Initial inverse problem in heat equation with Bessel operator

We propose and investigate an application of the method of fundamental solutions (MFS) to the radially symmetric and axisymmetric backward heat conduction problem (BHCP) in a solid or hollow cylinder. In the BHCP, the initial temperature is to be determined from the temperature measurements at a later time. This is an inverse and ill-posed problem, and we employ and generalize the MFS regularization approach [B.T. Johansson and D. Lesnic, A method of fundamental solutions for transient heat conduction, Eng. Anal. Boundary Elements 32 (2008), pp. 697-703] for the time-dependent heat equation to obtain a stable and accurate numerical approximation with small computational cost.

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“…In the above expressions, via (27), the normal (radial) derivative of the fundamental solution (3) is required. Straightforward calculations give…”

Section: The Mfs For the Radial Heat Equationmentioning

confidence: 99%

A method of fundamental solutions for radially symmetric and axisymmetric backward heat conduction problems

Johansson

Lesnic

Reeve

2012

International Journal of Computer Mathematics

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“…More recently, Al Masood et al considered a well conditioned damped heat equation [7] and a Bessel operator [8] to estimate the initial temperature distribution in a diffusion field. Nakamura et al [9] used transform techniques to solve the initial inverse problem in heat conduction, while Takeuchi et al [10] proved the existence of the solution and gave a numerical method to find point sources distributed on a 2-D domain.…”

Section: A Prior Artmentioning

confidence: 99%

“…Solutions to (8) are sin(ωx), cos(ωx) and e iωx . The periodic boundary condition forces ω's into the form ω n = n, so the homogeneous solution to (6) corresponding to ω n is…”

Section: A Eigenfunction Solutionmentioning

confidence: 99%

Sensor networks for diffusion fields: Detection of sources in space and time

Ranieri

Chebira

Vetterli

2011

2011 49th Annual Allerton Conference on Communication, Control, and Computing (Allerton)

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Abstract-We consider the problem of reconstructing a diffusion field, such as temperature, from samples collected by a sensor network. Motivated by the fast decay of the eigenvalues of the diffusion equation, we approximate the field by a truncated series. We show that the approximation error decays rapidly with time. On the other hand, the information content in the field also decays with time, suggesting the need for a proper choice of the sampling strategy. We propose two algorithms for sampling and reconstruction of the field. The first one reconstructs the distribution of point sources appearing at known times using the finite rate of innovation (FRI) framework. The second algorithm addresses a more difficult problem of estimating the unknown times at which the point sources appear, in addition to their locations and magnitudes. It relies on the assumption that the sources appear at distinct times. We verify that the algorithms are capable of reconstructing the field accurately through a set of numerical experiments. Specifically, we show that the second algorithm successfully recovers an arbitrary number of sources with unknown release times, satisfying the assumption. For simplicity, we develop the 1-D theory, noting the possibility of extending the framework to more general domains.

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“…has to be specified, compare (Yamamoto & Zou, 2001;Masood et al, 2002). In some papers instead of the condition (9) the temperature measurements on a part of the boundary are used, see e.g.…”

Section: Initial Value Determination Inverse Problemsmentioning

confidence: 99%

Inverse Heat Conduction Problems

Grysa¹

2011

Heat Conduction - Basic Research

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Initial inverse problem in heat equation with Bessel operator

Cited by 25 publications

References 3 publications

A method of fundamental solutions for radially symmetric and axisymmetric backward heat conduction problems

A method of fundamental solutions for radially symmetric and axisymmetric backward heat conduction problems

Sensor networks for diffusion fields: Detection of sources in space and time

Inverse Heat Conduction Problems

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