On a final value problem for the time-fractional diffusion equation with inhomogeneous source

Tuan, Nguyen Huy; Long, Le Dinh; Nguyen, Van Thinh; Tran, Thanh

doi:10.1080/17415977.2016.1259316

Cited by 33 publications

(14 citation statements)

References 26 publications

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“…By replacing many differential operators of fractional order with different type of PDEs of integer order, we formulate various types of boundary value problems with fractional order. Let us refer to many papers [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

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On a time fractional diffusion with nonlocal in time conditions

Tuấn

Triet²,

Luc³

et al. 2021

Adv Differ Equ

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In this work, we consider a fractional diffusion equation with nonlocal integral condition. We give a form of the mild solution under the expression of Fourier series which contains some Mittag-Leffler functions. We present two new results. Firstly, we show the well-posedness and regularity for our problem. Secondly, we show the ill-posedness of our problem in the sense of Hadamard. Using the Fourier truncation method, we construct a regularized solution and present the convergence rate between the regularized and exact solutions.

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Section: Introductionmentioning

confidence: 99%

“…• If a = 1, b = 0, Problem (1.1) is called backward in time problem and is well known to be ill-posed in the sense of Hadamard. Some papers [9,37] gave the ill-posedness and regularization results.…”

Section: Introductionmentioning

confidence: 99%

On a time fractional diffusion with nonlocal in time conditions

Tuấn

Triet²,

Luc³

et al. 2021

Adv Differ Equ

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show abstract

“…For

normalγ = 1

, the problem is a classical PDE and has been studied for many decades, and various regularization methods have been proposed, for example, see, and the references therein. However, there are very scarce studies on the

{IPFE}_{t}

for

normalγ \in false(0, 1 false)

.…”

Section: Introductionmentioning

confidence: 99%

Regularization of a terminal value problem for time fractional diffusion equation

Triet

Long

et al. 2020

Math Methods in App Sciences

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In this article, we study an inverse problem with inhomogeneous source to determine an initial data from the time fractional diffusion equation. In general, this problem is ill‐posed in the sense of Hadamard, so the quasi‐boundary value method is proposed to solve the problem. In the theoretical results, we propose a priori and a posteriori parameter choice rules and analyze them. Finally, two numerical results in the one‐dimensional and two‐dimensional case show the evidence of the used regularization method.

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“…In many practical applications, we can just measure density of the diffusing substance at positive moment without knowing the initial density in many practical situations, and so it reflects the advantages of the backward problem to deal with this kind of difficult situation. For more details of the fractional backward problem, one can see the papers in previous studies [18][19][20][21] and the references cited therein.…”

Section: Introductionmentioning

confidence: 99%

Existence and regularity results of a backward problem for fractional diffusion equations

Zhou

Ahmad

et al. 2019

Math Methods in App Sciences

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In this paper, we study a backward problem for an inhomogeneous fractional diffusion equation in a bounded domain. By applying the properties of Mittag-Leffler functions and the method of eigenvalue expansion, we establish some results about the existence, uniqueness, and regularity of the mild solutions as well as the classical solutions of the proposed problem in a weighted Hölder continuous function space. KEYWORDS backward problem, existence, fractional diffusion equation, regularity MSC CLASSIFICATION 26A33; 35R11 d dt ∫ t 0 (t − s) −1 u(s, x)ds, t > 0, Math Meth Appl Sci. 2019;42:6775-6790.wileyonlinelibrary.com/journal/mma

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On a final value problem for the time-fractional diffusion equation with inhomogeneous source

Cited by 33 publications

References 26 publications

On a time fractional diffusion with nonlocal in time conditions

On a time fractional diffusion with nonlocal in time conditions

Regularization of a terminal value problem for time fractional diffusion equation

Existence and regularity results of a backward problem for fractional diffusion equations

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