On a backward problem for nonlinear fractional diffusion equations

Tuan, Nguyen Huy; Huynh, Lam K.; Ngoc, Tran Bao; Zhou, Yong

doi:10.1016/j.aml.2018.11.015

Cited by 55 publications

(27 citation statements)

References 15 publications

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“…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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“…For

normalγ = 1

{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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show abstract

“…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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“…However, by adding some given data, we can discuss inverse problems such as the backward problem (recovering the initial data) or the source identification problem (recovering the source function). Initial inverse problems for fractional Riemann-Liouville or Caputo diffusion equations were discussed in the literature [20,26,27]. However, little is known on the initial inverse problem for the diffusion equation with a conformable derivative.…”

Section: Introductionmentioning

confidence: 99%

On an initial inverse problem for a diffusion equation with a conformable derivative

et al. 2019

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On a backward problem for nonlinear fractional diffusion equations

Cited by 55 publications

References 15 publications

Regularization of a terminal value problem for time fractional diffusion equation

Regularization of a terminal value problem for time fractional diffusion equation

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

On an initial inverse problem for a diffusion equation with a conformable derivative

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