Determination of space–time-dependent heat source in a parabolic inverse problem via the Ritz–Galerkin technique

Rashedi, Kamal; Adibi, Hojatollah; Dehghan, Mehdi

doi:10.1080/17415977.2013.854354

Cited by 25 publications

(17 citation statements)

References 50 publications

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“…A survey in the literature shows that inverse problems have been studied so far in various issues such as source identification in the heat conduction [1][2][3][4][5] and wave propagation problems, [6][7][8][9] determination of material parameters [10][11][12][13][14] and inverse boundary conditions reconstruction. [15][16][17] The problem of inverse source identification on thin plates is also another interesting topic which has been investigated in numerous studies.…”

Section: Introductionmentioning

confidence: 99%

Inverse identification of time-harmonic loads acting on thin plates using approximated Green’s functions

Movahedian

Boroomand

2015

Inverse Problems in Science and Engineering

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Section: Introductionmentioning

confidence: 99%

Inverse identification of time-harmonic loads acting on thin plates using approximated Green’s functions

Movahedian

Boroomand

2015

Inverse Problems in Science and Engineering

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“…Due to the instability of the inverse problem, the computational treatment of such problems needs applying more accurate numerical methods to obtain a stable approximate solution. In recent years, IHCPs have been investigated from the analytical and numerical point of view, in the literature, e.g., [3][4][5][6][8][9][10].…”

Section: Introductionmentioning

confidence: 99%

Numerical investigation of an inverse problem based on regularization method

et al. 2019

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In this study, a numerical approach of the spectral collocation method coupled with a regularization technique is applied for solving an inverse parabolic problem of the heat equation in a quarter plane. The problem includes the estimation of an unknown boundary condition from an overspecified condition. The stable solution of the problem exists and is proved by Tikhonov regularization technique. The algorithm works without any mesh points or elements, and accurate results can be obtained efficiently. By employing the numerical algorithm on the problem, the resultant matrix equation is ill-condition. To regularize this matrix equation, we apply regularization technique, with the L-curve and general cross-validation criteria for choosing the regularization parameter. For demonstrating the performance and ability of the proposed algorithm, a test example is presented. The numerical results showed that the solution obtained with the algorithm designed in this paper is stable with the noisy data and the unknown boundary condition was recovered very well.

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“…In conclusion, the large sets of collocation points are not needed for applying the supplemented boundary conditions which naturally leads to a system of algebraic equations of smaller size and hence reduces the computation time. Although the authors in [20,21] reported satisfactory results with relatively low-cost computational efforts, their solution could suffer from propagation of errors because improperly posed problems are always involved with noisy input data. Since the satisfier function incorporates all initial and boundary conditions including the possibly erroneous ones, the technique has some shortcomings.…”

Section: Introductionmentioning

confidence: 99%

Stable recovery of the time-dependent source term from one measurement for the wave equation

Rashedi

Sini

2015

Inverse Problems

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In this work, we discuss the inverse problem which consists of the determination of an unknown time-dependent force function from one time-dependent measurement collected in any space point for the one-dimensional wave equation. This problem is motivated by the question of estimating the timedependent body force which needs to be exerted on a given string to reach a desired shape at the final time. We prove its unique solvability using as data a linear combination of displacement and flux measured at one arbitrary fixed point of the string. We also derive a conditional Hölder stability estimate of this inverse problem. The numerical solution of the problem is investigated by means of the Ritz-Galerkin technique along with the application of the satisfier function to obtain cost-effective and stable results. Some numerical examples are provided to show the performance of the proposed scheme.

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Determination of space–time-dependent heat source in a parabolic inverse problem via the Ritz–Galerkin technique

Cited by 25 publications

References 50 publications

Inverse identification of time-harmonic loads acting on thin plates using approximated Green’s functions

Inverse identification of time-harmonic loads acting on thin plates using approximated Green’s functions

Numerical investigation of an inverse problem based on regularization method

Stable recovery of the time-dependent source term from one measurement for the wave equation

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