scholarly journals On a time fractional diffusion with nonlocal in time conditions

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Nguyen Hoang Tuan ◽  
Nguyen Anh Triet ◽  
Nguyen Hoang Luc ◽  
Nguyen Duc Phuong
Keyword(s):  
Fourier Series ◽  
Diffusion Equation ◽  
Convergence Rate ◽  
Exact Solutions ◽  
Mild Solution ◽  
Integral Condition ◽  
Fractional Diffusion ◽  
Well Posedness ◽  
Nonlocal Integral Condition ◽  
Regularized Solution

AbstractIn 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.

scholarly journals An inverse problem for a fractional diffusion equation with fractional power type nonlinearities

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Li Li
Keyword(s):  
Inverse Problem ◽  
Diffusion Equation ◽  
Approximation Property ◽  
Fractional Power ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Well Posedness ◽  
Power Type ◽  
First Order ◽  
Dirichlet To Neumann Map

<p style='text-indent:20px;'>We study the well-posedness of a semi-linear fractional diffusion equation and formulate an associated inverse problem. We determine fractional power type nonlinearities from the exterior partial measurements of the Dirichlet-to-Neumann map. Our arguments are based on a first order linearization as well as the parabolic Runge approximation property.</p>

scholarly journals Determination of source term for fractional heat equation on the sphere

2020 ◽  
Vol 24 (Suppl. 1) ◽  
pp. 361-370
Author(s):  
Nguyen Phuong ◽  
Tran Binh ◽  
Nguyen Luc ◽  
Nguyen Can
Keyword(s):  
Diffusion Equation ◽  
Heat Equation ◽  
Exact Solutions ◽  
Source Term ◽  
Fractional Diffusion ◽  
Inverse Source Problem ◽  
Fractional Heat Equation ◽  
Ill Posed ◽  
Time Fractional Diffusion Equation

In this work, we study a truncation method to solve a time fractional diffusion equation on the sphere of an inverse source problem which is ill-posed in the sense of Hadamard. Through some priori assumption, we present the error estimates between the regularized and exact solutions.

scholarly journals Identifying the space source term problem for time-space-fractional diffusion equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Erdal Karapinar ◽  
Devendra Kumar ◽  
Rathinasamy Sakthivel ◽  
Nguyen Hoang Luc ◽  
N. H. Can
Keyword(s):  
Diffusion Equation ◽  
Source Term ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Choice Rule ◽  
Parameter Choice ◽  
Time Space ◽  
Space Fractional Diffusion Equation ◽  
Ill Posed ◽  
Regularized Solution

Abstract In this paper, we consider an inverse source problem for the time-space-fractional diffusion equation. Here, in the sense of Hadamard, we prove that the problem is severely ill-posed. By applying the quasi-reversibility regularization method, we propose by this method to solve the problem (1.1). After that, we give an error estimate between the sought solution and regularized solution under a prior parameter choice rule and a posterior parameter choice rule, respectively. Finally, we present a numerical example to find that the proposed method works well.

scholarly journals A nonlinear fractional Rayleigh–Stokes equation under nonlocal integral conditions

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Nguyen Hoang Luc ◽  
Le Dinh Long ◽  
Ho Thi Kim Van ◽  
Van Thinh Nguyen
Keyword(s):  
Exact Solution ◽  
Convergence Rate ◽  
Stokes Equation ◽  
Mild Solution ◽  
Existence And Uniqueness ◽  
Integral Conditions ◽  
Initial Value ◽  
Truncation Method ◽  
Nonlocal Integral Conditions ◽  
Regularized Solution

AbstractIn this paper, we study the fractional nonlinear Rayleigh–Stokes equation under nonlocal integral conditions, and the existence and uniqueness of the mild solution to our problem are considered. The ill-posedness of the mild solution to the problem recovering the initial value is also investigated. To tackle the ill-posedness, a regularized solution is constructed by the Fourier truncation method, and the convergence rate to the exact solution of this method is demonstrated.

Inverse source problem for a distributed-order time fractional diffusion equation

2020 ◽  
Vol 28 (1) ◽  
pp. 17-32 ◽  
Author(s):  
Xiaoliang Cheng ◽  
Lele Yuan ◽  
Kewei Liang
Keyword(s):  
Diffusion Equation ◽  
Source Term ◽  
Series Representation ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Conditional Stability ◽  
Inverse Source Problem ◽  
Time Fractional Diffusion Equation ◽  
Distributed Order ◽  
Regularized Solution

AbstractThis paper studies an inverse source problem for a time fractional diffusion equation with the distributed order Caputo derivative. The space-dependent source term is recovered from a noisy final data. The uniqueness, ill-posedness and a conditional stability for this inverse source problem are obtained. The inverse problem is formulated into a minimization functional with Tikhonov regularization method. Further, based on the series representation of the regularized solution, we give convergence rates of the regularized solution under an a-priori and an a-posteriori regularization parameter choice rule. With an adjoint technique for computing the gradient of the regularization functional, the conjugate gradient method is applied to reconstruct the space-dependent source term. Two numerical examples illustrate the effectiveness of the proposed method.

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