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 Numerical Solution of Nonstationary Problems for a Space-Fractional Diffusion Equation

2016 ◽  
Vol 19 (1) ◽  
Author(s):  
Petr N. Vabishchevich
Keyword(s):  
Diffusion Equation ◽  
Elliptic Operator ◽  
Fractional Power ◽  
Finite Element Approximation ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Evolutionary Equation ◽  
Element Approximation ◽  
First Order ◽  
Space Fractional Diffusion Equation

AbstractAn unsteady problem is considered for a space-fractional diffusion equation in abounded domain. A first-order evolutionary equation containing a fractional power of an elliptic operator of second order is studied for general boundary conditions of Robin type. Finite element approximation in space is employed. To construct approximation in time, regularized two-level schemes are used. The numerical implementation is based on solving the equation with the fractional power of the elliptic operator using an auxiliary Cauchy problem for a pseudo-parabolic equation. The results of numerical experiments are presented for a model two-dimensional problem.

scholarly journals Inverse problem for fractional diffusion equation

2011 ◽  
Vol 14 (1) ◽  
Author(s):  
Vu Tuan
Keyword(s):  
Inverse Problem ◽  
Diffusion Coefficient ◽  
Lower Bound ◽  
Diffusion Equation ◽  
Spectral Data ◽  
A Priori ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
One Dimensional ◽  
Initial Distributions

AbstractWe prove that by taking suitable initial distributions only finitely many measurements on the boundary are required to recover uniquely the diffusion coefficient of a one dimensional fractional diffusion equation. If a lower bound on the diffusion coefficient is known a priori then even only two measurements are sufficient. The technique is based on possibility of extracting the full boundary spectral data from special lateral measurements.

Stability estimate for an inverse problem of the convection-diffusion equation

2020 ◽  
Vol 28 (1) ◽  
pp. 71-92
Author(s):  
Mourad Bellassoued ◽  
Imen Rassas
Keyword(s):  
Inverse Problem ◽  
Diffusion Equation ◽  
Scalar Potential ◽  
Stability Estimate ◽  
Inverse Boundary ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
First Order ◽  
Order Coefficient ◽  
Dirichlet To Neumann Map

AbstractWe consider the inverse boundary value problem for the dynamical steady-state convection-diffusion equation. We prove that the first-order coefficient and the scalar potential are uniquely determined by the Dirichlet-to-Neumann map. More precisely, we show in dimension {n\geq 3} a log-type stability estimate for the inverse problem under consideration. The method is based on reducing our problem to an auxiliary inverse problem and the construction of complex geometrical optics solutions of this problem.

scholarly journals Regularization of a sideways problem for a time-fractional diffusion equation with nonlinear source

2020 ◽  
Vol 28 (2) ◽  
pp. 211-235
Author(s):  
Tran Bao Ngoc ◽  
Nguyen Huy Tuan ◽  
Mokhtar Kirane
Keyword(s):  
Inverse Problem ◽  
Diffusion Equation ◽  
Regularization Method ◽  
A Priori ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Nonlinear Source ◽  
Numerical Example ◽  
Ill Posed ◽  
Time Fractional Diffusion Equation

AbstractIn this paper, we consider an inverse problem for a time-fractional diffusion equation with a nonlinear source. We prove that the considered problem is ill-posed, i.e., the solution does not depend continuously on the data. The problem is ill-posed in the sense of Hadamard. Under some weak a priori assumptions on the sought solution, we propose a new regularization method for stabilizing the ill-posed problem. We also provide a numerical example to illustrate our results.

scholarly journals An inverse source problem for a two dimensional time fractional diffusion equation with involution

2021 ◽  
Author(s):  
Batirkhan Turmetov ◽  
B. J. Kadirkulov
Keyword(s):  
Inverse Problem ◽  
Parabolic Equation ◽  
Diffusion Equation ◽  
Fourier Method ◽  
Dirichlet Condition ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Inverse Source Problem ◽  
Two Dimensional ◽  
Time Fractional Diffusion Equation

In this paper, we consider a two-dimensional generalization of the parabolic equation. Using the Fourier method, we study the solvability of the inverse problem with the Dirichlet condition and periodic conditions.

Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation

2009 ◽  
Vol 25 (11) ◽  
pp. 115002 ◽  
Author(s):  
Jin Cheng ◽  
Junichi Nakagawa ◽  
Masahiro Yamamoto ◽  
Tomohiro Yamazaki
Keyword(s):  
Inverse Problem ◽  
Diffusion Equation ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
One Dimensional
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