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Regularization method for the radially symmetric inverse heat conduction problem
Boundary Value Problems volumeÂ 2017, ArticleÂ number:Â 159 (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.
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 mathematical model of our problem can be described by the axisymmetric heat equation
with the boundary conditions
and the initial condition
where r is the radial coordinate, and \(0< a< b\).
Our purpose is to determine the boundary condition
from the measured Cauchy data \((g,h)\). It is known that this problem is severely illposed in the sense that if the solution exists, then it does not depend continuously on the dataÂ g. Indeed, a small perturbation in the data may cause dramatically aÂ large error in the solution \(u(\cdot,t)\). Hence, a regularization method is needed.
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 [1â€“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.Â [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 articleÂ [13].
In this paper, we formulate problemÂ (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 authorâ€™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 SectionÂ 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 SectionÂ 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 SectionÂ 4, we present a conclusion.
2 Direct problem
ProblemÂ (1)(2) can be reduced to an integral equation. For this, we assume that \(u(a,t)=f(t)\) is known and (for simplicity) \(h(t)=0\). We consider the following direct problem: givenÂ f, find \(u(x,t)\) such that
We have the following uniqueness theorem.
Theorem 1
ProblemÂ (5) has at most one solution in the space
Proof
Let \(u(r,t)\) be a solution of the homogeneous problemÂ (5) (with \(f=0\)). Multiplying PDE by ru and integrating by parts over the interval \((a,b)\), we obtain the identity
We set \(E(t)=\int_{a}^{b}u^{2}r\,dr\), \(t\geq 0\), which leads to
and therefore \(E(t)=E(0)=0\) for all \(t\geq 0\), and \(u(r,t)=0\), \(r\in (a,b)\), \(t\geq 0\).â€ƒâ–¡
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 \(t>0\).
The Laplace transform \(F(s)=\mathcal{L}f(s)\) of \(f(t)\) is defined by
which is an analytic function in the halfplane \(\Re (s)>\sigma \), and the inverse Laplace transform is given by the complex integral [14]
Let \(U(r,s)=\mathcal{L}u(r,\cdot )\) and \(F(s)=\mathcal{L}f(s)\). ProblemÂ (5) can be formulated as follows:
where \(U^{\prime }=\frac{\partial U}{\partial r}\).
The first equation in (6) is the modified Bessel differential equation with the general solution
Then the solution of problem (6) is given by
with
where \(I_{\nu }\) and \(K_{\nu }\) are the modified Bessel functions of the first and second kind, respectively [15].
Applying the inverse Laplace transform and the convolution theoremÂ [14], we obtain:
with the kernel
Lemma 1
The heat kernel \(k(r,t)\) satisfies the properties

(i)
$$ k(r,t)=\sum_{n=1}^{\infty } \beta_{n}\frac{W(r,s_{n})}{W^{\prime }(a,s_{n})}e ^{s_{n}t} \quad \textit{for } r>a, t>0, $$(11)
with
$$ \begin{aligned} &W(r,s_{n})=J_{0} \biggl( \frac{r}{a}\beta_{n} \biggr)Y_{1}(\lambda \beta_{n})J_{1}(\lambda \beta_{n})Y_{0} \biggl( \frac{r}{a}\beta_{n} \biggr),\quad \lambda = \frac{b}{a}, \\ &W^{\prime }(a,s_{n})=J_{1}(\beta_{n})Y_{1}( \lambda \beta_{n})J _{1}(\lambda \beta_{n})Y_{1}( \beta_{n}) \\ &\hphantom{W^{\prime }(a,s_{n})=}{}+\lambda \bigl[J_{0}(\lambda \beta _{n})Y_{0}(\beta_{n})J_{0}( \beta_{n})Y_{0}(\lambda \beta_{n}) \bigr], \end{aligned} $$(12)where \(J_{\nu }\) and \(Y_{\nu }\) denote the Bessel functions of the first and second kind, respectively, and \(s_{n}=\beta_{n}^{2}\), \(n=1,2, \ldots \)â€‰, is the sequence of the zeros of \(W(a,s)\).

(ii)
for all \(n\in \mathbf{N}\) and \(a< r\leq b\),
$$ \frac{\partial^{n} k}{\partial t^{n}}(r,0)=0. $$(13)
Proof
(i) Considering the contour \(C_{R}=[iR,iR]\cup \{Re^{i\theta }, \theta \in [\frac{\pi }{2},\frac{3\pi }{2}]\}\) and using the asymptotic formula \(\vert \frac{W(r,s)}{W(a,s)}e^{st} \vert =O(e^{\mu \vert s \vert ^{1/2}})\) as \(\vert s \vert \rightarrow \infty \) (see AppendixÂ 1), we can express the previous integral for \(t>0\) and \(a< r\leq b\) as follows:
The function \(W(a,s)\) is analytic with respect to the variable \(p=\sqrt{s}\) and possesses a sequence of simple roots \(p_{n}\) located on the imaginary axis such that \(p_{n}=i\beta_{n}\) where \(\beta_{n} \simeq (n\frac{1}{2})\frac{a\pi }{ba}\), \(n=1,2,\dots \) (see AppendixÂ 2). FormulaÂ (11) follows from the Cauchy theorem.
(ii) Deriving the integralÂ (10) with respect to t, for all \(n\in \mathbf{N}\), we obtain
The function \(G_{n}(r,s)=s^{n}\frac{W(r,s)}{W(a,s)}\) is analytic in the halfplane \(\Re (s)\geq 0\) except at the origin, where \(\lim_{s\rightarrow 0}G_{0}(r,s)=1\) and \(\lim_{s\rightarrow 0}G_{n}(r,s)=0\) for \(n\geq 1\), and has the behavior \(\vert G_{n}(r,s) \vert =O(\rho^{n}\exp[(ra)\rho^{1/2}\cos \frac{\theta}{2}])\) as \(\rho =\vert s \vert \rightarrow +\infty \) uniformly for \(\theta \in [\frac{\pi }{2},\frac{\pi }{2}]\). Then we can use the Cauchy theorem with an adequate contour to have
â€ƒâ–¡
As a consequence, the integralÂ (9) is written as the formal series
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 seriesÂ (17) converges in \(L^{2}(]a,b[)\) for all \(t\geq 0\) and defines a solution of problemÂ (5) belonging to \(\mathcal{H}\).
Proof
Integrating by parts inÂ (17), we can write u in the form
Using the asymptotic behavior (see AppendixÂ 1)
and the estimate
we see that the first series in (18) is uniformly convergent and the second term is of the same nature as the series \(\sum \frac{ \alpha_{n}}{\sqrt{r}}\cos (n(\lambda \frac{r}{a}))\), \(\alpha_{n}=\frac{(1)^{n}}{ \beta_{n}}\). Let us show that last series converges in \(L^{2}(]a,b[,r\,dr)\). Indeed, the partial sum \(S_{N}(r)=\sum_{n=0}^{N}\frac{ \alpha_{n}}{\sqrt{r}}\cos (\frac{(br)n}{a})\) is a Cauchy sequence, that is, for all \(q\geq 1\),
In the last step, we used the orthogonality of the system \(\{\cos (nz) \}\) in \(L^{2}(]0,\pi [)\), and l is an integer chosen such that \(\lambda 1\leq l\pi \). Furthermore,
which converges as the series \(\sum {\alpha_{n}} \cos {\frac{(br)n}{a}}\) with \(\vert \alpha_{n} \vert \leq M^{\prime }/\beta _{n}\), \(M^{\prime }=\sup \vert f^{\prime }(t) \vert \). Therefore \(\frac{\partial u}{\partial t}\in L^{2}(]a,b[)\). Now we show that u is a weak solution of the PDE in problemÂ (5). For this, let us consider the sequence
with
Defining the differential operator \(Pv:=\frac {\partial v}{\partial t}\frac {\partial^{2}v}{\partial r^{2}}\frac {1}{r}\frac {\partial v}{\partial r}\), we have
However, \(Pk_{N}(r,t)=0\), and then \(Pu_{N}(r,t)=f(t)k_{N}(r,0)\). From LemmaÂ 1 it follows that
On the other hand, \(P:\mathcal{D}^{\prime }(\mathbf{Q})\rightarrow \mathcal{D}^{\prime }(\mathbf{Q})\), \(\mathbf{Q}=\,]a,b[\,\times \, ]0,+ \infty [\), is a continuous operator. Then \(Pu=\lim_{N\rightarrow \infty }Pu_{N}=0\). We now show that \(u\in \mathcal{H}\). Since \(\frac{\partial }{\partial r}(r u_{r})=ru_{rr}+u _{r}=ru_{t}\) in \(\mathcal{D}^{\prime }(]a,b[)\) at fixed t, we have \(u_{r}\in H^{1}(]a,b[)\) and \(u(\cdot ,t)\in H^{2}(]a,b[)\). Finally, it is easy to verify that \(t\mapsto u(\cdot ,t)\) is \(C^{1}\) from \(]0,T[\) to \(L^{2}(]a,b[)\).â€ƒâ–¡
Remark 1

If \(f\in L^{2}(\mbox{R}_{+})\) (not smooth), then the differentiation of seriesÂ (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, where
$$ \varphi (t)= \textstyle\begin{cases} 1+\frac{t}{h},& 1\leq t\leq 0, \\ 1\frac{t}{h},& 0< t\leq 1, \end{cases} $$is a basic function, and \(f_{i}=f(t_{i})\), then \(g_{n}\) is approximated by
$$ g_{n}^{h}(t)=\sum_{i=1}^{M+1}g_{n,i} \varphi (tt_{i}),\quad g_{n,i}= \sum _{j< i}c_{ij}(s_{n})f_{j} $$with \(c_{ij}(s)=\frac{2}{hs^{2}}e^{(ij)hs}[\cosh (sh)1]\).
2.2 Approximation by finite difference method (FDM)
Problem (5) can be discretized by replacing the derivatives by difference quotients as follows. Consider a uniform grid points in the \((r,t)\) plane:
Letting \(u_{i}^{n}=u(r_{i},t_{n})\), systemÂ (5) is discretized by the following finite difference scheme:
SchemeÂ (23) is explicit, and the solution \(u_{i}^{n+1}\) is easily found:
We can prove the following result concerning the stability of schemeÂ (23).
Theorem 3

(1)
The finite difference schemeÂ (23) is consistent of order \(O(\tau +h^{2})\).

(2)
If
$$ R=\frac{\tau }{h^{2}}\leq \frac{1}{2}\quad (\textit{with } h< 1), $$then scheme (23) is \(L^{\infty }\)stable.
Proof
(1) Assuming that the solution is fairly regular \((C^{4})\), we use the Taylor expansion.
(2) This can be proved in a similar way as in the lectureÂ [16]. If \(R=\frac{\tau }{h^{2}}\leq \frac{1}{2}\), then we see from (24) that \(u_{i}^{n+1}\) is a convex combination of \(u_{i1}^{n}\), \(u_{i}^{n}\), \(u_{i+1}^{n}\). Letting \(M^{n}=\max_{i=1,\ldots,N}(u_{i}^{n})\), we have
from which it follows that \(u_{i}^{n+1}\leq M^{n}\). Taking the maximum, we deduce that \(M^{n+1}\leq M^{n}\). In the same way, we set \(m^{n}=\min_{i=1,\ldots,N}(u_{i}^{n})\). Then
and \(u_{i}^{n+1}\geq m^{n}\). Taking the minimum, we obtain \(m^{n+1} \geq m^{n}\). Hence
which establishes the stability.â€ƒâ–¡
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 \(f(t)\).
In FigureÂ 1, we call \(g_{\mathrm{ex}}\) the solution given by the truncated seriesÂ (17) (with a rank \(N\geq 40\)) and \(g_{\mathrm{ap}}\) the approximate solution computed by FDM with parameters \(N=30\) and \(M=12\text{,}000\).
3 Resolution of the inverse problem
3.1 Integral equation
As the function \(u(b,t)=g(t)\) can be known, the resolution of the inverse problem (1)(2) can be reduced to the resolution of the Volterra integral equation of the firstkind
with the kernel
The function \(k(t)\) is continuous on \([0,+\infty [\) satisfying \(f(0)=0\), \(\vert k(t) \vert \leq C_{1}e^{C_{2}t}\), where \(C_{1}\), \(C_{2}\) two constants, for \(t\geq 1\) and \(k\in C^{\infty }(]0,+\infty [)\). Then \(A\in \mathcal{L}(H)\), \(H=L^{2}([0,T])\). The range \(\mathcal{R}(A)\) of A is nonclosed in H (A is compact and nondegenerate). This means that equationÂ (25) is illposed. The problem is, moreover, severely illposed since all derivatives of the kernel \(k(t)\) vanish to zero according to LemmaÂ 1(ii) (see also [17]). Therefore, some kind of regularization procedure will be necessary to solve the problem in the case of a perturbed data \(g^{\delta }\).
For the numerical resolution of equation (25), we approximate the kernel k by the truncated series
Remark 2
The truncation error (the rest \(R_{N}=kk_{N}\) of the series) is estimated by
with \(C_{2}=(\frac{\pi }{\lambda 1})^{2}\). This means that, for t close to 0, we need more terms in the series. For computation, we choose \(N=40\) for \(h< t\leq 0.1\), \(N=20\) for \(0.1\leq t\leq 1\), and \(N=10\) for \(t\geq 1\). For \(0\leq t\leq h\), we can take \(k(t)=0\), since we know from (13) that \(k(b,t)\) is close to zero as \(t\rightarrow 0\) (see FigureÂ 2). The parameter \(h=\frac{T}{M}\) is the step time (we assume that \(h\geq \frac{\epsilon }{N}\)).
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
Using the asymptotic formulaÂ (19) with \(r=b\), we can write
with \(\gamma_{n}\) bounded. Then
Using (27) with \(\epsilon =1\), it follows that
which leads to
and \(\Vert AA_{N} \Vert _{\mathcal{L}(X)}=O(1/N)\).â€ƒâ–¡
3.2 Tikhonov regularization
The approximate equation \(A_{N}f=g\) is solved by the Tikhonov regularization method. Recall the principle of the method.
Suppose that \(A\in \mathcal{L}(H)\) is a compact operator in Hilbert space, injective and with dense range. The equation \(Af=g\) is ill posed, that is, \(A^{1}:\mathcal{R}(A)\rightarrow H\) is not bounded (a small error in data g generates an important perturbation on the computed solution f). The Tikhonov regularization method consists in solving the normal equation
where \(\alpha > 0\) is a regularization parameter, and I is the identity operator. Equivalently, \(f^{\alpha }\) is the unique minimum of the Tikhonov functional
The solution \(f^{\alpha }\) can be written as follows:
which can be expressed in terms of the singular system \((\mu_{j}, u _{j},v_{j})_{j\in \mbox{N}^{*}}\) as
Now let \(g\in \mathcal{R}(A)\), and let \(g^{\delta }\in H\) be a measured data with \(\Vert gg^{\delta } \Vert _{H}\leq \delta \). We define \(f^{\alpha , \delta }=R_{\alpha }g^{\delta }\). A posteriori discrepancy principle gives a selection of Î± (aÂ solution of the equation \(\Vert Af^{\alpha ,\delta }g^{\delta } \Vert _{H}=\delta \)). Standard Tikhonov regularization theory (see [18, Theorem 2.17]) shows the convergence \(f^{\alpha ,\delta }\rightarrow f\) in H as \(\delta \rightarrow 0\) with the strategy \(\alpha =\alpha (\delta )\). For other methods, we indicate the papers [17, 19], which give a survey of regularization methods for firstkind Volterra equations.
3.3 Numerical experiment
We consider the following examples.
Test 1
As a first example, we consider the couple \(\{f,g \}\), where
and \(g(t)=u(b,t)\) is computed by finite difference scheme (see SectionÂ 2.2).
Test 2
We consider the example given by
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 â€˜\(\operatorname{randn}(\cdot )\)â€™ 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 HansenÂ [20].
3.4 Results and discussion
FiguresÂ 3 and 4 show the numerical results that confirm the stability of the method with respect to the noise level \(\delta \leq 0.001\). However, the rank in \(k_{N}\) must be large enough (here \(N\geq 30\)) to ensure the convergence of the Tikhonov algorithm. FigureÂ 5(a) shows that, for \(\delta =0.01\), the oscillations increase, which requires a new regularization. Indeed, if we use the mollification method [4], then the oscillations are damped (see FigureÂ 5(b)). The operation consists in taking the convolution \(g_{\nu }=\rho_{\nu }*g\) with \(\rho_{\nu }(t)=\frac{1}{\nu \sqrt{\pi }}\exp {(\frac{t^{2}}{ \nu^{2}})}\), where \(\nu \rightarrow 0\) is the radius of mollification. If g vanishes near the ends of the interval \([0,T]\), then \(g_{\nu }\) is a smooth function and is a good approximation of g; this fact is realized if the observation time T is large enough (in practice, we take \(T=2T_{0}\) if \(\mbox{supp} f\subset [0,T_{0}]\)). In the presence of the noise, according to the analysis in [4], we choose \(\nu =c\sqrt{\delta }\) with \(c=1\) estimated by test.
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 dataÂ \(g^{\delta }\), but with time observation T large enough.
References
EldÃ©n, L, Berntsson, F, RegiÅ„ska, T: Wavelet and Fourier methods for solving the sideways heat equation. SIAM J.Â Sci. Comput. 21(6), 21872205 (2000)
Berntsson, F: A spectral method for solving the sideways heat equation. Inverse Probl. 15, 891906 (1999)
Fu, CL: Simplified Tikhonov and Fourier regularization methods on a general sideways parabolic equation. J.Â Comput. Appl. Math. 167(2), 449463 (2004)
Murio, DA: The Mollification Method and the Numerical Solution of IllPosed Problems. Wiley, New York (1993)
Yaparova, N: Numerical methods for solving a boundaryvalue inverse heat conduction problem. Inverse Probl. Sci. Eng. 22(5), 832847 (2014)
Cheng, W, Fu, CL: Solving the axisymmetric inverse heat conduction problem by a wavelet dual leastsquares method. Boundary Value Problems 2009, Article ID 260941 (2009). doi:10.1155/2009/260941
Cheng, W, Fu, CL: Two regularization methods for an axisymmetric inverse heat conduction problem. J.Â Inverse IllPosed Probl. 17(2), 159172 (2009)
Cheng, W: Regularization and stability estimates for an inverse source problem of the radially symmetric parabolic equation. J.Â Inequal. Appl. 2015, 136 (2015)
Cheng, W, Zhao, LL, Fu, CL: Source term identification for an axisymmetric inverse heat conduction problem. Comput. Math. Appl. 59, 142148 (2010)
Xiong, XT: On a radially symmetric inverse heat conduction problem. Appl. Math. Model. 34, 520529 (2010)
Cheng, W, Fu, CL: A modified Tikhonov regularization method for an axisymmetric backward heat equation. Acta Math. Sin. Engl. Ser. 26(11), 21572164 (2010)
Johansson, BT, Lesnic, D, Reeve, T: A method of fundamental solutions for the radially symmetric inverse heat conduction problem. Int. Commun. Heat Mass Transf. 39, 887895 (2012)
Tuan, NH, Kirane, M, Luu, VCH, BinMohsin, B: A regularization method for timefractional linear inverse diffusion problems. Electron. J.Â Differ. Equ. 2016, 290 (2016)
Ditkine, V, Proudnikov, A: Transformation IntÃ©grales et Calcul OpÃ©rationnel. Traduit du Russe. Ã‰ditions MIR, Moscou (1978)
Abramowitz, M, Stegun, IA: Handbook of Mathematical Functions. Dover, New York (1972)
Herbin, R: Analyse numÃ©rique des Ã©quations aux dÃ©rivÃ©es partielles. Engineering school, Marseille 2011. https://cel.archivesouvertes.fr/cel00637008
Lamm, PK: A survey of regularization methods for firstkind Volterra equations. In: Colton, D, Engl, HW, Louis, AK, McLaughlin, JR, Rundell, W (eds.) Surveys on Solution Methods for Inverse Problems, pp.Â 5382. Springer, Vienna (2000). doi:10.1007/9783709162965_4
Kirsh, A: An Introduction to the Mathematical Theory of Inverse Problems. Applied Mathematical Sciences Book Series, vol.Â 120. Springer, Berlin (2011)
Lamm, PK, EldÃ©n, L: Numerical solution of firstkind Volterra equations by sequential Tikhonov regularization. SIAM J.Â Numer. Anal. 34(4), 14321450 (1997)
Hansen, PC: Regularization tools version 4.0 for Matlab 7.3. Numerical Algorithms 46, 189194 (2007). doi:10.1007/s1107500791369
Acknowledgements
The work is supported by the National Research Foundation (CNEPRU) of Algeria (No. B01120120016).
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Appendices
Appendix 1: Asymptotic expansions
For fixed Î½ and \(\vert z \vert \rightarrow \infty \), we haveÂ [15]
We consider the rapport \(G(r,s)=\frac{W(r,s)}{W(a,s)}\) with \(W(r,s)=I_{1}(b\sqrt{s} )K_{0}(r \sqrt{s})+K_{1}(b\sqrt{s})I_{0}(r\sqrt{s})\). Due to (30), we have, for \(r\in\, ]a,b]\) and \(\vert s \vert \rightarrow +\infty \),
where \(\rho =\vert s \vert \) and \(\theta =\arg (s)\in [\frac{\pi }{2},\frac{3 \pi }{2}]\). Then there exist positive constants C and Î¼ such that, for \(\vert s \vert \) large enough, say \(\vert s \vert \geq \rho_{0}\),
Appendix 2: Zeros of crossproduct
We want to seek the roots \(s_{n}\) of the following function (crossproduct):
We show that \(s_{n}=\beta_{n}^{2}\) with \(\beta_{n}\rightarrow + \infty \) as \(n\rightarrow +\infty \). For this, we consider the selfadjoint operator defined in \(H=L^{2}((a,b);r\,dr)\) by
If \(AU=\lambda U\), then \((AU,U)=\Vert U^{\prime } \Vert ^{2}=\lambda \Vert U \Vert ^{2}\). Since \(A^{1}\) is compact, A has a sequence of positive eigenvalues \(\lambda_{n}\rightarrow +\infty \). On the other hand \(\lambda_{n}=s _{n}\) coincides with the roots of \(W(s)\). Indeed, this can be seen by solving the SturmLiouville problem (system (6) in SectionÂ 2) with \(\lambda =s\) and \(F=0\).
Now we give the behavior of \(s_{n}\). Using the relations between the Bessel functions
we obtain, for \(s=\beta^{2}\),
Using the asymptotic expansions (29) for large Î², we get
and hence the zeros of \(W(s)\) are a sequence \(s_{n}=\beta_{n}^{2}\), \(n=1,2,\ldots \)â€‰, such that
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Djerrar, I., Alem, L. & Chorfi, L. Regularization method for the radially symmetric inverse heat conduction problem. Bound Value Probl 2017, 159 (2017). https://doi.org/10.1186/s136610170890x
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DOI: https://doi.org/10.1186/s136610170890x