above satisfy the relation ω j (z k ) = δ jk . Theorem 1 Let μ be defined as (7) where the Laplace transformation of f with c ≥ 2(b −a) exists for all s in + , and let F(s) = O (s) as s → ∞ in + . Let g ∈ H 1 (D ), and let α and α f , be positive constants such that 0 < α ≤ 1. Set τ P(r, τ ) =
f (r + τ − η)g(η)dη, a
and assume that P(r, .) ∈ Mα (D ), uniformly, for r ∈ [0, b − a], and also that |Pr (r, τ )| ≤ c2
[ρ f (r )]α f φ f (r ) [1 + ρ f (r )]2α f
,
for all r ∈ [0, b − a], and for all τ ∈ D , with c2 a constant independent of r and τ . Let 1 h = ( απNd ) 2 , then there exists a constant c3 which is independent of N such that 1
1 2
μ − {F(Am )G m }T W ∞ ≤ c3 N 2 e−(π dα N ) .
Proof Stenger (1993)
3 Integral representation of the problems The inverse Problems I and II can be transformed into an equivalent set of integral equations. Consider the heat conduction problem (1)–(4) and assume that the source function S(x, t) is bounded over the domain Q = (0, 1) × (0, tf ), and that S(x, t) is uniformly Hölder continuous on each compact subset of Q. Also, we shall assume that the initial function p is piecewise-continuous. So, the bounded unique solution T of the problem (1)–(4) is the form (Cannon 1984)
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t T (x, t) = w(x, t) − 2
θ (x, t − τ )σ (τ )ϕ0 (τ )dτ 0
t θ (x − 1, t − τ )σ (τ )ϕ1 (τ )dτ,
−2
(9)
0
where ϕ0 , ϕ1 and σ are piecewise-continuous solutions of the following system of integral equations t ϕ0 (t) = w(0, t) − 2
θ (0, t − τ )σ (τ )ϕ0 (τ )dτ 0
t −2
θ (1, t − τ )σ (τ )ϕ1 (τ )dτ,
(10)
0
t ϕ1 (t) = w(1, t) − 2
θ (1, t − τ )σ (τ )ϕ0 (τ )dτ 0
t θ (0, t − τ )σ (τ )ϕ1 (τ )dτ,
−2
(11)
0
with 1 w(x, t) =
t 1 G(x, t, ζ, 0) p(ζ )dζ +
0
G(x, t, ζ, τ )S(ζ, τ )dζ dτ 0
0
t
t θ (x, t − τ )g0 (τ )dτ + 2
+2 0
θ (x − 1, t − τ )g1 (τ )dτ, 0
G(x, t, ζ, τ ) = θ (x − ζ, t − τ ) + θ (x + ζ, t − τ ), and +∞ 1 (x + 2n)2 θ (x, t) = √ exp − . 4t 4πt n=−∞ On considering the additional information (5) or (6), then the reformulation of the Problems I and II becomes: Problem I We distinguish three cases: 1. In the case λ0 = 0 and λ1 = 0, we have ϕ0 (t) = λ10 χ(t), which is known. Therefore, the problem is reduced to the system of integral equations (10) and (11), where ϕ1 and σ are unknown. 2. In the case λ0 = 0 and λ1 = 0, we have ϕ1 (t) = λ11 χ(t), which is known. Therefore, the problem is reduced to (10) and (11), where ϕ0 and σ are unknown.
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Restoration of the heat transfer coefficient from boundary measurements
3. In the case λ0 , λ1 = 0, we may write ϕ1 (t) =
1 λ0 χ(t) − ϕ0 (t), t ∈ (0, tf ]. λ1 λ1
(12)
Substituting (12) in (10) and (11), yields the following integral equations t ϕ0 (t) + 2
{θ (0, t − τ ) − 0
+
2 λ1
t θ (1, t − τ )σ (τ )χ(τ )dτ = w(0, t),
2 λ1
(13)
0
λ0 ϕ0 (t) − 2 λ1 −
λ0 θ (1, t − τ )}σ (τ )ϕ0 (τ )dτ λ1
t {θ (1, t − τ ) − 0
λ0 θ (0, t − τ )}σ (τ )ϕ0 (τ )dτ λ1
t θ (0, t − τ )σ (τ )χ(τ )dτ = 0
1 χ(t) − w(1, t). λ1
(14)
So, in this case, the inverse problem is reduced to the system of integral equations (13) and (14) where σ and ϕ0 must be determined. Problem II This problem is reduced to (10) and (11) and ϕ02 (t) + ϕ12 (t) = χ(t),
(15)
where ϕ0 , ϕ1 and σ are unknown.
4 Approximation based on Sinc collocation In Sect. 3, we transformed the inverse Problems I and II into an equivalent set of integral equations. In this section, we will use the Sinc method in order to reduce the corresponding system of integral equations to a system of nonlinear algebraic equations. The Laplace transformation for θ (x, t), with c = ∞, can be determined as ∞ (x, s) =
t
θ (x, t)e− s dt =
∞
s s cos(kπ x) + , 2 1 + k2π 2s k=1
0
for s ∈ + , x ∈ [0, 1]. In particular, for x = 0, 1, we may write √ s 1 coth √ , s ∈ + , (0, s) = 2 s ∞ s s (1, s) = + (−1)k , s ∈ + . 2 1 + k2π 2s k=1
According to (8), for given x ∈ [0, 1], the square matrix (x, Am ) may be defined via the relation −1 . (x, Am ) = X m diag[(x, s−N ), . . . , (x, s N )]X m
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Let Wα (D ), with 0 < α ≤ 1, denote the family of all functions g ∈ H 1 (D ), such that |ϒr (r, x, τ )| ≤ C1
r ξ −1 , r ∈ (0, tf ), τ ∈ D , x ∈ [0, 1], (1 + r )2ξ
with C1 and ξ constants independent of r , τ and x, where τ θ (x, r + τ − η)g(η)dη,
ϒ(r, x, τ ) = 0
(D ),
uniformly, for r ∈ (0, tf ). and ϒ(r, .) ∈ Mα Assume that σ ϕk ∈ Wα (D ), by using the numerical procedure for convolution integrals described in Sect. 2, we may write xi θ (d, t − τ )σ (τ )ϕk (τ )dτ =
N
ϑidj σ (x j )ϕk (x j ) + E i (θd , σ ϕk ),
(16)
j=−N
0
for k, d = 0, 1 where i = −N , . . . , N and ϑidj is the (i, j)th entry of the matrix (d, Am ). By Theorem 1, we have 1 2
1
E i (θ j , σ ϕk ) ≤ C jk N 2 e−(π dα N ) ,
j, k = 0, 1, i = −N , . . . , N ,
which C jk is independent of N . At t = xi , i = −N , . . . , N , we have, on substituting (16) in (10) and (11), ϕ0 (xi ) = w(0, xi ) − 2
N
ϑi0j σ (x j )ϕ0 (x j )
j=−N
−2
N
ϑi1j σ (x j )ϕ1 (x j ) + 2E i (θ0 , σ ϕ0 ) + 2E i (θ1 , σ ϕ1 ),
(17)
j=−N
ϕ1 (xi ) = w(1, xi ) − 2
N
ϑi1j σ (x j )ϕ0 (x j )
j=−N
−2
N
ϑi0j σ (x j )ϕ1 (x j ) + 2E i (θ1 , σ ϕ0 ) + 2E i (θ0 , σ ϕ1 ).
(18)
j=−N
Using equations (17) and (18), the numerical discretization of the equivalent system of integral equations for Problems I and II becomes: Problem I 1. In the case λ0 = 0 and λ1 = 0, the corresponding approximating equations are as follows: N N 2 0 1 ϑi j χ(x j )σ j + 2 ϑi1j σ j ϕ 1j = w(0, xi ) − χ(xi ), λ0 λ0 j=−N
N N 2 1 ϑi j χ(x j )σ j + ϕi1 + 2 ϑi0j σ j ϕ 1j = w(1, xi ), λ0 j=−N
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(19)
j=−N
j=−N
(20)
Restoration of the heat transfer coefficient from boundary measurements
where ϕk1 and σk , k = −N , . . . , N must be determined. ϕk1 and σk have been found, we obtain the approximations TN (x, t) and σ N (t) for the inverse problem by N
σ N (t) =
σ j ω j (t), t ∈ (0, tf ),
(21)
j=−N
TN (x, t) = w(x, t) − 2{(x, Am )X + (x − 1, Am )Y }T W (t),
(22)
where T 1 χ(x−N )σ−N , . . . , χ(x N )σ N , λ0 T 1 Y = σ−N ϕ−N , . . . , σ N ϕ 1N .
X =
2. In the case λ0 = 0 and λ1 = 0, the corresponding approximating equations are as follows: ϕi0 + 2
N
ϑi0j σ j ϕ 0j +
j=−N
2
N
N 2 1 ϑi j χ(x j )σ j = w(0, xi ), λ1
(23)
N 2 0 1 ϑi j χ(x j )σ j = w(1, xi ) − χ(xi ), λ1 λ1
(24)
j=−N
ϑi1j σ j ϕ 0j +
j=−N
j=−N
where ϕk0 and σk , k = −N , . . . , N must be determined. In this case, the approximations TN (x, t) and σ N (t) for the inverse problem can be written by (21) and (22) where 0 , . . . , σ N ϕ 0N ]T , X = [σ−N ϕ−N 1 Y = [χ(x−N )σ−N , . . . , χ(x N )σ N ]T . λ1
3. In the case λ0 , λ1 = 0, the corresponding approximating equations are as follows: ϕi0 + 2
N
ϑi0j −
j=−N
N λ0 1 2 1 ϑi j σ j ϕ 0j + ϑi j χ(x j )σ j = w(0, xi ), λ1 λ1
(25)
j=−N
N N λ0 0 λ0 2 0 1 ϑi1j − ϑi0j σ j ϕ 0j − ϕi − 2 ϑi j χ(x j )σ j = χ(xi ) − w(1, xi ), λ1 λ1 λ1 λ1 j=−N
j=−N
(26) where ϕk0 and σk , k = −N , . . . , N must be determined. Then, the approximations TN (x, t) and σ N (t) for the inverse problem can be written by (21) and (22) where 0 , . . . , σ N ϕ 0N ]T , X = [σ−N ϕ−N T 1 λ0 χ(x−N )σ−N , . . . , χ(x N )σ N − Y = X. λ1 λ1
Problem II Using equations (15), (17) and (18), the system of algebraic equations for this problem is as follows: ϕi0 + 2
N j=−N
ϑi0j σ j ϕ 0j + 2
N
ϑi1j σ j ϕ 1j = w(0, xi ),
j=−N
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(27)
R. Zolfaghari, A. Shidfar
2
N
ϑi1j σ j ϕ 0j + ϕi1 + 2
j=−N
N
ϑi0j σ j ϕ 1j = w(1, xi ),
(28)
j=−N
(ϕi0 )2 + (ϕi1 )2 = χ(xi ), i = −N , . . . , N ,
(29)
and σi , i = −N , . . . , N are unknown. By solving the nonlinear system of where algebraic equations (27), (28) and (29), we obtain approximate solutions TN (x, t) and σ N (t) for the inverse problem by (21) and (22) where ϕi0 ,
ϕi1
0 X = [σ−N ϕ−N , . . . , σ N ϕ 0N ]T , 1 Y = [σ−N ϕ−N , . . . , σ N ϕ 1N ]T .
5 Numerical results In this section, we present some results of numerical comparison of the approximations TN and σ N given in Sect. 4 and the corresponding analytical solutions T and σ of the problem. In all examples we take α = 1 and d = π2 , which yields h = √π . 2N In practical applications, data contain random noise. We will illustrate the effect of the solution in virtue of the noisy data pδ (x) = p(x)(1 + δ sin 50x), giδ (t) = gi (t)(1 + δ sin 50t), i = 0, 1, χδ (t) = χ(t)(1 + δ sin 50t), where δ is the noise parameter. Example 1 Consider (1)–(4) with tf = 1, and p(x) = cos x + x 2 + 2, g0 (t) = (2 + e−t + 2t)(et − t − sin t), g1 (t) = 2 − e−t sin 1 + (3 + 2t + e−t cos 1)(et − t − sin t), S(x, t) = 0, and for the additional information, consider T (0, t) = e−t + 2t + 2.
(30)
The analytical solution for this problem is T (x, t) = e−t cos x + x 2 + 2t + 2, and σ (t) = et − sin t − t. This problem has been solved by taking different values of N . We report the relative errors of TN (1, t) and σ N (t) for N = 5, 10, 15 and 20 in Tables 1 and 2, respectively. As seen from the tables that approximations are improved by increasing the number of nodes. In order to test the stability of the numerical method, T (1, t) and σ (t) together with their numerical solutions for various values of the noise parameter δ = 0, 1 and 2 % are shown in Figs. 1 and 2, respectively. Example 2 Consider the inverse problem given in Example 1, except with the additional information (30) replaced by 0.7T (0, t) + 0.3T (1, t) = 2.3 + 2t + e−t (0.7 + 0.3 cos 1).
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Restoration of the heat transfer coefficient from boundary measurements Table 1 Relative errors of TN (1, t) with various values of N from Example1
t
T (1, t) Exact
N =5 Error
N = 10 Error
N = 20 Error
0.1
3.68889
1.44 × 10−3
1.82 × 10−4
4.85 × 10−6
0.2
3.84236
2.07 × 10−4
4.17 × 10−5
6.04 × 10−7
0.3
4.00027
2.58 × 10−4
1.72 × 10−5
2.69 × 10−6
4.16218
2.43 × 10−4
2.86 × 10−5
1.59 × 10−6
0.5
4.32771
6.42 × 10−5
5.50 × 10−6
2.09 × 10−8
0.6
4.49652
1.62 × 10−4
1.06 × 10−5
8.11 × 10−7
4.66831
1.43 × 10−6
1.10 × 10−5
8.39 × 10−7
0.8
4.84277
1.91 × 10−5
8.31 × 10−6
7.14 × 10−7
0.9
5.01967
2.38 × 10−5
4.01 × 10−6
1.72 × 10−7
t
σ (t) Exact
N =5 Error
N = 15 Error
N = 20 Error
0.1
0.905338
6.35 × 10−4
9.19 × 10−5
4.62 × 10−6
0.2
0.822733
5.49 × 10−4
7.70 × 10−5
4.30 × 10−6
0.754339
2.55 × 10−4
3.86 × 10−5
3.00 × 10−6
0.4
0.702406
8.55 × 10−5
1.47 × 10−5
3.41 × 10−7
0.5
0.669296
2.29 × 10−5
1.56 × 10−6
1.64 × 10−7
0.657476
2.91 × 10−5
2.77 × 10−6
8.11 × 10−7
0.7
0.669535
2.59 × 10−5
2.76 × 10−6
1.03 × 10−6
0.8
0.708185
4.46 × 10−5
4.10 × 10−6
9.16 × 10−7
0.776276
2.76 × 10−5
3.43 × 10−6
1.01 × 10−6
0.4
0.7
Table 2 Relative errors of σ N (t) with various values of N from Example1
0.3
0.6
0.9
T 1,t
5.0
4.5
Exact δ =0 δ =1% δ =2%
4.0
3.5
—— - - - - - - - - - -
t 0.0
0.2
0.4
0.6
0.8
1.0
Fig. 1 The numerical and analytical boundary temperature T (1, t) in Example 1 when N = 5
In Figs. 3 and 4 we show the comparison between analytical heat fluxes q(0, t) = − ∂∂Tx (0, t) and q(1, t) = ∂∂Tx (1, t) and numerical heat fluxes obtained by the proposed method with N = 5.
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R. Zolfaghari, A. Shidfar σ t 1.0 Exact =0 =1% =2%
0.9
—— - - - - - - - - - -
0.8
0.7
t 0.0
0.2
0.4
0.6
0.8
1.0
Fig. 2 The numerical and analytical heat transfer coefficient σ (t) in Example 1 when N = 5 q 0,t 1.0
0.5
Exact —— =0 - - - =0.5% - - - =1% - - - -
0.0
t 0.0
0.2
0.4
0.6
0.8
1.0
1.0
Fig. 3 The numerical and exact heat flux q(0, t) in Example 2 when N = 5 q 1,t 1.7
1.6
Exact —— =0 - - - =0.5% - - - =1% - - - -
1.5
1.4
1.3
t 0.2
0.4
0.6
0.8
1.0
Fig. 4 The numerical and exact heat flux q(1, t) in Example 2 when N = 5
The numerical method is accurate also for approximating the temperature T (x, t) for other values of the time t and the space x. To illustrate this, we show in Figs. 5 and 6, a twodimensional surface representing the absolute value of the difference between the analytical
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Restoration of the heat transfer coefficient from boundary measurements 1.0
x 0.5 0.0 0.020 0.015 0.010 0.005 0.000 0.0 0.5 t
1.0
Fig. 5 The difference |T (x, t) − T2 (x, t)| in Example 2 1.0
x 0.5 0.0 0.010
0.005
0.0
0.000
0.5 t 1.0
Fig. 6 The difference |T (x, t) − T5 (x, t)| in Example 2 EN 0.00030
T(0,0.5)
0.00025 0.00020
T(1,0.5)
0.00015 0.00010 0.00005 N 5
10
15
20
Fig. 7 The reduction in the error for boundary temperatures as a function of N in Example 3
solution and the numerical one over the solution domain, obtained with N = 2 and N = 5. From this figure it can be seen that the described computational technique provides accurate results even when we employ a small number of collocation points.
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R. Zolfaghari, A. Shidfar EN q(1,0.5) 0.005
q(0,0.5)
0.004 0.003 0.002 0.001
N 5
10
15
20
Fig. 8 The reduction in the error for heat fluxes as a function of N in Example 3 EN σ 0.5 0.001 5 10
4
1 10
4
5 10
5
1 10
5
5 10
6
N 5
10
15
20
Fig. 9 Log plot of |σ (0.5) − σ N (0.5)| σ t 1.0 Exact =0 =1% =2%
0.9
—— - - - - - - - - - -
0.8
0.7
t 0.0
0.2
0.4
0.6
0.8
1.0
Fig. 10 The effect of noisy data in σ5 (t) from Example 3
Example 3 Consider the inverse problem given in Example 1, except with the additional information (30) replaced by T 2 (0, t) + T 2 (1, t) = (2 + e−t + 2t)2 + (3 + 2t + e−t cos 1)2 .
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(32)
Restoration of the heat transfer coefficient from boundary measurements
Figures 7 and 8, show the reduction in the error for boundary temperatures and heat fluxes at t = 0.5 by increasing the value of N . Finally, the log plot E N {σ (0.5)} = |σ (0.5) − σ N (0.5)| as a function of N , and the effect of noisy data in σ5 (t), are shown in Figs. 9 and 10, respectively.
6 Conclusion The exponential convergence rate of Sinc approximation makes this approach very attractive and contributes to the good agreement between approximate and exact values. In this paper, identifying the temperature and the heat transfer coefficient in an IHCP subject to the standard or nonstandard boundary measurements, has been recast as an equivalent system of convolution-type integral equations. Then, the Sinc-collocation procedure for approximating convolution integrals was successfully employed for obtaining an accurate approximate solution to the inverse problem. Results were presented both on the boundary and inside the solution domain to illustrate that the numerical method developed here provides an efficient technique, in terms of accuracy and convergence, to investigate this IHCP numerically. In addition, the numerical tests showed that the described computational technique provides accurate results even when we employ a small number of collocation points. The numerical strategy introduced in this paper can be extended to higher dimensions, and this numerical development will be undertaken in a future work.
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