Analysis of the inverse problem in a time fractional parabolic equation with mixed boundary conditions

Boundary value problems

Ebru Ozbilge¹  Ali Demir² 

[1] Department of Mathematics, Faculty of Science and Literature, Izmir University of Economics, Balcova, Izmir, Turkey;Department of Mathematics, Kocaeli University, Umuttepe, Turkey

关键词: Inverse Problem;    Fractional Differential Equation;    Parabolic Problem;    Unknown Coefficient;    Mixed Boundary Condition;   

DOI  :  10.1186/1687-2770-2014-134

学科分类：数学（综合）

来源: SpringerOpen

【 摘 要 】

This article deals with the mathematical analysis of the inverse coefficient problem of identifying the unknown coefficientk(x)in the linear time fractional parabolic equationDtαu(x,t)=(k(x)ux)x,0<α≤1, with mixed boundary conditionsu(0,t)=ψ0(t),ux(1,t)=ψ1(t). By defining the input-output mappingsΦ[⋅]:K→C1[0,T]andΨ[⋅]:K→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 mappingsΦ[⋅]andΨ[⋅]. This work shows that the input-output mappingsΦ[⋅]andΨ[⋅]have the distinguishability property. Moreover, the valuek(0)of the unknown diffusion coefficientk(x)atx=0can be determined explicitly by making use of measured output data (boundary observation)k(0)ux(0,t)=f(t), which brings greater restriction on the set of admissible coefficients. It is also shown that the measured output dataf(t)andh(t)can be determined analytically by a series representation, which implies that the input-output mappingsΦ[⋅]:K→C1[0,T]andΨ[⋅]:K→C[0,T]can be described explicitly.

【 授权许可】

CC BY