Inverse source problem in a space fractional diffusion equation from the final overdetermination

Salehi Shayegan, Amir Hossein, et al. "Inverse source problem in a space fractional diffusion equation from the final overdetermination." Applications of Mathematics 64.4 (2019): 469-484. <http://eudml.org/doc/294454>.

@article{SalehiShayegan2019,
abstract = {We consider the problem of determining the unknown source term $ f=f(x,t) $ in a space fractional diffusion equation from the measured data at the final time $ u(x,T) = \psi (x) $. In this way, a methodology involving minimization of the cost functional $ J(f) = \int _0^l ( u(x,t;f )|_\{t = T\} - \psi (x)) ^2 \{\rm d\}x$ is applied and shown that this cost functional is Fréchet differentiable and its derivative can be formulated via the solution of an adjoint problem. In addition, Lipschitz continuity of the gradient is proved. These results help us to prove the monotonicity and convergence of the sequence $\lbrace J^\{\prime \}(f^\{(n)\}) \rbrace $, where $ f^\{(n)\} $ is the $ n$th iteration of a gradient like method. At the end, the convexity of the Fréchet derivative is given.},
author = {Salehi Shayegan, Amir Hossein, Bayat Tajvar, Reza, Ghanbari, Alireza, Safaie, Ali},
journal = {Applications of Mathematics},
keywords = {inverse source problem; space fractional diffusion equation; weak solution theory; adjoint problem; Lipschitz continuity},
language = {eng},
number = {4},
pages = {469-484},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Inverse source problem in a space fractional diffusion equation from the final overdetermination},
url = {http://eudml.org/doc/294454},
volume = {64},
year = {2019},
}

TY - JOUR
AU - Salehi Shayegan, Amir Hossein
AU - Bayat Tajvar, Reza
AU - Ghanbari, Alireza
AU - Safaie, Ali
TI - Inverse source problem in a space fractional diffusion equation from the final overdetermination
JO - Applications of Mathematics
PY - 2019
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 64
IS - 4
SP - 469
EP - 484
AB - We consider the problem of determining the unknown source term $ f=f(x,t) $ in a space fractional diffusion equation from the measured data at the final time $ u(x,T) = \psi (x) $. In this way, a methodology involving minimization of the cost functional $ J(f) = \int _0^l ( u(x,t;f )|_{t = T} - \psi (x)) ^2 {\rm d}x$ is applied and shown that this cost functional is Fréchet differentiable and its derivative can be formulated via the solution of an adjoint problem. In addition, Lipschitz continuity of the gradient is proved. These results help us to prove the monotonicity and convergence of the sequence $\lbrace J^{\prime }(f^{(n)}) \rbrace $, where $ f^{(n)} $ is the $ n$th iteration of a gradient like method. At the end, the convexity of the Fréchet derivative is given.
LA - eng
KW - inverse source problem; space fractional diffusion equation; weak solution theory; adjoint problem; Lipschitz continuity
UR - http://eudml.org/doc/294454
ER -