 Research
 Open Access
Regularization method for the radially symmetric inverse heat conduction problem
 I Djerrar^{1},
 L Alem^{1} and
 L Chorfi^{1}Email authorView ORCID ID profile
 Received: 12 July 2017
 Accepted: 21 October 2017
 Published: 2 November 2017
Abstract
We consider an axisymmetric inverse problem for the heat equation inside the cylinder \(a\leq r\leq b\). We wish to determine the surface temperature on the interior surface \(\{r=a\}\) from the Cauchy data on the exterior surface \(\{r=b\}\). This problem is illposed. Using the Laplace transform, we solve the direct problem. Then the inverse problem is reduced to a Volterra integral equation of the first kind. A standard Tikhonov regularization method is applied to the approximation of this integral equation when the data is not exact. Some numerical examples are given to illustrate the stability of the proposed method.
Keywords
 illposed problem
 radially symmetric heat equation
 Laplace transform
 Tikhonov regularization
MSC
 35K05
 65N06
 35R30
 44A10
1 Introduction
The inverse heat conduction problems (IHCPs) have many applications in different branches of science and technology. It consists in determining the temperature and heat flux on both sides of the boundary when one side is inaccessible to measurements.
The standard problem of the heat conduction in Cartesian coordinates \(u_{t}=u_{xx}\), \(x\in [0,1]\), \(t>0\), with the data \(u(0,t)=f(t)\) and \(u_{x}(0,t)=0\) is well studied by various methods. The Fourier method was used in [1–3]. The mollification method and projection regularization based on the Laplace and Fourier transforms are applied respectively in [4] and [5]. For axisymmetric problems, we should mention recent articles. In [6, 7], the authors consider an axisymmetric IHCP of determining the surface temperature from a fixed location inside a cylinder. In [8, 9], the authors investigated the case of identifying a source from the final data. Xiong [10] studied the problem of identifying a boundary condition by the method of quasireversibility. A modified Tikhonov regularization method was applied for an axisymmetric backward heat equation in [11]. Lesnic et al. [12] applied the method of fundamental solutions (MFS) (with a Tikhonov regularization) to the radially symmetric inverse heat conduction problem (IHCP) analogous to our problem. Inverse problems for fractional diffusion equations are studied by many authors; for example, we mention the recent article [13].
In this paper, we formulate problem (1)(4) as an integral equation of the firstkind of Volterra type. Then we use the Tikhonov regularization method to approximate this equation. To the author’s knowledge, there are no papers devoted to IHCP with radial axisymmetry using the Laplace transform. Our contribution can therefore be considered as a generalization of the paper [5] to the axisymmetric case.
This paper is organized as follows. In Section 2, we give a representation of the solution of the direct problem using the Laplace transform. Then finite difference method (FDM) is applied to give numerical approximation. In Section 3, our inverse problem is reduced to the integral equation of Volterra type; then we apply the Tikhonov regularization method to compute the boundary temperature \(u(a,t)=f(t)\) from the Cauchy data \(u(b,t)=g(t)\), \(u_{r}(b,t)=0\) and give some numerical results. Finally, in Section 4, we present a conclusion.
2 Direct problem
Theorem 1
Proof
2.1 Reconstruction of the solution
We use the Laplace transform (with respect to the variable t) for the representation of the solution.
Let \(f(t)\), \(t\geq 0\), be a continuous function of slow growth, which means that there exist two constants \(C\geq 0\) and \(\sigma \geq 0\) such that \(\vert f(t) \vert \leq Ce^{\sigma t}\) for \(t>0\).
Lemma 1
 (i)with$$ k(r,t)=\sum_{n=1}^{\infty } \beta_{n}\frac{W(r,s_{n})}{W^{\prime }(a,s_{n})}e ^{s_{n}t} \quad \textit{for } r>a, t>0, $$(11)where \(J_{\nu }\) and \(Y_{\nu }\) denote the Bessel functions of the first and second kind, respectively, and \(s_{n}=\beta_{n}^{2}\), \(n=1,2, \ldots \) , is the sequence of the zeros of \(W(a,s)\).$$ \begin{aligned} &W(r,s_{n})=J_{0} \biggl( \frac{r}{a}\beta_{n} \biggr)Y_{1}(\lambda \beta_{n})J_{1}(\lambda \beta_{n})Y_{0} \biggl( \frac{r}{a}\beta_{n} \biggr),\quad \lambda = \frac{b}{a}, \\ &W^{\prime }(a,s_{n})=J_{1}(\beta_{n})Y_{1}( \lambda \beta_{n})J _{1}(\lambda \beta_{n})Y_{1}( \beta_{n}) \\ &\hphantom{W^{\prime }(a,s_{n})=}{}+\lambda \bigl[J_{0}(\lambda \beta _{n})Y_{0}(\beta_{n})J_{0}( \beta_{n})Y_{0}(\lambda \beta_{n}) \bigr], \end{aligned} $$(12)
 (ii)for all \(n\in \mathbf{N}\) and \(a< r\leq b\),$$ \frac{\partial^{n} k}{\partial t^{n}}(r,0)=0. $$(13)
Proof
Theorem 2
Assume that \(f(t)\in C^{1}([0,+\infty[)\) is such that \(f(0)=0\) and \(f(t)=0\) for \(t\geq T\). Then series (17) converges in \(L^{2}(]a,b[)\) for all \(t\geq 0\) and defines a solution of problem (5) belonging to \(\mathcal{H}\).
Proof
Remark 1

If \(f\in L^{2}(\mbox{R}_{+})\) (not smooth), then the differentiation of series (17) with respect to the variable r or t presents some difficulties. We can only say from the previous proof that u is a weak solution in \(L^{2}(]a,b[)\). However, if we know that u is differentiable with respect to t, then u is regular in both variables \((r,t)\).

For the numerical computation, the integral \(g_{n}(t)= \int_{0}^{t}f(\tau )e^{s_{n}(t\tau )}\,d\tau \) is approximated by the trapezoidal rule. More precisely, if \(\{t_{i}=ih, i=\overline{1,M+1} \}\) is a subdivision of \([0,T]\) and \(f^{h}(t)= \sum_{i=1}^{M+1}f_{i}\varphi (tt_{i})\) is an interpolation of f, whereis a basic function, and \(f_{i}=f(t_{i})\), then \(g_{n}\) is approximated by$$ \varphi (t)= \textstyle\begin{cases} 1+\frac{t}{h},& 1\leq t\leq 0, \\ 1\frac{t}{h},& 0< t\leq 1, \end{cases} $$with \(c_{ij}(s)=\frac{2}{hs^{2}}e^{(ij)hs}[\cosh (sh)1]\).$$ g_{n}^{h}(t)=\sum_{i=1}^{M+1}g_{n,i} \varphi (tt_{i}),\quad g_{n,i}= \sum _{j< i}c_{ij}(s_{n})f_{j} $$
2.2 Approximation by finite difference method (FDM)
Theorem 3
Proof
(1) Assuming that the solution is fairly regular \((C^{4})\), we use the Taylor expansion.
2.3 Numerical examples
Put \(a=1\), \(b=2\), \(T=3\).
We consider the following examples.
Test 1
Data: \(u(a,t)=f(t)=\chi_{[1,2]}\) (\(\chi_{I}\) denotes the characteristic function of an interval I).
Test 2
In the following figures, we show the response \(g(t)=u(b,t)\) to the source \(f(t)\).
3 Resolution of the inverse problem
3.1 Integral equation
Remark 2
We denote by \(A_{N}\) the operator with kernel \(k_{N}\).
Proposition 1
\(A_{N}\) converge to A in the Banach space \(\mathcal{L}(X)\), \(X=C[0,T]\) equipped with the norm \(\Vert \cdot \Vert _{\infty }\).
Proof
3.2 Tikhonov regularization
The approximate equation \(A_{N}f=g\) is solved by the Tikhonov regularization method. Recall the principle of the method.
3.3 Numerical experiment
We consider the following examples.
Test 1
Test 2
To check the efficiency of the proposed algorithm, we choose in numerical experiments the parameters \(a=1\), \(b=2\), \(T=3\) or \(T=5\) as required, and the rank of truncation \(30\leq N\leq 40\).
For an exact data function \(g(t)=u(b,t)\), we use a finite difference scheme with \(N=30\) points in the interval \([1,2]\) and \(M=12\text{,}000\) points in \([0,T]\). The discrete noisy version is \(g^{\delta }=g+\delta \operatorname{randn}(\operatorname{size}(g))\), the command ‘\(\operatorname{randn}(\cdot )\)’ generates arrays of random numbers whose elements are normally distributed with mean 0, variance \(\sigma^{2}=1\), and standard deviation \(\sigma =1\). For the singular decomposition and TikhonovMorozov algorithms, we used the Matlab package developed by Hansen [20].
3.4 Results and discussion
4 Conclusion
In this paper, we considered the inverse boundaryvalue problem of heat conduction with radial variable (in the cylindrical domain). The problem is solved by the approach based on the direct and inverse Laplace transforms. This leads to the Volterra equation of the firstkind with a special kernel. The Tikhonov method is applied to solve numerically this equation for perturbed data. The numerical results show that the method is efficient when the noise level \(\delta =0.001\). If \(\delta =0.01\), then the mollification procedure is applied (before the regularization) to the data \(g^{\delta }\), but with time observation T large enough.
Declarations
Acknowledgements
The work is supported by the National Research Foundation (CNEPRU) of Algeria (No. B01120120016).
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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