Identification of the diffusion coefficient in a time fractional diffusion equation

Amir Hossein Salehi Shayegan; Ali Zakeri; Soheila Bodaghi; M. Heshmati

doi:10.1515/jiip-2018-0109

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Identification of the diffusion coefficient in a time fractional diffusion equation

Amir Hossein Salehi Shayegan , Ali Zakeri , Soheila Bodaghi und M. Heshmati

Veröffentlicht/Copyright: 10. März 2020

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Aus der Zeitschrift Journal of Inverse and Ill-posed Problems Band 28 Heft 2

Abstract

In this paper, we study the existence and uniqueness of a quasi solution to a time fractional diffusion equation related to DtαC⁢u-∇⋅(k⁢(x)⁢∇⁡u)=f, where the function k=k⁢(x) is unknown. We consider a methodology, involving minimization of a least squares cost functional, to identify the unknown function k. At the first step of the methodology, we give a stability result corresponding to connectivity of k and u which leads to the continuity of the cost functional. We next construct an appropriate class of admissible functions and show that a solution of the minimization problem exists for the continuous cost functional. At the end, convexity of the introduced cost functional and subsequently the uniqueness theorem of the quasi solution are given.

Keywords: Inverse coefficient problem; quasi solution; time fractional diffusion equation

MSC 2010: 35R30

A Appendix: Existence and uniqueness of the weak solution of the adjoint problem

In this section, the existence and uniqueness of the weak solution of adjoint problem (2.5) are proved by using the Lax–Milgram lemma. To do so, we recall that the weak formulation of (2.5) is to find ϕ∈Bα2⁢(QT) such that

(A.1)A⁢(ϕ,v)=F⁢(v),v∈Bα2⁢(QT),

where the bilinear forms A⁢(⋅,⋅) and F⁢(⋅) are defined by

A⁢(ϕ,v)=(DαtC⁢ϕ,v)L2⁢(QT)+(k⁢∇⁡ϕ,∇⁡v)L2⁢(QT),

F(v)=(q(⋅)ϱ(⋅-T),v)L2⁢(QT).

It is easy to prove the continuities of the bilinear form A⁢(⋅,⋅) and the right-hand functional F⁢(⋅). To be exact, for the bilinear form, we have

A⁢(ϕ,v)=(DαtC⁢ϕ,v)L2⁢(QT)+(k⁢∇⁡ϕ,∇⁡v)L2⁢(QT)=(DtR⁢ϕ,v)L2⁢(QT)-(ϕ⁢(x,T)Γ⁢(1-α)⁢(T-t)α,v)L2⁢(QT)+(k⁢∇⁡ϕ,∇⁡v)L2⁢(QT)=(Dα2tR⁢ϕ,Dtα2R⁢v)L2⁢(QT)+(k⁢∇⁡ϕ,∇⁡v)L2⁢(QT).

In addition, we obtain

|A⁢(ϕ,v)|≤|(Dα2tR⁢ϕ,Dtα2R⁢v)L2⁢(QT)|+M⁢|(∇⁡ϕ,∇⁡v)L2⁢(QT)|≤∥Dα2tR⁢ϕ∥L2⁢(QT)⁢∥Dtα2R⁢v∥L2⁢(QT)+M⁢∥∇⁡ϕ∥L2⁢(QT)⁢∥∇⁡v∥L2⁢(QT)=∥ϕ∥Hα2⁢((0,T),L2⁢(Ω))⁢∥v∥Hα2⁢((0,T),L2⁢(Ω))+M⁢∥ϕ∥L2⁢((0,T),H01⁢(Ω))⁢∥v∥L2⁢((0,T),H01⁢(Ω))≤C⁢∥ϕ∥Bα2⁢(QT)⁢∥v∥Bα2⁢(QT).

For the right-hand functional, we get

|F⁢(v)|=|(q(⋅)ϱ(⋅-T),v)L2⁢(QT)|=∫Ω∫0Tq(x)ϱ(t-T)v(x,t)dtdx=∫Ωq(x)v(x,T)dx≤∥q∥L2⁢(Ω)∥v(⋅,T)∥L2⁢(Ω)≤∥q∥L2⁢(Ω)max0≤t≤T∥v(⋅,t)∥L2⁢(Ω)=∥q∥L2⁢(Ω)⁢∥v∥L∞⁢((0,T),L2⁢(Ω))≤C⁢∥q∥L2⁢(Ω)⁢∥v∥L2⁢(QT).

We next prove the coercivity of the bilinear operator A⁢(⋅,⋅) on Bα2⁢(QT). We have

A⁢(v,v)=(DαtC⁢v,v)L2⁢(QT)+(k2⁢∇⁡v,∇⁡v)L2⁢(QT)≥(Dα2tR⁢v,Dtα2R⁢v)L2⁢(QT)+m⁢(∇⁡v,∇⁡v)L2⁢(QT)=cos⁡(π⁢α2)⁢∥Dtα2R⁢v∥L2⁢(QT)2+∥∇⁡v∥L2⁢(QT)2≥C⁢∥v∥Bα2⁢(QT)2,

where we applied the Poincaré inequalities in the last inequality. By using the well-known Lax–Milgram lemma, there exists a unique solution ϕ∈Bα2⁢(QT) such that (A.1) holds.

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Received: 2018-11-13

Revised: 2020-01-14

Accepted: 2020-02-01

Published Online: 2020-03-10

Published in Print: 2020-04-01

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Artikel in diesem Heft

https://doi.org/10.1515/jiip-2018-0109

Schlagwörter für diesen Artikel

Inverse coefficient problem; quasi solution; time fractional diffusion equation