The method of fundamental solutions for the backward heat conduction problem

Mera, N. S.

doi:10.1080/10682760410001710141

Cited by 102 publications

(65 citation statements)

References 17 publications

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“…The third example we investigate is a typical benchmark case of one-dimensional BHCP [6,8], whose exact solution is And the boundary conditions are also given by u(0, t) = g 1 (t) = 0, u(1, t) = g 2 (t) = 0. The BHCP retrieves the initial temperature from the measurement data at final time.…”

Section: Examplementioning

confidence: 99%

“…Equation (8), together with (9) and (10), builds an over-determined system of linear algebraic equations with N × M unknowns and (N × M + N 1 + N b × M 1 ) equations, as follows:…”

Section: The Rbcmmentioning

confidence: 99%

“…However, in many practical situations, not all the necessary conditions are available due to incomplete understanding of the physical process, or technical difficulties associated with direct measurement. And thus the IHCP problems arise, which may involve the estimation of the system geometry [1], the thermal properties [2], the boundary conditions [3][4][5] and the initial temperature distribution [6][7][8] with the assistance of auxiliary temperature measurements. These problems are inherently ill-posed so that small input errors in the measured data may produce arbitrarily large deviations in the results [9,10].…”

Section: Introductionmentioning

confidence: 99%

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Global space–time multiquadric method for inverse heat conduction problem

Mao

2010

Numerical Meth Engineering

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SUMMARYIn this paper, a radial basis collocation method (RBCM) based on the global space-time multiquadric (MQ) is proposed to solve the inverse heat conduction problem (IHCP). The global MQ is simply constructed by incorporating time dimension into the MQ function as a new variable in radial coordinate. The method approximates the IHCP as an over-determined linear system with the use of two sets of collocation points: one is satisfied with the governing equation and another is for the given conditions. The least-square technique is introduced to find the solution of the over-determined linear system. The present work investigates two types of the ill-posed heat conduction problems: the IHCP to recover the surface temperature and heat flux history on a source point from the measurement data at interior locations, and the backward heat conduction problem (BHCP) to retrieve the initial temperature distribution from the known temperature distribution at a given time. Numerical results of four benchmark examples show that the proposed method can provide accurate and stable numerical solutions for one-dimensional and two-dimensional IHCP problems. The sensitivity of the method with respect to the measured data, location of measurement, time step, shape parameter and scaling factor is also investigated.

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Section: Examplementioning

confidence: 99%

“…Equation (8), together with (9) and (10), builds an over-determined system of linear algebraic equations with N × M unknowns and (N × M + N 1 + N b × M 1 ) equations, as follows:…”

Section: The Rbcmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Global space–time multiquadric method for inverse heat conduction problem

Mao

2010

Numerical Meth Engineering

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“…A BHCP is severely ill-posed problem [1]. To overcome this difficulty, many scholars proposed some regularization techniques for the BHCP, such as the kernel-based method [2], the mollification method [3], the Fourier regularization method [4], optimal filtering method [5], the iterative method [6], the quasi-reversibility method [7][8][9], the central difference method [10], the filter regularization method [11], the method of fundamental solutions [12,13], the boundary element method [14,15], the group preserving scheme [16], modified Tikhonov regularization method [17], Quasi-boundary value method [18] and so on. But these references about BHCP, there are some drawbacks as follows: firstly, the regularization parameter is a prior choice rule, according to this choice rule, the parameter depends on the prior bound of the exact solution.…”

Section: Introductionmentioning

confidence: 99%

The truncation regularization method for identifying the initial value of heat equation on a spherical symmetric domain

Yang

Sun

et al. 2018

Bound Value Probl

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In this paper, identifying the initial value for high dimension heat equation with inhomogeneous source on a spherical symmetric domain is investigated. The truncation regularization method is a powerful technique for solving this inverse problem. We prove the convergence estimates between the regularization solution and the exact solution under the prior and the posterior regularization parameter choice rulers. A numerical example is presented to validate the effectiveness of this method.

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“…Schröter and Tautenhahn [5] established an optimal error estimate for a special BHCP. Mera and Jourhmane used many numerical methods with regularization techniques to approximate the problem in [6][7][8], etc. A mollification method has been studied by Haö in [9].…”

Section: Introductionmentioning

confidence: 99%

Quasi-Reversibility Regularization Method for Solving a Backward Heat Conduction Problem

Qian¹,

Mao²

2011

AJCM

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Non-standard backward heat conduction problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the data. In this paper, we propose a regularization strategy-quasi-reversibility method to analysis the stability of the problem. Meanwhile, we investigate the roles of regularization parameter in this method. Numerical result shows that our algorithm is effective and stable.

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The method of fundamental solutions for the backward heat conduction problem

Cited by 102 publications

References 17 publications

Global space–time multiquadric method for inverse heat conduction problem

Global space–time multiquadric method for inverse heat conduction problem

The truncation regularization method for identifying the initial value of heat equation on a spherical symmetric domain

Quasi-Reversibility Regularization Method for Solving a Backward Heat Conduction Problem

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