Numerical Solutions of Direct and Inverse Problems for a Time Fractional Viscoelastoplastic Equation

Tatar, Salih; Tnaztepe, Ramazan; Zeki, Mustafa

doi:10.1061/(asce)em.1943-7889.0001239

Cited by 9 publications

(5 citation statements)

References 49 publications

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“…The article [11] proposes a new mathematical model for a viscoelastic-plastic process by formulating a temporary fractional equation containing an elliptic operator, which depends on the gradient of the solution. In [12], the authors construct and study backward Euler and locally one dimensional schemes approximating Dirichlet problem with Caputo time-fractional derivative and a quasilinear elliptic part without mixed derivatives.…”

Section: Introductionmentioning

confidence: 99%

The Regularized Mesh Scheme to Solve Quasilinear Parabolic Equation with Time-Fractional Derivative

Лапин

Laitinen

2021

Lobachevskii J Math

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A quasilinear parabolic problem with a time fractional derivative of the Caputo type and mixed boundary conditions is considered. The coefficients of the elliptic operator depend on the gradient of the solution, and this operator is uniformly monotone and Lipschitz-continuous. For this problem, unconditionally stable linear regularized semi-discrete scheme is constructed based on the $$L1$$-approximation of the fractional time derivative. Stability estimates are obtained by the variational method. Accuracy estimates are given provided that the initial data and the solution to the differential problem are sufficiently smooth. The proved result of stability of the semi-discrete scheme is valid for the mesh scheme obtained from the semi-discrete problem using the finite element method in spatial variables.

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Section: Introductionmentioning

confidence: 99%

The Regularized Mesh Scheme to Solve Quasilinear Parabolic Equation with Time-Fractional Derivative

Лапин

Laitinen

2021

Lobachevskii J Math

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“…The difference of our study from the studies in the references [31][32][33][34][35][36][37][38][39][40][41][42][43] is that the unknown of the inverse problem is non-linear, i.e., it depends on the solution u. This is a relatively new topic and there are only a few works, see [44][45][46]. In [44], the unknown coefficient depends on the gradient of the solution and belongs to a set of admissible coefficients.…”

Section: −β Tmentioning

confidence: 95%

“…In [44], the numerical solutions of the direct and the inverse problems have been introduced and in [45] an application of the governing equation in the materials sciences has been mentioned. An inverse problem for the nonlinear time-fractional diffusion (12) is studied in [46].…”

Section: −β Tmentioning

confidence: 99%

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

et al. 2023

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In this paper, we study direct and inverse problems for a nonlinear time fractional diffusion equation. We prove that the direct problem has a unique weak solution and the solution depends continuously on the coefficient. Then we show that the inverse problem has a quasi-solution. The direct problem is solved by the method of lines using an operator approach. A quasi-Newton optimization method is used for the numerical solution to the inverse problem. The Tikhonov regularization is used to overcome the ill-posedness of the inverse problem. Numerical examples with noise-free and noisy data illustrate the applicability and accuracy of the proposed method to some extent.

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“…Then, existence of a quasi-solution of the inverse problem is obtained in the appropriate class of admissible coefficients. 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].…”

Section: Introductionmentioning

confidence: 99%

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

Zeki

Tinaztepe

Tatar

et al. 2022

Preprint

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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.

show abstract

Numerical Solutions of Direct and Inverse Problems for a Time Fractional Viscoelastoplastic Equation

Cited by 9 publications

References 49 publications

The Regularized Mesh Scheme to Solve Quasilinear Parabolic Equation with Time-Fractional Derivative

The Regularized Mesh Scheme to Solve Quasilinear Parabolic Equation with Time-Fractional Derivative

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

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

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