Identification of the unknown boundary condition in a linear parabolic equation

R E S E A R C H

Open Access

Identification of the unknown boundary

condition in a linear parabolic equation

Ali Demir

_{and Ebru Ozbilge}

*_{Correspondence:}

[email protected]

2_{Department of Mathematics,}

Faculty of Science and Literature, Izmir University of Economics, Sakarya Caddesi, No.156, Balcova, Izmir, 35330, Turkey

Full list of author information is available at the end of the article

Abstract

In this article, a semigroup approach is presented for the mathematical analysis of the inverse problems of identifying the unknown boundary conditionu(1,t) =f(t) in a linear parabolic equationut(x,t) = (k(u(x,t))ux(x,t))xwith Dirichlet boundary conditions

u(0,t) =

ψ

ψ

ux(x0,t) =

ψ

1 Introduction

Consider the following initial boundary value problem for the linear diffusion equation:

ut(x,t) = (k(x)ux(x,t))x, (x,t)∈T,

u(x, ) =g(x),  <x< ,

u(,t) =ψ, u(,t) =f(t),  <t<T,

()

whereT={(x,t)∈R:  <x< ,  <t≤T}. The left boundary valueψis assumed to be constant. The functionsc>k(x)≥c>  andg(x) satisfy the following conditions:

(C) k(x)∈H,[, ];

(C) g(x)∈H,_{[, ],g() =ψ}

,g() =f().

The initial boundary value problem () has the unique solutionu(x,t) satisfyingu(x,t)∈

H,_{[, ]∩H},_{[, ] [–] under these conditions.}

In physics, many applications of this problem can be found. The simple model of flame propagation and the spread of a biological populations, whereu=u(x,t),k(x) denote the temperature and density respectively, are given by the equation in problem (). Especially

k=k(x) represents the density-dependent coefficient in the problems of the spread of biological populations [–].

Let us consider the inverse problems[] of determining boundary u(,t) at x=  in problem () from Dirichlet type of measured output data at the boundariesx=x:

u(x,t) =ψ, t∈(,T]; ()

and from Neumann type of measured output data at the boundariesx=x:

ux(x,t) =ψ, t∈(,T]. ()

Here the solution of the parabolic problem () is denoted byu=u(x,t). In this context, the parabolic problem () will be referred to as adirect (forward) problemwith theinputs g(x),k(x) andf(t). Notice thatu(x, ) =ψandux(x, ) =ψ. Therefore it is assumed that the functionsu(x,t) =ψandux(x,t) =ψrespectively satisfy the consistency conditions

ψ=g(x) andψ=g(x).

The purpose of this paper is to determine the boundary functionu(,t) atx=  via the semigroup approach which is studied by [–].

The paper is organized as follows. In Section , an analysis of the semigroup approach is given for the inverse problem with the single measured output datau(x,t) =ψgiven at x=x. A similar analysis is applied to the inverse problem with the single measured output dataux(x,t) =ψgiven at the pointx=xin Section . Some concluding remarks are given in Section .

2 Analysis of the inverse problem of the boundary condition by Dirichlet type of over measured datau(x0,t) =

ψ

Consider now the inverse problem with one measured output datau(x,t) =ψatx=x. In order to formulate the solution of the parabolic problem () in terms of semigroup, let us first arrange the parabolic equation as follows:

ut(x,t) –

k()ux(x,t)

x=

k(x) –k()ux(x,t)

x, (x,t)∈T.

Then the initial boundary value problem () can be rewritten in the following form:

ut(x,t) –k()uxx(x,t) =

k(x) –k()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 conditionu(,t) =f(t), we need to deter-mine the solution of the following parabolic problem:

ut(x,t) –k()uxx(x,t) =

k(x) –k()ux(x,t)

x, (x,t)∈T,

u(x, ) =g(x),  <x< ,

u(,t) =ψ, u(x,t) =ψ,  <t<T.

()

To formulate the solution of the above problem in terms of semigroup, we need to define a new function

v(x,t) =u(x,t) +(ψ–ψ)x

x

–ψ, x∈[, ], ()

which satisfies the following parabolic problem:

vt(x,t) +A

v(x,t)=k(x) –k()vx(x,t) –

(ψ–ψ)

x

x

, (x,t)∈T,

v(x, ) =g(x) +(ψ–ψ)x

x

–ψ,  <x< ,

v(,t) = , v(x,t) = ,  <t<T.

Here,A[·] := –k()d_dx[·]is a second-order differential operator and its domain isDA={v∈

H,_(,x

)∩H,[,x] :v() =v(x) = }, whereH,(,x) =C(,x) andH,[,x] =

C

[,x] are Sobolev spaces. Obviously, by completiong(x)∈DAsince the initial value

function g(x) belongs to C_{[, ]. HenceD}

Ais dense inH,[, ], which is a necessary 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 a semigroup of linear operators by T(t) generated by the operator –A

[, ]. We can easily find the eigenvalues and eigenfunctions of the differential opera-torA. Moreover, the semigroupT(t) can be easily constructed by using the eigenvalues and eigenfunctions of the infinitesimal generatorA. Hence we first consider the following eigenvalue problem:

Aφ(x) =λφ(x),

φ() = ; φ(x) = .

()

It can easily be determined that the eigenvalues areλn=k()n _π x

 for alln= , , . . . and

hence the corresponding eigenfunctions areφn(x) =sin(n_xπx). This allows us to represent

the semigroupT(t) in the following way:

T(t)U(x,s) =

∞

n=

φn(x),U(x,s)

e–λnt_φ

n(x), ()

whereφn(x),U(x,s) =



φn(x)U(x,s)dx. Under this representation, the null space of the semigroupT(t) of the linear operators can be defined as follows:

N(T) =U(x,s) :φn(x),U(x,s) = , for alln= , , , , . . .

.

From the construction of the semigroupT(t), it can be concluded that the null space of it consists of only zero function,i.e.,N(T) ={}. This result implies the uniqueness of the unknown boundary conditionu(,t).

The unique solution of the initial-boundary value problem () in terms of semigroup

T(t) can be represented in the following form:

v(x,t) =T(t)v(x, ) +

t



T(t–s)k(x) –k()vx(x,s) –

(ψ–ψ)

x

x

ds.

Now, by using the identity () and taking the initial valueu(x, ) =g(x) into account, the integral equation for the solutionu(x,t) of the parabolic problem () in terms of semigroup can be written in the following form:

u(x,t) =ψ–

(ψ–ψ)x

x

+T(t)

g(x) +(ψ–ψ)x

x

–ψ

+

t



T(t–s)k(x) –k()ux(x,s)

In order to arrange the above integral equation, let us define the following:

ζ(x) =

g(x) +(ψ–ψ)x

x

–ψ

,

ξ(x,t) =k(x) –k()ux(x,t)

x.

Then we can rewrite the integral equation in terms ofζ(x) andξ(x,s) in the following form:

u(x,t) =ψ–

(ψ–ψ)x

x

+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 () on T ={(x,t)∈R :  <x<x,  <t≤T}. It is obvious from the eigenfunctions

φn(x), the domain of eigenfunctions can be extended to a closed interval [, ].

More-over, they are continuous on [, ]. Under this extension, the uniqueness of the initial-boundary value problems () and () imply that the integral representation () becomes the integral representation of a solution of the initial-boundary value problem () on

T={(x,t)∈R:  <x< ,  <t≤T}.

At this stage, it is obvious that the solution of the inverse problem can easily be obtained by substitutingx=  into the integral representation () of the solutionu(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 thatf(t) can be determined analytically. The right-hand side of identity () definesthe semigroup representation of the unknown boundary condition u(,t)at x= .

Substitutingt=  into equation () yields

u(, ) =f() =g(), ()

which is the condition we have for the initial-boundary value problems (). Similarly, plug-gingt=  into equation () produces the following:

u(x, ) =g(x), ()

which is the initial condition we have.

3 Analysis of the inverse problem of the boundary condition by Neumann type of over measured dataux(x0,t) =

ψ

Consider now the inverse problem with one measured output dataux(x,t) =ψatx=x. In order to formulate the solution of the parabolic problem () in terms of semigroup, as before, we arrange the parabolic equation as follows:

ut(x,t) –

k()ux(x,t)

x=

k(x) –k()ux(x,t)

Then the initial boundary value problem () can be rewritten in the following form:

ut(x,t) –k()uxx(x,t) =

k(x) –k()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 conditionu(,t) =f(t), we need to deter-mine the solution of the following parabolic problem:

ut(x,t) –k()uxx(x,t) =

k(x) –k()ux(x,t)

x, (x,t)∈T,

u(x, ) =g(x),  <x< ,

u(,t) =ψ, ux(x,t) =ψ,  <t<T.

()

To formulate the solution of the above problem in terms of semigroup, we need to define a new function

v(x,t) =u(x,t) –ψx–ψ, x∈[, ], ()

which satisfies the following parabolic problem:

vt(x,t) +B

v(x,t)=k(x) –k()vx(x,t) +ψ

x, (x,t)∈T,

v(x, ) =g(x) –ψx–ψ,  <x< ,

v(,t) = , vx(x,t) = ,  <t<T.

()

HereB[·] := –k()d_dx[·] is a second-order differential operator, its domain isDB={v∈

H,(,x)∩H,[,x] :v() =vx(x) = }. It is clear from the definition ofDBthatDB⊂

H,_[,x ].

Let us denote the semigroup of linear operators byS(t) generated by the operator –B

[, ]. We can easily find the eigenvalues and eigenfunctions of the differential opera-torB. Moreover, the semigroupS(t) can be easily constructed by using the eigenvalues and eigenfunctions of the infinitesimal generatorB. Hence we first consider the following eigenvalue problem:

Bφ(x) =λφ(x),

φ() = ; φx(x) = .

()

The eigenvalues are determined asλn=k()(n+) _π

x 

for alln= , , . . . . Therefore the

cor-responding eigenfunctions areφn(x) =sin((n_+)_xπx). Under these eigenfunctions and

eigen-values, the semigroupS(t) can be represented in the following way:

S(t)U(x,s) =

∞

n=

φn(x),U(x,s)

e–λnt_φ

whereφn(x),U(x,s) =



φn(x)U(x,s)dx. Under this representation, the null space of the semigroupS(t) of the linear operators can be defined as follows:

N(S) =U(x,s) : φn(x),U(x,s)

= , for alln= , , , , . . ..

As in the previous section, the construction of the semigroupS(t) implies that the null space of it consists of only zero function,i.e.,N(S) ={}. The uniqueness of the unknown boundary conditionu(,t) follows from this result.

The unique solution of the initial-boundary value problem () in terms of semigroup

S(t) can be represented in the following form:

v(x,t) =S(t)v(x, ) +

t



S(t–s)k(x) –k()vx(x,s) +ψ

xds.

Now, by using the identity () and taking the initial valueu(x, ) =g(x) into account, the integral equation for the solutionu(x,t) of the parabolic problem () in terms of 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(x) –k()ux(x,s)

xds. ()

In order to arrange the above integral equation, let us define the following:

ζ(x) =g(x) –ψ–ψx

,

ξ(x,t) =k(x) –k()ux(x,t)

x.

Then we can rewrite the integral equation in terms ofζ(x) andξ(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 () on T ={(x,t)∈R:  <x<x,  <t≤T}. It is obvious from the eigenfunctions

φn(x), the domain of eigenfunctions can be extended to a closed interval [, ].

More-over, they are continuous on [, ]. Under this extension, the uniqueness of the initial-boundary value problems () and () imply that integral representation () becomes the integral representation of a solution of the initial-boundary value problem () on

T={(x,t)∈R:  <x< ,  <t≤T}.

Substitutingx=  into the integral representation () of the solutionu(x,t) yields

u(,t) =f(t) =ψ+ψ+

S(t)ζ(·)(,t) +

t



S(t–s)ξ(·,s)(,t,s)ds, ()

which implies thatf(t) can be determined analytically.

4 Conclusion

The purpose of this study is to identify the unknown boundary conditionu(,t) atx=  via the semigroup approach by using the over measured datau(x,t) =ψandux(x,t) =ψ. The crucial point here is the unique extensions of the solutions on{(x,t)∈R:  <x<

x,  <t≤T} to{(x,t)∈R:  <x< ,  <t≤T} which are implied by the uniqueness of the solutions. This key leads to the integral representation of the unknown boundary conditionu(,t) atx=  obtained analytically.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors contributed equally to the manuscript and read and approved the final manuscript.

Author details

1_{Department of Mathematics, Kocaeli University, Umuttepe, Izmit, Kocaeli 41380, Turkey.}2_{Department of Mathematics,}

Faculty of Science and Literature, Izmir University of Economics, Sakarya Caddesi, No.156, Balcova, Izmir, 35330, Turkey.

Acknowledgements

Dedicated to Professor Hari M Srivastava.

The research was supported by parts by the Scientific and Technical Research Council (TUBITAK) of Turkey and Izmir University of Economics.

Received: 27 December 2012 Accepted: 18 February 2013 Published: 7 March 2013 References

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doi:10.1186/1029-242X-2013-96