A Ashyralyev

A Demir

A Hasanov

E Ozbilge

Ebru Ozbilge

JR Cannon

M Renardy

MS Pilant

P DuChateau

R Showalter

V Isakov

Boundary Value Problems

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Ozbilge Boundary Value Problems 2013, 2013:2http://www.boundaryvalueproblems.com/content/2013/1/2RESEARCH Open AccessDetermination of the unknown boundarycondition of the inverse parabolic problemsvia semigroup methodEbru Ozbilge**Correspondence:ebru.ozbilge@ieu.edu.trDepartment of Mathematics,Faculty of Science and Literature,Izmir University of Economics,Sakarya Caddesi, No. 156, Izmir,Balcova 35330, TurkeyAbstractIn this article, a semigroup approach is presented for the mathematical analysis ofinverse problems of identifying the unknown boundary condition u(1, t) = f (t) in thequasi-linear parabolic equation ut(x, t) = (k(u(x, t))ux(x, t))x , with Dirichlet boundaryconditions u(0, t) =ψ0, u(1, t) = f (t), by making use of the over measured datau(x0, t) =ψ1 and ux(x0, t) =ψ2 separately. The purpose of this study is to identify theunknown boundary condition u(1, t) at x = 1 by using the over measured datau(x0, t) =ψ1 and ux(x0, t) =ψ2. First, by using over measured data as a boundarycondition, we deﬁne the problem onT0 = {(x, t) ∈ R2 : 0 < x < x0, 0 < t ≤ T}, then theintegral representation of this problem via a semigroup of linear operators isobtained. Finally, extending the solution uniquely to the closed interval [0, 1], wereach the result. The main point here is the unique extensions of the solutions on[0, x0] to the closed interval [0, 1] which are implied by the uniqueness of thesolutions. This point leads to the integral representation of the unknown boundarycondition u(1, t) at x = 1.1 IntroductionConsider the following initial boundary value problem for quasilinear diﬀusion equation:⎧⎪⎪⎨⎪⎪⎩ut(x, t) = (k(u(x, t))ux(x, t))x, (x, t) ∈T ,u(x, ) = g(x),  < x < ,u(, t) =ψ, u(, t) = f (t),  < t < T ,()where T = {(x, t) ∈ R :  < x < ,  < t ≤ T}. The left boundary value ψ is assumed to beconstant. The functions c > k(u(x, t))≥ c >  and g(x) satisfy the following conditions:(C) |ku(u) – ku(u)| < d|u – u|;(C) g(x) ∈ C[, ], g() =ψ, g() = f ().The initial boundary value problem () has a unique solution u(x, t) satisfying u(x, t) ∈H,[, ]∩H,[, ] [–].In physics, many applications of this problem can be found. The simple model of ﬂamepropagation and the spread of biological populations, where u = u(x, t) denotes the tem-perature and density respectively, are given by the equation in the problem (). Especially© 2013 Ozbilge; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribu-tion License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in anymedium, provided the original work is properly cited.Ozbilge Boundary Value Problems 2013, 2013:2 Page 2 of 7http://www.boundaryvalueproblems.com/content/2013/1/2k = k(u(x, t)) represents the density-dependent coeﬃcient in the problems of the spreadof biological populations [–].We consider the inverse problems [] of determining boundary u(, t) at x =  in theproblem () from Dirichlet type of measured output data at the boundaries x = xu(x, t) =ψ, t ∈ (,T], ()and from Neumann type of measured output data at the boundaries x = xux(x, t) =ψ, t ∈ (,T]. ()Here u = u(x, t) is the solution of the parabolic problem (). In this context, the parabolicproblem () will be referred to as a direct (forward) problem with the inputs g(x), k(u(x, t))and f (t). It is assumed that the functions u(x, t) =ψ and ux(x, t) =ψ respectively satisfythe consistency conditions ψ = g(x) and ψ = g ′(x).The semigroup approach [] for inverse problems for the identiﬁcation of an unknowncoeﬃcient in a quasi-linear parabolic equation was studied by Demir and Ozbilge [].The study in this paper is based on the philosophy similar to that in [–].The paper is organized as follows. In Section , the analysis of the semigroup approachis given for the inverse problem with the single measured output data u(x, t) = ψ givenat x = x. The similar analysis is applied to the inverse problem with the single measuredoutput data ux(x, t) =ψ given at the point x = x in Section . Some concluding remarksare given in Section .2 Analysis of the inverse problem of the boundary condition by Dirichlet typeof over measured data u(x0, t) =ψ1Consider now the inverse problem with one measured output data u(x, t) = ψ at x = x.In order to formulate the solution of the parabolic problem () in terms of a semigroup,let us ﬁrst arrange the parabolic equation as follows:ut(x, t) –(k(u(, ))ux(x, t))x =([k(u) – k(u(, ))]ux(x, t))x, (x, t) ∈T .Then the initial boundary value problem () can be rewritten in the following form:ut(x, t) – k(u(, ))uxx(x, t) =((k(u) – k(u(, )))ux(x, t))x, (x, t) ∈T ,u(x, ) = g(x),  < x < ,u(, t) =ψ, u(, t) = f (t),  < t < T .()In order to determine the unknown boundary condition u(, t) = f (t), we need to deter-mine the solution of the following parabolic problem:ut(x, t) – k(u(, ))uxx(x, t) =((k(u) – k(u(, )))ux(x, t))x, (x, t) ∈T ,u(x, ) = g(x),  < x < x,u(, t) =ψ, u(x, t) =ψ,  < t < T ,()Ozbilge Boundary Value Problems 2013, 2013:2 Page 3 of 7http://www.boundaryvalueproblems.com/content/2013/1/2where T = {(x, t) ∈ R :  < x < x,  < t ≤ T}. To formulate the solution of the aboveproblem in terms of a semigroup, we need to deﬁne a new functionv(x, t) = u(x, t) + (ψ –ψ)xx–ψ, x ∈ [,x], ()which satisﬁes the following parabolic problem:vt(x, t) +A[v(x, t)]=((k(v(x, t) +ψ –(ψ –ψ)xx)– k(u(, ))) (vx(x, t) –(ψ –ψ)x))x, (x, t) ∈T ,v(x, ) = g(x) + (ψ –ψ)xx–ψ,  < x < x,v(, t) = , v(x, t) = ,  < t < T .()Here, A[·] := –k(u(, )) d[·]dx is a second-order diﬀerential operator and its domainis DA = {v(x) ∈ H, (, ) ∩ H, [, ] : v() =  = v()}, where H, (, ) = C(, ) andH, [, ] = C[, ] are Sobolev spaces. Obviously, by completion g(x) ∈DA, since the ini-tial value function g(x) belongs to C[, ]. Hence, DA is dense in H, [, ], which is anecessary condition for being an inﬁnitesimal generator.In the following, despite doing the calculations in the smooth function space, by com-pletion they are valid in the Sobolev space.Let us denote the semigroup of linear operators by T(t) generated by the operator A[, ]. We can easily ﬁnd the eigenvalues and eigenfunctions of the diﬀerential opera-tor A. Moreover, the semigroup T(t) can be easily constructed by using the eigenvaluesand eigenfunctions of the inﬁnitesimal generator A. Hence, we ﬁrst consider the followingeigenvalue problem:Aφ(x) = λφ(x),φ() = ; φ(x) = .()We can easily determine that the eigenvalues are λn = k(u(, )) nπxfor all n = , . . . andthe corresponding eigenfunctions are φn(x) = sin( nπxx ). In this case, the semigroup T(t)can be represented in the following way:T(t)U(x, s) =∞∑n=〈φn(x),U(x, s)〉e–λntφn(x), ()where 〈φn(x),U(x, s)〉 =∫  φn(x)U(x, s)dx. Under this representation, the null space of thesemigroup T(t) of the linear operators can be deﬁned as follows:N(T) ={U(x, s) :〈φn(x),U(x, s)〉= , for all n = , , , , . . .}.From the deﬁnition of the semigroup T(t), we can say that the null space of it consists ofonly zero function, i.e., N(T) = {}. This result is very important for the uniqueness of theunknown boundary condition u(, t).Ozbilge Boundary Value Problems 2013, 2013:2 Page 4 of 7http://www.boundaryvalueproblems.com/content/2013/1/2The unique solution of the initial-boundary value problem () in terms of the semigroupT(t) can be represented in the following form:v(x, t) = T(t)v(x, ) +∫ tT(t – s)((k(v(x, s) +ψ –(ψ –ψ)xx)– k(u(, ))) (vx(x, s) –(ψ –ψ)x))xds.Now, by using the identity () and taking the initial value u(x, ) = g(x) into account, the in-tegral equation for the solution u(x, t) of the parabolic problem () in terms of a semigroupcan be written in the following form:u(x, t) = ψ –(ψ –ψ)xx+ T(t)(g(x) + (ψ –ψ)xx–ψ)+∫ tT(t – s)((k(u(x, s))– k(u(, )))ux(x, s))x ds. ()In order to arrange the above integral equation, let us deﬁne the following:ζ (x) =(g(x) + (ψ –ψ)xx–ψ),ξ (x, t) =((k(u(x, t))– k(u(, )))ux(x, t))x.Thenwe can rewrite the integral equation in terms of ζ (x) and ξ (x, s) in the following form:u(x, t) =ψ –(ψ –ψ)xx+(T(t)ζ (·))(x, t) +∫ t(T(t – s)ξ (·, s))(x, t, s)ds. ()This is the integral representation of a solution of the initial-boundary value problem ()onT = {(x, t) ∈ R :  < x < x,  < t ≤ T}. It is obvious from the eigenfunctions φn(x), thedomain of eigenfunctions can be extended to the closed interval [, ]. Moreover they arecontinuous on [, ]. Under this extension, the uniqueness of the solutions of the initial-boundary value problems () and () imply that the integral representation () becomesthe integral representation of a solution of the initial-boundary value problem () onT ={(x, t) ∈ R :  < x < ,  < t ≤ T}.At this stage, it is obvious that the solution of the inverse problem can easily be obtainedby substituting x =  into the integral representation () of the solution u(x, t),u(, t) = f (t) =ψ –(ψ –ψ)x+(T(t)ζ (·))(, t) +∫ t(T(t – s)ξ (·, s))(, t, s)ds, ()which implies that f (t) can be determined analytically.The right-hand side of the identity () deﬁnes the semigroup representation of the un-known boundary condition u(, t) at x = .3 Analysis of the inverse problem of the boundary condition by Neumann typeof over measured data ux(x0, t) =ψ2Consider now the inverse problem with one measured output data ux(x, t) =ψ at x = x.In order to formulate the solution of the parabolic problem () in terms of a semigroup,Ozbilge Boundary Value Problems 2013, 2013:2 Page 5 of 7http://www.boundaryvalueproblems.com/content/2013/1/2we arrange the parabolic equation as follows:ut(x, t) –(k(u(, ))ux(x, t))x =([k(u) – k(u(, ))]ux(x, t))x, (x, t) ∈T .Then the initial boundary value problem () can be rewritten in the following form:ut(x, t) – k(u(, ))uxx(x, t) =((k(u) – k(u(, )))ux(x, t))x, (x, t) ∈T ,u(x, ) = g(x),  < x < ,u(, t) =ψ, u(, t) = f (t),  < t < T .()In order to determine the unknown boundary condition u(, t) = f (t), we need to deter-mine the solution of the following parabolic problem:ut(x, t) – k(u(, ))uxx(x, t) =((k(u) – k(u(, )))ux(x, t))x, (x, t) ∈T ,u(x, ) = g(x),  < x < x,u(, t) =ψ, ux(x, t) =ψ,  < t < T ,()where T = {(x, t) ∈ R :  < x < x,  < t ≤ T}.To formulate the solution of the above problem in terms of a semigroup, we need todeﬁne a new functionv(x, t) = u(x, t) –ψx –ψ, x ∈ [,x], ()which satisﬁes the following parabolic problem:vt(x, t) + B[v(x, t)]=((k(v(x, t) +ψ +ψx)– k(u(, )))(vx(x, t) +ψ))x,(x, t) ∈T ,v(x, ) = g(x) –ψx –ψ,  < x < x,v(, t) = , vx(x, t) = ,  < t < T .()Here B[·] := –k(u(, )) d[·]dx is a second-order diﬀerential operator, its domain is DB ={v ∈ C(,x)∩C[,x] : v() = vx(x) = }. It is clear from the deﬁnition of DB that DB ⊂C[,x].Let us denote the semigroup of linear operators by S(t) generated by the operator B[, ]. We can easily ﬁnd the eigenvalues and eigenfunctions of the diﬀerential opera-tor B. Moreover, the semigroup S(t) can be easily constructed by using the eigenvaluesand eigenfunctions of the inﬁnitesimal generator B. Hence, we ﬁrst consider the followingeigenvalue problem:Bφ(x) = λφ(x),φ() = ; φx(x) = .()We can easily determine that the eigenvalues are λn = k(u(, )) (n+)πxfor all n = , , . . .and the corresponding eigenfunctions are φn(x) = sin( (n+)πxx ). In this case, the semigroupOzbilge Boundary Value Problems 2013, 2013:2 Page 6 of 7http://www.boundaryvalueproblems.com/content/2013/1/2S(t) can be represented in the following way:S(t)U(x, s) =∞∑n=〈φn(x),U(x, s)〉e–λntφn(x), ()where 〈φn(x),U(x, s)〉 =∫  φn(x)U(x, s)dx. Under this representation, the null space of thesemigroup S(t) of the linear operators can be deﬁned as follows:N(S) ={U(x, s) :〈φn(x),U(x, s)〉= , for all n = , , , , . . .}.From the deﬁnition of the semigroup S(t), we can say that the null space of it consists ofonly zero function, i.e., N(S) = {}. This result is very important for the uniqueness of theunknown boundary condition u(, t).The unique solution of the initial-boundary value problem () in terms of the semi-group S(t) can be represented in the following form:v(x, t) = S(t)v(x, )+∫ tS(t – s)((k(v(x, s) +ψ +ψx)– k(u(, )))(vx(x, s) +ψ))x ds.Now, by using the identity () and taking the initial value u(x, ) = g(x) into account, theintegral equation for the solution u(x, t) of the parabolic problem () in terms of a semi-group can be written in the following form:u(x, t) = ψ +ψx + S(t)(g(x) –ψ –ψx)+∫ tS(t – s)((k(u(x, s))– k(u(, )))ux(x, s))x ds. ()In order to arrange the above integral equation, let us deﬁne the following:ζ (x) =(g(x) –ψ –ψx),ξ (x, t) =((k(u(x, t))– k(u(, )))ux(x, t))x.Thenwe can rewrite the integral equation in terms of ζ (x) and ξ (x, s) in the following form:u(x, t) =ψ +ψx +(S(t)ζ (·))(x, t) +∫ t(S(t – s)ξ (·, s))(x, t, s)ds. ()This is the integral representation of a solution of the initial-boundary value problem ()onT = {(x, t) ∈ R :  < x < x,  < t ≤ T}. It is obvious from the eigenfunctions φn(x), thedomain of eigenfunctions can be extended to the closed interval [, ]. Moreover, they arecontinuous on [, ]. Under this extension, the uniqueness of the solutions of the initial-boundary value problems () and () imply that the integral representation () becomesthe integral representation of a solution of the initial-boundary value problem () onT = {(x, t) ∈ R :  < x < ,  < t ≤ T}.Ozbilge Boundary Value Problems 2013, 2013:2 Page 7 of 7http://www.boundaryvalueproblems.com/content/2013/1/2Substituting x =  into the integral representation () of the solution u(x, t) yieldsu(, t) = f (t) =ψ +ψ +(S(t)ζ (·))(, t) +∫ t(S(t – s)ξ (·, s))(, t, s)ds, ()which implies that f (t) can be determined analytically.The right-hand side of the identity () deﬁnes the semigroup representation of the un-known boundary condition u(, t) at x = . . . .4 ConclusionThe goal of this study is to identify the unknown boundary condition u(, t) at x =  byusing the over measured data u(x, t) = ψ and ux(x, t) = ψ. The key point here is theunique extensions of solutions on [,x] to the closed interval [, ] which are implied bythe uniqueness of the solutions. This key point leads to the integral representation of theunknown boundary condition u(, t) at x =  obtained analytically. . . .Competing interestsThe author declares that they have no competing interests.AcknowledgementsDedicated to my father and mother Yusuf/Sevim Ozbilge.The research was supported by parts by the Scientiﬁc and Technical Research Council (TUBITAK) of Turkey and IzmirUniversity of Economics. . . .Received: 23 November 2012 Accepted: 17 December 2012 Published: 4 January 2013References1. DuChateau, P: Monotonicity and invertibility of coeﬃcient-to-data mappings for parabolic inverse problems. SIAMJ. Math. Anal. 26, 1473-1487 (1995)2. Isakov, V: On uniqueness in inverse problems for quasilinear parabolic equations. Arch. Ration. Mech. Anal. 124, 1-13(1993)3. 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Demir, A, Hasanov, A: Identiﬁcation of the unknown diﬀusion coeﬃcient in a linear parabolic equation by thesemigroup approach. J. Math. Anal. Appl. 340, 5-15 (2008)doi:10.1186/1687-2770-2013-2Cite this article as: Ozbilge: Determination of the unknown boundary condition of the inverse parabolic problemsvia semigroup method. Boundary Value Problems 2013 2013:2.

Determination of the unknown boundary condition of the inverse parabolic problems via semigroup method

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Determination of the unknown boundary condition of the inverse parabolic problems via semigroup method

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