Identification and regularization for unknown source for a time-fractional diffusion equation

Tuan, Nguyen Huy; Kirane, Mokhtar; Hoan, Luu Vu Cam; Long, Le Dinh

doi:10.1016/j.camwa.2016.10.002

Cited by 22 publications

(9 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In Ahmad et al, there are good references to publications on related issues. We note from recent papers close to the theme of our article. In these papers, different variants of direct and inverse initial‐boundary value problems for evolutionary equations are considered, including problems with nonlocal boundary conditions and problems for equations with fractional derivatives.…”

Section: Reduction To a Mathematical Problemmentioning

confidence: 85%

On an inverse problem of reconstructing a subdiffusion process from nonlocal data

Kirane

Sadybekov

Sarsenbi

2019

Math Methods in App Sciences

Self Cite

View full text Add to dashboard Cite

We consider a problem of modeling the thermal diffusion process in a closed metal wire wrapped around a thin sheet of insulation material. The layer of insulation is assumed to be slightly permeable. Therefore, the temperature value from one side affects the diffusion process on the other side. For this reason, the standard heat equation is modified, and a third term with an involution is added. Modeling of this process leads to the consideration of an inverse problem for a one‐dimensional fractional evolution equation with involution and with periodic boundary conditions with respect to a space variable. This equation interpolates heat equation. Such equations are also called nonlocal subdiffusion equations or nonlocal heat equations. The inverse problem consists in the restoration (simultaneously with the solution) of the unknown right‐hand side of the equation, which depends only on the spatial variable. The conditions for overdefinition are initial and final states. Existence and uniqueness results for the given problem are obtained via the method of separation of variables.

show abstract

Section: Reduction To a Mathematical Problemmentioning

confidence: 85%

On an inverse problem of reconstructing a subdiffusion process from nonlocal data

Kirane

Sadybekov

Sarsenbi

2019

Math Methods in App Sciences

Self Cite

View full text Add to dashboard Cite

show abstract

“…The noisy data are generated by adding a random perturbation as follows:

{\overset{u_{1}}{true}}^{ϵ} = u_{1} + ϵ ξ .

We choose R ( β , ω ) as in Corollary . Let us define a regularized solution for the system as follows:

{\overset{f}{true}}^{ϵ} false(x false) = - \frac{1}{2 π} \int_{- ω_{\max}}^{+ ω_{\max}} \frac{\int_{0}^{1} {\overset{u_{1}}{true}}^{ϵ} false(x false) \cos false(ω x false) d x - E_{α, 1} false(- ω^{2} false) \int_{0}^{1} u_{0} false(x false) \cos false(ω x false) d x}{\int_{0}^{1} s^{α - 1} E_{α, α} false(- ω^{2} s^{α} false) φ false(1 - s false) d s} e^{i ω x} d ω .

Using, we obtain

true \int_{0}^{1} E_{α, 1} false(y s^{α} false) {false(1 - s false)}^{β - 1} E_{α, β} false[z {false(1 - s false)}^{α} false] d s = \frac{y E_{α, 1 + β} false(y false) - z E_{α, β + 1} false(z false)}{y - z} \cdot

Let y =−1, β = α and z =− ω 2 into , we have

\int_{0}^{1} s^{α - 1} E_{α, α} false(- ω^{2} s^{α} false) φ false(1 - s false) d<...>

…”

Section: Numerical Experimentsmentioning

confidence: 99%

Identification of an inverse source problem for time‐fractional diffusion equation with random noise

Thach

Tuan

Tam

et al. 2018

Math Methods in App Sciences

View full text Add to dashboard Cite

Communicated by: M. Kirane MSC Classification: 35K05; 35K99; 47J06; 47H10In this paper, we consider an inverse source problem for a time fractional diffusion equation. In general, this problem is ill posed, therefore we shall construct a regularized solution using the filter regularization method in the random noise case. We will provide appropriate conditions to guarantee the convergence of the approximate solution to the exact solution. Then, we provide examples of filters in order to obtain error estimates for their approximate solutions. Finally, we present a numerical example to show efficiency of the method. KEYWORDSdiffusion process, fractional derivative, inverse source problem, random noise, regularization 204 We start by recalling the definition of the Mittag-Leffler function, a special function that plays an important role in understanding diffusion processes. For more information, see 12 and references therein.Definition 2.1. For any > 0 and ∈ R , the two-parameter Mittag-Leffler function, a special function that plays an important role in understanding diffusion processes. For more information, see 13 and references therein, z ∈ C.The asymptotic growth of the Mittag-Leffler function is characterized in the following lemma.

show abstract

“…Therefore the main result on the existence and uniqueness of the solution of the problem of type I (1), (2), (24) in classical and generalized senses follows from Theorem 1 on corresponding solvability of boundary value problems with conditions of the Sturm type. We will formulate this main result at once for all the four types of not strongly regular boundary conditions at the end of the paper.…”

Section: Case Of Sturm-type Boundary Conditionsmentioning

confidence: 99%

On one problem for restoring the density of sources of the fractional heat conductivity process with respect to initial and final temperatures

Erdogan¹,

Florida²,

Kusmangazinova³

et al. 2018

Bul. Kar. Univ. - Math.

View full text Add to dashboard Cite

On one problem for restoring the density of sources of the fractional heat conductivity process with respect to initial and final temperatures In this paper we consider inverse problems for a fractional heat equation, where the fractional time derivative is taken into account in Riemann-Liouville sense. For the solution of this equation, we have to find an unknown right-hand side depending only on a spatial variable. The problem modeling the process of determining the temperature and density of sources in the process of fractional heat conductivity with respect to given initial and final temperatures is considered. Problems with general boundary conditions with respect to the spatial variable that are not strongly regular are investigated. The existence and uniqueness of classical solution to the problem are proved. The problem is considered independent from a corresponding spectral problem for an operator of multiple differentiation with not strongly regular boundary conditions has the basis property of root functions.

show abstract

Identification and regularization for unknown source for a time-fractional diffusion equation

Cited by 22 publications

References 11 publications

On an inverse problem of reconstructing a subdiffusion process from nonlocal data

On an inverse problem of reconstructing a subdiffusion process from nonlocal data

Identification of an inverse source problem for time‐fractional diffusion equation with random noise

On one problem for restoring the density of sources of the fractional heat conductivity process with respect to initial and final temperatures

Contact Info

Product

Resources

About