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

In this paper, we consider an inverse problem of recovering the initial value for a generalization of time‐fractional diffusion equation, where the time derivative is replaced by a regularized hyper‐Bessel operator. First, we investigate the existence and regularity of our terminal value problem. Then we show that the backward problem is ill‐posed, and we propose a regularizing scheme using a fractional Tikhonov regularization method. We also present error estimates between the regularized solution and the exact solution using two parameter choice rules.

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“…Then it is a Banach space endowed with the usual supremum norm. For any κ >0, we introduce the following Hölder continuous space of exponent κ

C^{κ} (I; X) = \{v \in C (I; X) : \sup_{s \neq t \in I} \frac{‖ v (t) - v (s) {false‖}_{double-struckX}}{| t - s {false|}^{κ}} < \infty\},

which equips with the

‖ v {false‖}_{C^{κ} (I; X)} = \sup_{t \in I} ‖ v (t) {false‖}_{double-struckX} + \sup_{s \neq t \in I} \frac{‖ v (t) - v (s) {false‖}_{double-struckX}}{| t - s {false|}^{κ}} .

Since Zhou et al, we recall the space

F^{η, κ} false(double-struckI \ ({}, 0); double-struckX false)

with the following norm

‖ u {false‖}_{{double-struckF}^{η, κ} (I \ \{0\}; X)} = \sup_{t \in I \ \{0\}} {sans-serift}^{η} ‖ u (

…”

Section: Preliminariesmentioning

confidence: 99%

On a terminal value problem for a generalization of the fractional diffusion equation with hyper‐Bessel operator

Tuan

Huynh

Băleanu

et al. 2019

Math Methods in App Sciences

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“…In, Sakamoto and Yamamoto considered fractional diffusion ‐ wave equations, Yang et al solved a final value fractional diffusion problem by boundary condition regularization, and Denche et al modified quasi‐boundary value method for ill‐posed problems. Very recently, it has been considered by some other authors, such as …”

Section: Introductionmentioning

confidence: 99%

“…Very recently, it has been considered by some other authors, such as. 2,[19][20][21][22][23][24][25][26] Motivated by above reasons, in this study, we apply the quasi-boundary value regularization method to solve the (IPFE) t (1). We estimate a convergence rate under a priori bound assumption of the exact solution and a priori parameter choice rule.…”

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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“…The growing interest in the subject is due to its extensive applications in diverse fields such as physics, fluid mechanics, viscoelasticity, heat conduction in materials with memory, chemistry and engineering. Much of the work is devoted to the existence and uniqueness of solutions for fractional differential equations; see, for example, Kilbas et al [10], Miller and Ross [13], Podlubny [14], Zhou [22] and [1,5,19,21,23,24] and the references cited therein. Since Hilfer [9] proposed the generalized Riemann-Liouville fractional derivative (Hilfer fractional derivative), there has been shown some interest in studying evolution equations involving Hilfer fractional derivatives (see [2,4,7,8,18,20]).…”

Section: Introductionmentioning

confidence: 99%

Attractivity for Hilfer fractional stochastic evolution equations

et al. 2020

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This article is devoted to the study of the attractivity of solutions to a class of stochastic evolution equations involving Hilfer fractional derivative. By employing the semigroup theory, fractional calculus and the fixed point technique, we establish new alternative criteria to ensure the existence of globally attractive solutions for the Cauchy problem when the associated semigroup is compact.Keywords: Stochastic evolution equations; Hilfer fractional derivative; Attractivity Recently, the stability of fractional-order systems has been discussed by several authors. Abbas et al. [2] investigated the Ulam stability for Hilfer fractional differential inclusions via the weakly Picard operators theory. Chen et al. [6] established the global attractivity for nonlinear fractional differential equations. Losada, et al. [11] studied the attractivity of solutions for a class of multi-term fractional functional differential equations. Rajivganthi, Rihan et al. [15][16][17] studied the stability of a fractional-order prey-predator system. Moreover, stochastic perturbation is unavoidable in nature and sometimes useful in research, for example, the existence of stochastic perturbation in the mathematical model has been found to be quite effective in preventing the explosion of the population. Hence, it is important and necessary to consider stochastic perturbation into the investigation of fractional differential equations (see [3,12]. However, it seems that there is less literature related to the stability of Hilfer fractional stochastic evolution equations. Therefore, the attractivity of solutions of Hilfer fractional stochastic evolution equations might be a fascinating and useful problem.

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

Cited by 55 publications

References 26 publications

On a terminal value problem for a generalization of the fractional diffusion equation with hyper‐Bessel operator

On a terminal value problem for a generalization of the fractional diffusion equation with hyper‐Bessel operator

Regularization of a terminal value problem for time fractional diffusion equation

Attractivity for Hilfer fractional stochastic evolution equations

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