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

Ebru OZBİLGE; Ali DEMİR

Article Details

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

Authors:

Ebru OZBİLGE | Ali DEMİR

Publisher:

Hacettepe University

Main Topics:

Mathematics

Journal:

Hacettepe Journal of Mathematics and Statistics 2018 Vol.47 Issue 6, pp.1503-1511

Document Type:

Article

Publication Date:

12/2018

Document Language:

English

Document ID (Org No.):

2000451715

Frequency:

Bimonthly

eISSN:

2651-477X

pISSN:

2651-477X

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 Dαtu(x,t)=(k(x)ux)x+F(x,t)0<α≤1Dtαu(x,t)=(k(x)ux)x+F(x,t)0<α≤1D_{t}^{\alpha }u(x,t)=(k(x)u_{x})_{x}+F(x,t) \quad 0<\alpha \leq 1, with Robin boundary conditions u(0,t)=ψ0(t)u(0,t)=ψ0(t)u(0,t)=\psi _{0}(t), ux(1,t)=γ(u(1,t)−ψ1(t))ux(1,t)=γ(u(1,t)−ψ1(t))u_{x}(1,t)=\gamma(u(1,t)-\psi _{1}(t)). By defining the input-output mappings Φ[⋅]:K→C1[0,T]Φ[⋅]:K→C1[0,T]\Phi [\cdot ]:\mathcal{K}\rightarrow C^1[0,T] and Ψ[⋅]:K→C[0,T]Ψ[⋅]:K→C[0,T]\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 Φ[⋅]Φ[⋅]\Phi[\cdot ] and Ψ[⋅]Ψ[⋅]\Psi [\cdot ]. Moreover, the measured output data  f(t)f(t)f(t) and h(t)h(t)h(t) can be determined analytically by a series representation, which implies that the input-output  mappings Φ[⋅]:K→C1[0,T]Φ[⋅]:K→C1[0,T]\Phi [\cdot ]:\mathcal{K}\rightarrow C^1[0,T] and Ψ[⋅]:K→C[0,T]Ψ[⋅]:K→C[0,T]\Psi [\cdot]:\mathcal{K}\rightarrow C[0,T] can be described explicitly.

Article Details

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

Authors: Ebru OZBİLGE | Ali DEMİR

Publisher: Hacettepe University

Main Topics: Mathematics

Journal: Hacettepe Journal of Mathematics and Statistics 2018 Vol.47 Issue 6, pp.1503-1511

Document Type: Article

Publication Date: 12/2018

Document Language: English

Document ID (Org No.): 2000451715

Frequency: Bimonthly

eISSN: 2651-477X

pISSN: 2651-477X

Issues

Authors:

Ebru OZBİLGE | Ali DEMİR

Publisher:

Hacettepe University

Main Topics:

Mathematics

Journal:

Hacettepe Journal of Mathematics and Statistics 2018 Vol.47 Issue 6, pp.1503-1511

Document Type:

Article

Publication Date:

12/2018

Document Language:

English

Document ID (Org No.):

2000451715

Frequency:

Bimonthly

eISSN:

2651-477X

pISSN:

2651-477X