On a Riesz–Feller space fractional backward diffusion problem with a nonlinear source

Tuan, Nguyen Huy; Hai, Dinh Nguyen Duy; Long, Le Dinh; Nguyen, Van Thinh; Kirane, Mokhtar

doi:10.1016/j.cam.2016.01.003

Cited by 20 publications

(10 citation statements)

References 20 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%

See 1 more Smart Citation

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%

“…Thus, in what follows, we will consider the inverse problems (1) to (3) and (25). Similarly, as before, the formal solution of this problem can be constructed in the form of series…”

Section: Construction Of a Formal Solution Of The Problemmentioning

confidence: 99%

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

“…In previous study, Li developed the Crank‐Nicolson scheme with the finite volume method for the Riesz space distributed‐order advection‐diffusion equation. Besides, Tuan proposed a modified regularization solution for a Riesz‐Feller space fractional backward diffusion problem with a nonlinear source. Jiang used analytical solutions for the multiterm time‐space Caputo‐Riesz fractional advection‐diffusion equations on a finite domain.…”

Section: Introductionmentioning

confidence: 99%

A space‐time spectral collocation method for the 2‐dimensional nonlinear Riesz space fractional diffusion equations

Jiang

2018

Math Methods in App Sciences

View full text Add to dashboard Cite

The space-time spectral collocation method was initially presented for the 1-dimensional sine-Gordon equation. In this article, we introduce a space-time spectral collocation method for solving the 2-dimensional nonlinear Riesz space fractional diffusion equations. The method is based on a Legendre-Gauss-Lobatto spectral collocation method for discretizing spatial and the spectral collocation method for the time nonlinear first-order system of ordinary differential equation. Optimal priori error estimates in L 2 norms for the semidiscrete formulation and the uniqueness of the approximate solution are derived. The method has spectral accuracy in both space and time, and the numerical results confirm the statement. KEYWORDSerror estimates, nonlinear initial-value problems, Riesz fractional derivative, space-time spectral method, spectral collocation method 6130

show abstract

“…In [21], a new modified regularization method can be used to solve the backward problem for the nonlinear space-fractional diffusion equation. In [22], two new modified regularization methods are applied to solve the backward problem for a nonlinear Riesz-Feller space fractional diffusion equation. However, in [21,22], the fractional diffusion equation is a space-fractional diffusion equation.…”

Section: Introductionmentioning

confidence: 99%

Fourier Truncation Regularization Method for a Time-Fractional Backward Diffusion Problem with a Nonlinear Source

Yang¹,

Fan²,

Li³

et al. 2019

Mathematics

View full text Add to dashboard Cite

In present paper, we deal with a backward diffusion problem for a time-fractional diffusion problem with a nonlinear source in a strip domain. We all know this nonlinear problem is severely ill-posed, i.e., the solution does not depend continuously on the measurable data. Therefore, we use the Fourier truncation regularization method to solve this problem. Under an a priori hypothesis and an a priori regularization parameter selection rule, we obtain the convergence error estimates between the regular solution and the exact solution at 0 ≤ x < 1 .

show abstract

On a Riesz–Feller space fractional backward diffusion problem with a nonlinear source

Cited by 20 publications

References 20 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

A space‐time spectral collocation method for the 2‐dimensional nonlinear Riesz space fractional diffusion equations

Fourier Truncation Regularization Method for a Time-Fractional Backward Diffusion Problem with a Nonlinear Source

Contact Info

Product

Resources

About