Distinguishability of a source function in a time fractional inhomogeneous parabolic equation with Robin boundary condition

Authors : Ebru OZBİLGE, Ali DEMİR

Pages : 1503-1511

View : 16 | Download : 5

Publication Date : 2018-12-12

Article Type : Research Paper

Abstract :This article deals with the mathematical analysis of the inverse problem of identifying the distinguishability of input-output mappings in the linear time fractional inhomogeneous parabolic equation $D_{t}^{\alpha }uinsert ignore into journalissuearticles values(x,t);=insert ignore into journalissuearticles values(kinsert ignore into journalissuearticles values(x);u_{x});_{x}+Finsert ignore into journalissuearticles values(x,t); \quad 0<\alpha \leq 1$, with Robin boundary conditions $uinsert ignore into journalissuearticles values(0,t);=\psi _{0}insert ignore into journalissuearticles values(t);$, $u_{x}insert ignore into journalissuearticles values(1,t);=\gammainsert ignore into journalissuearticles values(uinsert ignore into journalissuearticles values(1,t);-\psi _{1}insert ignore into journalissuearticles values(t););$. By defining the input-output mappings $\Phi [\cdot ]:\mathcal{K}\rightarrow C^1[0,T]$ and $\Psi [\cdot ]:\mathcal{K}\rightarrow C[0,T]$ the inverse problem is reduced to the problem of their invertibility. Hence, the main purpose of this study is to investigate the distinguishability of the input-output mappings $\Phi[\cdot ]$ and $\Psi [\cdot ]$. Moreover, the measured output data  $finsert ignore into journalissuearticles values(t);$ and $hinsert ignore into journalissuearticles values(t);$ can be determined analytically by a series representation, which implies that the input-output  mappings $\Phi [\cdot ]:\mathcal{K}\rightarrow C^1[0,T]$ and $\Psi [\cdot]:\mathcal{K}\rightarrow C[0,T]$ can be described explicitly.
Keywords : Inverse problem, time fractional parabolic equation, distinguishability