Recovering the source term for parabolic equation with nonlocal integral condition

Phương, Nguyễn Duy; Băleanu, Dumitru; Phong, Tran Thanh; Long, Le Dinh

doi:10.1002/mma.7331

Cited by 3 publications

(3 citation statements)

References 13 publications

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“…Inverse problems for timefractional diffusion equations seek to retrieve initial data, source function, diffusion coefficient, and other parameters through additional data. However, such problems have received little attention recently [3][4][5]. As far as we know, no previous studies have focused on (1)-(2) concerning random noise as depicted in (4).…”

Section: Introductionmentioning

confidence: 99%

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Recovering solution of the Reverse nonlinear time Fractional diffusion equations with fluctuations data

Doan Thi,

Vo Thi

2023

Electron. J. Appl. Math.

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In this study, our focus is on obtaining an estimated solution for the nonlinear fractional time diffusion equation. Specifically, we have utilized the Riemann Liouville fractional derivative. Additionally, we have concerned Gaussian white noise in the input data. As we are aware, this problem is considered ill-posed according to Hadamard's definition. To tackle this problem, we have proposed a regularized solution and demonstrated the convergence between the mild solution and the regularized solution.

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Section: Introductionmentioning

confidence: 99%

“…However, such problems have received little attention recently [3][4][5]. As far as we know, no previous studies have focused on (1)-(2) concerning random noise as depicted in (4).…”

Section: Introductionmentioning

confidence: 99%

Recovering solution of the Reverse nonlinear time Fractional diffusion equations with fluctuations data

Doan Thi,

Vo Thi

2023

Electron. J. Appl. Math.

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“…This condition is proposed in the paper by Dokuchaev [29]. Very recently, Tuan and coauthors used this condition to solve some nonlocal problem, for example [30][31][32], and [33]. Motivated by this above reason, in this paper, we apply the fractional Tikhonov method to solve problem (1.1).…”

Section: Introductionmentioning

confidence: 99%

On backward problem for fractional spherically symmetric diffusion equation with observation data of nonlocal type

et al. 2021

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The main target of this paper is to study a problem of recovering a spherically symmetric domain with fractional derivative from observed data of nonlocal type. This problem can be established as a new boundary value problem where a Cauchy condition is replaced with a prescribed time average of the solution. In this work, we set some of the results above existence and regularity of the mild solutions of the proposed problem in some suitable space. Next, we also show the ill-posedness of our problem in the sense of Hadamard. The regularized solution is given by the fractional Tikhonov method and convergence rate between the regularized solution and the exact solution under a priori parameter choice rule and under a posteriori parameter choice rule.

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An Approximate Solution for a Class of Ill-Posed Nonhomogeneous Cauchy Problems

Teniou

Djezzar

2021

Journal of Mathematics

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In this paper, we consider a nonhomogeneous differential operator equation of first order u ′ t + A u t = f t . The coefficient operator A is linear unbounded and self-adjoint in a Hilbert space. We assume that the operator does not have a fixed sign. We associate to this equation the initial or final conditions u 0 = Φ or u T = Φ . We note that the Cauchy problem is severely ill-posed in the sense that the solution if it exists does not depend continuously on the given data. Using a quasi-boundary value method, we obtain an approximate nonlocal problem depending on a small parameter. We show that regularized problem is well-posed and has a strongly solution. Finally, some convergence results are provided.

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Recovering the source term for parabolic equation with nonlocal integral condition

Cited by 3 publications

References 13 publications

Recovering solution of the Reverse nonlinear time Fractional diffusion equations with fluctuations data

Recovering solution of the Reverse nonlinear time Fractional diffusion equations with fluctuations data

On backward problem for fractional spherically symmetric diffusion equation with observation data of nonlocal type

An Approximate Solution for a Class of Ill-Posed Nonhomogeneous Cauchy Problems

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