A physical model considered here is a hollow sphere, and R and \(r_{0}\) denote its external and internal radius, respectively. Let the hollow sphere be adiabatic at its external surface, and a thermocouple is installed inside the hollow sphere at the radius \(r=r_{1}\), \(r_{0}< r_{1}< R\), as illustrated in Figure 1. Assuming a spherically symmetric temperature distribution of the model, the correspondingly mathematical model can be described as the following radially symmetric heat conduction problem:
$$ \begin{aligned} &u_{t}=u_{rr}+\frac{2}{r}u_{r}, \quad r_{0}< r< R, t>0, \\ &u(r,0)=0, \quad r_{0}\le r\leq R, \\ &u(r_{1},t)=g(t), \quad t\geq0, \\ &u_{r}(R,t)=0,\quad t\geq0, \end{aligned} $$
(1.1)
where r denotes the radial coordinate, \(g(t)\) is the temperature history at one fixed radius \(r_{1}\), \(r_{0}< r_{1}< R\). We want to recover the temperature distribution \(u(r,\cdot)\) (\(r_{0}\le r< r_{1}\)) based on the measured data of \(g (\cdot)\). This is an inverse heat conduction problem.
The inverse heat conduction problem (IHCP) has numerous important applications in various sciences and engineering [1]. For example, determination of thermal fields at surfaces without access, obtaining the force applied to a complex structure from knowledge of the response and transfer function which describes the system, or the diagnosis of a disease by computerized tomography [2]. In all cases, the boundary conditions of these problems are inaccessible to measurements or not known. Usually sensors are installed beneath the surface and the unknown boundary conditions of these problems are estimated.
The solutions of inverse heat conduction problems (IHCPs) are very challenging, because IHCPs are severely ill-posed in the Hadamard sense that the solution (if it exists) does not depend continuously on the given data, i.e., a small measurement error in the given data can cause an enormous error in the solution [3, 4]. Therefore, an appropriate regularization method needs to be applied. These methods include the filtering method [5], the spectral method [6], the mollification method [7, 8], the boundary element method [9], the fundamental solution method [10], the wavelet and wavelet-Galerkin method [11–14], the Fourier method [15], the differential-difference method [16], the global time method [17], the modified Tikhonov method [18], the iterative method [19] etc. However, the results already published in the works on IHCP are mainly devoted to the heat equation with constant coefficient. The works presented for heat equation with variable coefficient are still limited. A few works have been developed for the inverse problems on heat equation with variable coefficient [20–23]. In [20], Fu used a simplified Tikhonov and a Fourier regularization methods to deal with an IHCP on heat equation with variable coefficient and provided two kinds of convergence rates. Grabski et al. [21] applied the method of fundamental solutions for identifying a time-dependent perfusion coefficient in the bioheat equation. A non-iterative inverse determination of temperature-dependent thermal conductivity was solved by Mierzwiczak and Kolodziej [22]. Chang and Chang [23] investigated the determination of spatially- and temperature- dependent thermal conductivity by a semi-discretization method. In this work, we will use a wavelet method to deal with IHCP (1.1) (\(r_{0}\le r< r_{1}\)) with variable coefficient and to obtain a quite sharp error estimate between the approximate solution and the exact solution.
The wavelet method has become a powerful method for solving partial differential equations (PDEs). And the method has been applied to direct problems as well as to various types of inverse problems such as the IHCP [24], the Cauchy problem of Laplace equation [25, 26], the backward heat conduction problem [27], the inverse source identification problems [28, 29] and the Cauchy problem for the modified Helmholtz equation [30, 31]. It is worth mentioning that Feng and Ning [32] used a Meyer wavelet regularization method for solving numerical analytic continuation and presented the Hölder-type stability estimates. In this paper, we solve the radially symmetric inverse heat conduction problem (1.1) in the interval \([r_{0}, r_{1})\) by determining the temperature distribution using a wavelet dual least squares method generated by the family of Shannon wavelets.
When we deal with problem (1.1) in \(L^{2}(\mathbb{R})\) with respect to variable t, we extend all functions of variable t appearing in the paper to be zero for \(t<0\). Since the measurement data of \(g(t)\) contain noises, the solutions have to be sought from the data function \(g^{\delta}(t)\in L^{2}(\mathbb{R})\), which satisfy
$$ \bigl\Vert g-g^{\delta}\bigr\Vert \leq\delta, $$
(1.2)
where the constant \(\delta>0\) denotes a bound on the measurement error, and \(\Vert \cdot \Vert \) represents the \(L^{2}(\mathbb{R})\) norm. It is also assumed that there exists an a priori bound for function \(u(r_{0},t)\)
$$ \bigl\Vert u(r_{0},\cdot) \bigr\Vert _{H^{p}}\leq E, \quad p\geq0, $$
(1.3)
where \(\Vert u(r_{0},\cdot) \Vert _{H^{p}}\) denotes the norm in the Sobolev space \(H^{p}(\mathbb{R})\) defined by
$$ \bigl\Vert u(r_{0},\cdot) \bigr\Vert _{H^{p}}:= \biggl( \int_{-\infty}^{\infty} \bigl(1+\xi^{2} \bigr)^{p} \bigl\vert \hat{f}(\xi) \bigr\vert ^{2}\,d\xi \biggr)^{\frac{1}{2}}. $$
Using the Fourier transform with respect to the variable t, problem (1.1) can be formulated in a frequency space as follows:
$$ \textstyle\begin{cases} i\xi\hat{u}(r,\xi)=\frac{\partial^{2}\hat{u}(r,\xi)}{\partial r^{2}}+\frac{2}{r}\frac{\partial\hat{u}(r,\xi)}{\partial r}, \quad r\in(r_{0}, R],\xi\in\mathbb{R},\\ \hat{u}(r_{1},\xi)=\hat{g}(\xi),\quad \xi\in\mathbb{R},\\ \hat{u}_{r}(R,\xi)=0,\quad \xi\in\mathbb{R}. \end{cases} $$
(1.4)
We can get a formal solution for problem (1.4), refer to [33],
$$ \hat{u}(r, \xi) =(r_{1}/r)\varphi(r,\xi) e^{(r_{1}-r) \sqrt{i\xi}} \hat{g}( \xi),\quad r\in[r_{0}, R),\xi\in\mathbb{R}, $$
(1.5)
where
$$ \varphi(r,\xi)=\frac{(\sqrt{i\xi}R+1)e^{2r\sqrt{i\xi}}+ (\sqrt{i\xi}R-1)e^{2R\sqrt{i\xi}}}{(\sqrt{i\xi}R+1)e^{2r_{1}\sqrt{i\xi}}+ (\sqrt{i\xi}R-1)e^{2R\sqrt{i\xi}}}. $$
According to Lemma 2.3 in [33], function \(\varphi(r,\xi)\) satisfies
$$ c_{1}\le \bigl\vert \varphi(r,\xi) \bigr\vert \le c_{2}, \quad r\in[r_{0},r_{1}),\xi\in \mathbb{R}, $$
(1.6)
where \(c_{1}\) and \(c_{2}\) are positive constants. Due to \(\vert (r_{1}/r)\varphi(r,\xi) e^{(r_{1}-r) \sqrt{i\xi}} \vert \) increases rapidly with exponential order as \(\vert \xi \vert \rightarrow\infty\), the Fourier transform of the exact data \(g(t)\) must decay rapidly at high frequencies for \(r_{1}>r\). But such a decay is not likely to occur in \(g^{\delta}(t)\). So, a small measurement error in the given data \(g^{\delta}(t)\) in high frequency components can completely destroy the solution of problem (1.1) for \(r\in[r_{0}, r_{1})\).
For problem (1.1), we define an operator \(A_{r}:u(r,\cdot)\longmapsto g(\cdot)\) in the space \(X=L^{2}(\mathbb{R})\). Then problem (1.1) can be rewritten as
$$ A_{r}u(r,t)=g(t),\quad\forall u(r,\cdot)\in X, r_{0} \le r< r_{1}. $$
(1.7)
According to expression (1.5), there holds
$$ \widehat{A_{r}u}(r,\xi)=\hat{g}(\xi)=(r/r_{1})e^{(r-r_{1}) \sqrt{i\xi}} \varphi^{-1}(r,\xi)\hat{u}(r,\xi), \quad r\in[r_{0},r_{1}). $$
(1.8)
Then we have \(\widehat{A_{r}u(r,\xi)}:=\widehat{A_{r}}\hat{u}(r,\xi)\) and a multiplication operator \(\widehat{A_{r}}:L^{2}(\mathbb{R})\longmapsto L^{2}(\mathbb{R})\) given by
$$ \widehat{A_{r}}\hat{u}(r,\xi)=(r/r_{1})e^{(r-r_{1}) \sqrt{i\xi}} \varphi^{-1}(r,\xi)\hat{u}(r,\xi). $$
(1.9)
Therefore, we have the following lemma.
Lemma 1.1
If
\(A^{*}_{r}\)
is the adjoint of
\(A_{r}\), then
\(A^{*}_{r}\)
corresponds to the following problem:
$$ \textstyle\begin{cases} -\frac{\partial U}{\partial t}=\frac{\partial^{2} U}{\partial r^{2}}+\frac{2}{r}\frac{\partial U}{\partial r}, \quad r_{0}< r\le R, t\ge0,\\ U(r,0)=0, \quad r_{0}\le r\le R,\\ U(r_{1},t)=g(t), \quad t\geq0,\\ U_{r}(R,t)=0,\quad t\geq0, \end{cases} $$
(1.10)
and
$$ \widehat{A^{*}_{r}} =(r/r_{1})e^{(r-r_{1}) \overline{\sqrt{i\xi}}} \overline{\varphi^{-1}(r,\xi)}. $$
(1.11)
Proof
Using the following relations and expression (1.9)
$$\langle A_{r}u,\upsilon\rangle=\langle \widehat{A_{r}} \hat{u},\hat{\upsilon}\rangle=\bigl\langle \hat{u},\widehat {A_{r}}^{*} \hat{\upsilon}\bigr\rangle =\bigl\langle u,A^{*}_{r}\upsilon \bigr\rangle =\bigl\langle \hat{u},\widehat{A^{*}_{r}}\hat{ \upsilon}\bigr\rangle , $$
where \(\langle\cdot,\cdot\rangle\) denotes the inner product, we can obtain the adjoint operator \(A^{*}_{r}\) of \(A_{r}\) in the frequency domain
$$ \widehat{A^{*}_{r}}=\widehat{A_{r}}^{*}=(r/r_{1})e^{(r-r_{1}) \overline{\sqrt{i\xi}}} \overline{\varphi^{-1}(r,\xi)}. $$
Applying the Fourier transform with respect to the variable t, we can rewrite problem (1.10) in the following form (in the frequency space):
$$ \textstyle\begin{cases} -i\xi\hat{U}(r,\xi)=\frac{\partial^{2}\hat{U}(r,\xi)}{\partial r^{2}}+\frac{2}{r}\frac{\partial\hat{U}(r,\xi)}{\partial r}, \quad r\in(r_{0}, R],\xi\in\mathbb{R},\\ \hat{U}(r_{1},\xi)=\hat{g}(\xi),\quad \xi\in\mathbb{R},\\ \vert \hat{U}_{r}(R,\xi) \vert =0,\quad \xi\in\mathbb{R}. \end{cases} $$
(1.12)
Taking the conjugate operator for problem (1.4), we know that \(\hat{U}(r,\xi)=\overline{\hat{u}(r,\xi)}\). So, combining with (1.5), we get
$$ \hat{U}(r,\xi)=\overline{\hat{u}(r,\xi)} =(r_{1}/r) e^{(r_{1}-r) \overline{\sqrt{i\xi}}} \overline{\varphi(r,\xi)} \hat{g}(\xi) $$
(1.13)
and
$$ \hat{g}(\xi)=(r/r_{1}) e^{(r-r_{1}) \overline{\sqrt{i\xi}}} \overline{ \varphi^{-1}(r,\xi)} \hat{U}(r,\xi)=\widehat{A^{*}_{r}} \hat{U}(r,\xi)=\widehat{A^{*}_{r}U}. $$
(1.14)
□
The outline of the paper is as follows. In Section 2, using Hölder’s inequality, we prove the conditional stability for IHCP (1.1) in the interval \([r_{0}, r_{1})\). The relevant properties of Shannon wavelets are summarized in Section 3. The last section presents error estimates via wavelet dual least squares method approximation.