An inverse problem for a nonlinear diffusion equation with time-fractional derivative

Tatar, Salih; Ulusoy, Süleyman

doi:10.1515/jiip-2015-0100

Cited by 17 publications

(24 citation statements)

References 28 publications

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“…The difference of the current study from the references [33]- [46] is that the unknown of the inverse problem is non-linear, i.e depends on the solution u. This is a relatively new topic and there are only few works, see [47]- [49]. In [47], the unknown coefficient depends on the gradient of the solution and belongs to a set of admissible coefficients.…”

Section: Introductionmentioning

confidence: 92%

“…In [48], the authors study the numerical solutions of the direct and the inverse problems in [47] and mention about an application of the governing equation in the materials sciences. An inverse problem for the nonlinear time-fractional diffusion equation (1.12) is studied in [49]. In this paper, the authors prove that the direct problem has a unique solution.…”

Section: Introductionmentioning

confidence: 97%

See 1 more Smart Citation

Determination of a Nonlinear Coefficient in a Time-Fractional Diffusion Equation

Zeki

Tinaztepe

Tatar

et al. 2022

Preprint

Self Cite

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Direct and inverse problems for a nonlinear time fractional equation are studied. It is proved that the direct problem has a unique weak solution and the solution depends continuously on the nonlinear coefficient. Then it is shown that the inverse problem has a quasi-solution. The direct problem is solved by method of lines using an operator approach. A quasi-Newton optimization method is used for the numerical solution to the inverse problem. Tikhonov regularization is used to overcome the ill-posedness of the inverse problem. Numerical examples with noise free and noisy data illustrate applicability and accuracy of the proposed method to some extent.

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Section: Introductionmentioning

confidence: 92%

Section: Introductionmentioning

confidence: 97%

Determination of a Nonlinear Coefficient in a Time-Fractional Diffusion Equation

Zeki

Tinaztepe

Tatar

et al. 2022

Preprint

Self Cite

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“…Rundell, Xu and Zuo [48] studied a similar inverse problem of determining a nonlinear term f in the boundary condition −∂ x y(1, t) = f (y(1, t)), 0 < t < T for the one-dimensional fractional diffusion equation. See Tatar and Ulusoy [54] for an optimization method for reconstructing a in ∂ α t y(x, t) = div (a(y(x, t))∇y(x, t)) + F (x, t), x ∈ Ω ⊂ R d , 0 < t < T.…”

Section: Determination Of Nonlinear Termsmentioning

confidence: 99%

Inverse problems of determining coefficients of the fractional partial differential equations

Li¹,

Yamamoto²

2019

Fractional Differential Equations

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When considering fractional diffusion equation as model equation in analyzing anomalous diffusion processes, some important parameters in the model, for example, the orders of the fractional derivative or the source term, are often unknown, which requires one to discuss inverse problems to identify these physical quantities from some additional information that can be observed or measured practically. This chapter investigates several kinds of inverse coefficient problems for the fractional diffusion equation.

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“…To solve inverse problems, it is necessary to have additional information about the solution of the equation. Most often, such additional information is information about the solution on a part of the boundary of the domain on which the equation is considered (see, for example, [25][26][27][28][29][30][31][32][33][34][35]). However, recently, the formulation of inverse problems for the equation of this type with additional information on the dynamics of the reaction front has become relevant because the reaction front is an experimentally easily distinguishable contrast structure (see, for example, [36,37]).…”

Section: Introductionmentioning

confidence: 99%

On Some Features of the Numerical Solving of Coefficient Inverse Problems for an Equation of the Reaction-Diffusion-Advection-Type with Data on the Position of a Reaction Front

et al. 2021

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The work continues a series of articles devoted to the peculiarities of solving coefficient inverse problems for nonlinear singularly perturbed equations of the reaction-diffusion-advection-type with data on the position of the reaction front. In this paper, we place the emphasis on some problems of the numerical solving process. One of the approaches to solving inverse problems of the class under consideration is the use of methods of asymptotic analysis. These methods, under certain conditions, make it possible to construct the so-called reduced formulation of the inverse problem. Usually, a differential equation in this formulation has a lower dimension/order with respect to the differential equation, which is included in the full statement of the inverse problem. In this paper, we consider an example that leads to a reduced formulation of the problem, the solving of which is no less a time-consuming procedure in comparison with the numerical solving of the problem in the full statement. In particular, to obtain an approximate numerical solution, one has to use the methods of the numerical diagnostics of the solution’s blow-up. Thus, it is demonstrated that the possibility of constructing a reduced formulation of the inverse problem does not guarantee its more efficient solving. Moreover, the possibility of constructing a reduced formulation of the problem does not guarantee the existence of an approximate solution that is qualitatively comparable to the true one. In previous works of the authors, it was shown that an acceptable approximate solution can be obtained only for sufficiently small values of the singular parameter included in the full statement of the problem. However, the question of how to proceed if the singular parameter is not small enough remains open. The work also gives an answer to this question.

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An inverse problem for a nonlinear diffusion equation with time-fractional derivative

Cited by 17 publications

References 28 publications

Determination of a Nonlinear Coefficient in a Time-Fractional Diffusion Equation

Determination of a Nonlinear Coefficient in a Time-Fractional Diffusion Equation

Inverse problems of determining coefficients of the fractional partial differential equations

On Some Features of the Numerical Solving of Coefficient Inverse Problems for an Equation of the Reaction-Diffusion-Advection-Type with Data on the Position of a Reaction Front

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