Two regularization methods to identify time-dependent heat source through an internal measurement of temperature

Yang, Fan; Fu, Chu-Li

doi:10.1016/j.mcm.2010.10.016

Cited by 20 publications

(8 citation statements)

References 32 publications

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“…The Fourier truncation regularization method is a very effective method for dealing with ill-posed problems. Many authors have used it to deal with different ill-posed problems, such as in [23][24][25][26][27][28][29][30][31][32]. In [33], the authors extended the Fourier method to the general filtering method and solved the semi-linear ill-posed problem in the general framework.…”

Section: Fourier Regularization Methods and Resultsmentioning

confidence: 99%

Fourier Truncation Regularization Method for a Time-Fractional Backward Diffusion Problem with a Nonlinear Source

Yang¹,

Fan²,

Li³

et al. 2019

Mathematics

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In present paper, we deal with a backward diffusion problem for a time-fractional diffusion problem with a nonlinear source in a strip domain. We all know this nonlinear problem is severely ill-posed, i.e., the solution does not depend continuously on the measurable data. Therefore, we use the Fourier truncation regularization method to solve this problem. Under an a priori hypothesis and an a priori regularization parameter selection rule, we obtain the convergence error estimates between the regular solution and the exact solution at 0 ≤ x < 1 .

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Section: Fourier Regularization Methods and Resultsmentioning

confidence: 99%

Fourier Truncation Regularization Method for a Time-Fractional Backward Diffusion Problem with a Nonlinear Source

Yang¹,

Fan²,

Li³

et al. 2019

Mathematics

Self Cite

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“…Problems of this type have been considered, e. g., in [10,14,76,77,78,79]. They arise in different engineering problems.…”

Section: Identification Of Heat Sourcesmentioning

confidence: 99%

“…They arise in different engineering problems. For example, an accurate estimation of the pollution source is crucial for environmental protection in cities with high population, see [77].…”

Section: Identification Of Heat Sourcesmentioning

confidence: 99%

Conditional Stability Estimates for Ill-Posed PDE Problems by Using Interpolation

Tautenhahn

Hämarik

Hofmann

et al. 2013

Numerical Functional Analysis and Optimization

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Abstract. The focus of this paper is on conditional stability estimates for illposed inverse problems in partial differential equations. Conditional stability estimates have been obtained in the literature by a couple different methods. In this paper we propose a method called interpolation method, which is based on interpolation in variable Hilbert scales. We are going to work out the theoretical background of this method and show that optimal conditional stability estimates are obtained. The capability of our method is illustrated by a comprehensive collection of different inverse and ill-posed PDE problems containing elliptic and parabolic problems, one source problem and the problem of analytic continuation.

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“…Recently, Fourier regularization method has been effectively applied to solve different inverse problem: The sideways heat equation [18,19], a more general sideways parabolic equation [20], numerical differentiation [21], a posteriori Fourier method for solving ill-posed problems [22], the unknown source in the Poisson equation [23], the time-dependent heat source for heat equation [24], the heat source problem for time fractional diffusion equation [25,26], the semi-linear backward parabolic problems [27], the unknown source for time-fractional diffusion equation in bounded domain [28], the Cauchy problem for the Helmholtz equation [29], the a posteriori truncation method for some Cauchy problems associated with Helmholtz-type equations [30], the Cauchy problem of the inhomogeneous Helmholtz equation [31].…”

Section: Introductionmentioning

confidence: 99%

Fourier Truncation Regularization Method for a Three-Dimensional Cauchy Problem of the Modified Helmholtz Equation with Perturbed Wave Number

Yang

Fan

2019

Mathematics

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In this paper, the Cauchy problem of the modified Helmholtz equation (CPMHE) with perturbed wave number is considered. In the sense of Hadamard, this problem is severely ill-posed. The Fourier truncation regularization method is used to solve this Cauchy problem. Meanwhile, the corresponding error estimate between the exact solution and the regularized solution is obtained. A numerical example is presented to illustrate the validity and effectiveness of our methods.

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Two regularization methods to identify time-dependent heat source through an internal measurement of temperature

Cited by 20 publications

References 32 publications

Fourier Truncation Regularization Method for a Time-Fractional Backward Diffusion Problem with a Nonlinear Source

Fourier Truncation Regularization Method for a Time-Fractional Backward Diffusion Problem with a Nonlinear Source

Conditional Stability Estimates for Ill-Posed PDE Problems by Using Interpolation

Fourier Truncation Regularization Method for a Three-Dimensional Cauchy Problem of the Modified Helmholtz Equation with Perturbed Wave Number

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