Research | Open | Published:

# Semigroup approach for identification of the unknown diffusion coefficient in a linear parabolic equation with mixed output data

*Boundary Value Problems***volume 2013**, Article number: 43 (2013)

## Abstract

This article presents a semigroup approach for the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient $k(x)$ in the linear parabolic equation ${u}_{t}(x,t)={(k(x){u}_{x}(x,t))}_{x}$ with mixed boundary conditions $k(0){u}_{x}(0,t)={\psi}_{0}$, $u(1,t)={\psi}_{1}$. The aim of this paper is to investigate the distinguishability of the input-output mappings $\mathrm{\Phi}[\cdot ]:\mathcal{K}\to {H}^{1,2}[0,T]$, $\mathrm{\Psi}[\cdot ]:\mathcal{K}\to {H}^{1,2}[0,T]$ via semigroup theory. In this paper, we show that if the null space of the semigroup $T(t)$ consists of only zero function, then the input-output mappings $\mathrm{\Phi}[\cdot ]$ and $\mathrm{\Psi}[\cdot ]$ have the distinguishability property. It is shown that the types of the boundary conditions and the region on which the problem is defined have a significant impact on the distinguishability property of these mappings. Moreover, in the light of *measured output data* (boundary observations) $f(t):=u(0,t)$ or/and $h(t):=k(1){u}_{x}(1,t)$, the values $k(0)$ and $k(1)$ of the unknown diffusion coefficient $k(x)$ at $x=0$ and $x=1$, respectively, can be determined explicitly. In addition to these, the values ${k}^{\prime}(0)$ and ${k}^{\prime}(1)$ of the unknown coefficient $k(x)$ at $x=0$ and $x=1$, respectively, are also determined via the input data. Furthermore, it is shown that *measured output data* $f(t)$ and $h(t)$ can be determined analytically by an integral representation. Hence the input-output mappings $\mathrm{\Phi}[\cdot ]:\mathcal{K}\to {H}^{1,2}[0,T]$, $\mathrm{\Psi}[\cdot ]:\mathcal{K}\to {H}^{1,2}[0,T]$ are given explicitly in terms of the semigroup.

## 1 Introduction

Consider the following initial boundary value problem:

where ${\mathrm{\Omega}}_{T}=\{(x,t)\in {R}^{2}:0<x<1,0<t\le T\}$. The left flux ${\psi}_{0}$ and the right boundary condition ${\psi}_{1}$ are assumed to be constants. The functions ${c}_{1}>k(x)\ge {c}_{0}>0$ and $g(x)$ satisfy the following conditions:

(C1) $k(x)\in {H}^{1,2}[0,1]$;

(C2) $g(x)\in {H}^{2,2}[0,1]$, ${g}^{\prime}(0)={\psi}_{0}$, $g(1)={\psi}_{1}$.

Under these conditions, the initial boundary value problem (1) has the unique solution $u(x,t)\in {H}^{2,2}[0,1]\cap {H}^{1,2}[0,1]$ [1–4].

Consider *the inverse problem* of determining the unknown coefficient $k=k(x)$ [5–9] from the following observations at the boundaries $x=0$ and $x=1$:

Here $u=u(x,t)$ is the solution of the parabolic problem (1). The functions $f(t)$, $h(t)$ are assumed to be *noisy free measured output data*. In this context, the parabolic problem (1) will be referred to as a *direct (forward) problem* with the *inputs* $g(x)$ and $k(x)$. It is assumed that the functions $f(t)$ and $h(t)$ belong to ${H}^{1,2}[0,T]$ and satisfy the consistency conditions $f(0)=g(0)$, ${h}^{\prime}(0)=k(1){g}^{\prime}(1)$.

We denote by $\mathcal{K}:=\{k(x)\in {H}^{1,2}[0,1]:{c}_{1}>k(x)\ge {c}_{0}>0,x\in [0,1]\}\subset {H}^{1,2}[0,1]$, the set of admissible coefficients $k=k(x)$ and introduce the input-output mappings $\mathrm{\Phi}[\cdot ]:\mathcal{K}\to {H}^{1,2}[0,T]$, $\mathrm{\Psi}[\cdot ]:\mathcal{K}\to {H}^{1,2}[0,T]$, where

Then the inverse problem [10] with the measured data $f(t)$ and $h(t)$ can be formulated as the following operator equations:

We denote by $\mathcal{K}:=\{k(x)\in {H}^{1,2}[0,1]:{c}_{1}>k(x)\ge {c}_{0}>0,x\in [0,1]\}\subset {H}^{1,2}[0,1]$, the set of admissible coefficients $k=k(x)$. The monotonicity, continuity and hence invertibility of the input-output mappings $\mathrm{\Phi}[\cdot ]:\mathcal{K}\to {H}^{1,2}[0,T]$ and $\mathrm{\Psi}[\cdot ]:\mathcal{K}\to {H}^{1,2}[0,T]$ are given in [3, 4].

The aim of this paper is to study a distinguishability of the unknown coefficient via the above input-output mappings. We say that the mapping $\mathrm{\Phi}[\cdot ]:\mathcal{K}\to {H}^{1,2}[0,T]$ (or $\mathrm{\Psi}[\cdot ]:\mathcal{K}\to {H}^{1,2}[0,T]$) has the distinguishability property if $\mathrm{\Phi}[{k}_{1}]\ne \mathrm{\Phi}[{k}_{2}]$ ($\mathrm{\Psi}[{k}_{1}]\ne \mathrm{\Psi}[{k}_{2}]$) implies ${k}_{1}(x)\ne {k}_{2}(x)$. This, in particular, means injectivity of the inverse mappings ${\mathrm{\Phi}}^{-1}$ and ${\mathrm{\Psi}}^{-1}$.

The purpose of this paper is to study the distinguishability of the unknown coefficient via the above input-output mappings. The results presented here are the first ones, to the knowledge of authors, from the point of view of semigroup approach [11] to inverse problems. This approach sheds more light on the identifiability of the unknown coefficient [12] and shows how much information can be extracted from the measured output data, in particular in the case of constant flux and boundary data [12–15].

The paper is organized as follows. In Section 2, the analysis of the semigroup approach is given for the inverse problem with the measured data $f(t)$. A similar analysis is applied to the inverse problem with the single measured output data $h(t)$ given at the point $x=1$ in Section 3. The inverse problem with two Neumann measured data $f(t)$ and $h(t)$ is discussed in Section 4. Finally, some concluding remarks are given in Section 5.

## 2 Analysis of the inverse problem with measured output data $f(t)$

Consider now the inverse problem with one measured output data $f(t)$ at $x=0$. In order to formulate the solution of the parabolic problem (1) in terms of a semigroup, let us first arrange the parabolic equation as follows:

Then the initial boundary value problem (1) can be rewritten in the following form:

Here we assume that $k(0)$ was known. Later we will determine the value $k(0)$. In order to formulate the solution of the parabolic problem (5) in terms of a semigroup, we need to define the following function:

which satisfies the following parabolic problem:

Here $A[v(x,t)]:=-k(0){d}^{2}v(x,t)/d{x}^{2}$ is a second-order differential operator, its domain is ${D}_{A}=\{u\in {H}^{2,2}(0,1)\cap {H}^{1,2}[0,1]:{u}_{x}(0)=u(1)=0\}$. Since the initial value function $g(x)$ belongs to ${C}^{2}[0,1]$, it is obvious that $g(x)\in {D}_{A}$.

Denote by $T(t)$ the semigroup of linear operators generated by the operator −*A* [5, 6]. Note that we can easily find the eigenvalues and eigenfunctions of the differential operator *A*. Furthermore, the semigroup $T(t)$ can be easily constructed by using the eigenvalues and eigenfunctions of a differential operator *A*. For this reason, we first consider the following eigenvalue problem:

This problem is called a Sturm-Liouville problem. We can easily determine that the eigenvalues are ${\lambda}_{n}=k(0){(2n-1)}^{2}{\pi}^{2}/4$ for all $n=1,\dots $ and the corresponding eigenfunctions are ${\varphi}_{n}(x)=\sqrt{2}cos((2n-1)x\pi /2)$. In this case, the semigroup $T(t)$ can be represented in the following way:

where $\u3008{\varphi}_{n}(x),U(x,s)\u3009={\int}_{0}^{1}{\varphi}_{n}(x)U(x,s)\phantom{\rule{0.2em}{0ex}}dx$. The null space of the semigroup $T(t)$ of the linear operators can be defined as follows:

From the definition of the semigroup $T(t)$, we can say that the null space of it is an empty set, *i.e.*, $N(T)=\{0\}$. This result is very important for the uniqueness of the unknown coefficient $k(x)$.

The unique solution of the initial value problem (7) in terms of a semigroup $T(t)$ can be represented in the following form:

Hence, by using identity (6), the solution $u(x,t)$ of the parabolic problem (5) in terms of a semigroup can be written in the following form:

In order to arrange the above solution representation, let us define the following:

Then we can rewrite the solution representation in terms of $\zeta (x)$ and $\xi (x,s)$ in the following form:

Substituting $x=0$ into this solution representation yields

Taking into account the overmeasured data $u(0,t)=f(t)$, we get

which implies that $f(t)$ can be determined analytically.

Differentiating both sides of the above identity with respect to *x* and using semigroup properties at $x=0$ yield

Using the boundary condition $k(0){u}_{x}(0,t)={\psi}_{0}$, we can write $k(0)={\psi}_{0}/{u}_{x}(0,t)$ for all $t\ge 0$ which can be rewritten in terms of a semigroup in the following form:

Taking limit as $t\to 0$ in the above identity, we obtain the following explicit formula for the value $k(0)$ of the unknown coefficient $k(x)$:

The right-hand side of identity (11) defines explicitly *the semigroup representation of the input-output mapping* $\mathrm{\Phi}[k]$ on the set of admissible unknown diffusion coefficients $\mathcal{K}$:

Let us differentiate now both sides of identity (8) with respect to *t*:

Using the semigroup property $-{\int}_{0}^{t}AT(s)u(x,s)\phantom{\rule{0.2em}{0ex}}ds=T(t)u(x,t)-T(0)u(x,t)$, we obtain

Taking $x=0$ in the above identity, we get

Since $u(0,t)=f(t)$, we have ${u}_{t}(0,t)={f}^{\prime}(t)$. Taking into account this and substituting $t=0$ yield

Solving this equation for ${k}^{\prime}(0)$ and substituting ${u}_{x}(0,0)={g}^{\prime}(0)/k(0)$, we obtain the following explicit formula for the value ${k}^{\prime}(0)$ of the first derivative ${k}^{\prime}(x)$ of the unknown coefficient at $x=0$:

Under the determined values $k(0)$ and ${k}^{\prime}(0)$, the set of admissible coefficients can be defined as follows:

The following lemma implies the relationship between the diffusion coefficients ${k}_{1}(x),{k}_{2}(x)\in \mathcal{K}$ at $x=0$ and the corresponding outputs ${f}_{j}(t):=u(0,t;{k}_{j})$, $j=1,2$.

**Lemma 2.1** *Let* ${u}_{1}(x,t)=u(x,t;{k}_{1})$ *and* ${u}_{2}(x,t)=u(x,t;{k}_{2})$ *be solutions of the direct problem* (5) *corresponding to the admissible coefficients* ${k}_{1}(x),{k}_{2}(x)\in \mathcal{K}$. *Suppose that* ${f}_{j}(t)=u(0,t;{k}_{j})$, $j=1,2$, *are the corresponding outputs and denote by* $\mathrm{\Delta}f(t)={f}_{1}(t)-{f}_{2}(t)$, $\mathrm{\Delta}\xi (x,t)={\xi}^{1}(x,t)-{\xi}^{2}(x,t)$. *If the condition*

*holds*, *then the outputs* ${f}_{j}(t)$, $j=1,2$, *satisfy the following integral identity*:

*for each* $\tau \in (0,T]$.

*Proof* The solutions of the direct problem (5) corresponding to the admissible coefficients ${k}_{1}(x),{k}_{2}(x)\in \mathcal{K}$ can be written at $x=0$ as follows:

respectively, by using representation (11). From identity (9) it is obvious that ${\zeta}^{1}(0,\tau )={\zeta}^{2}(0,\tau )$ for each $\tau \in (0,T]$. Hence the difference of these formulas implies the desired result. □

This lemma with identity (14) implies the following.

**Corollary 2.1** *Let conditions of Lemma* 2.1 *hold*. *Then* ${f}_{1}(t)={f}_{2}(t)$, $\mathrm{\forall}t\in [0,T]$, *if and only if*

Since the Strum-Liouville problem generates a complete orthogonal family of eigenfunctions, the null space of a semigroup contains only zero function, *i.e.*, $N(T)=\{0\}$. Thus Corollary 2.1 states that ${f}_{1}\equiv {f}_{2}$ if and only if ${\xi}^{1}(x,t)-{\xi}^{2}(x,t)=0$ for all $(x,t)\in {\mathrm{\Omega}}_{T}$. The definition of $\xi (x,t)$ implies that ${k}_{1}(x)={k}_{2}(x)$ for all $x\in [0,1]$.

The combination of the conclusions of Lemma 2.1 and Corollary 2.1 can be given by the following theorem which states the distinguishability of the input-output mapping $\mathrm{\Phi}[\cdot ]:{\mathcal{K}}_{0}\to {H}^{1,2}[0,T]$.

**Theorem 2.1** *Let conditions* (C1) *and* (C2) *hold*. *Assume that* $\mathrm{\Phi}[\cdot ]:{\mathcal{K}}_{0}\to {H}^{1,2}[0,T]$ *is the input*-*output mapping defined by* (3) *and corresponding to the measured output* $f(t):=u(0,t)$. *Then the mapping* $\mathrm{\Phi}[k]$ *has the distinguishability property in the class of admissible coefficients* ${\mathcal{K}}_{0}$, *i*.*e*.,

## 3 Analysis of the inverse problem with measured output data $h(t)$

Consider now the inverse problem with one measured output data $h(t)$ at $x=1$. As in the previous section, let us arrange the parabolic equation as follows:

Then the initial boundary value problem (1) can be rewritten in the following form:

In order to formulate the solution of the above parabolic problem in terms of a semigroup, let us use the same variable $v(x,t)$ in identity (6), which satisfies the following parabolic problem:

Here $B[v(x,t)]:=-k(1){d}^{2}v(x,t)/d{x}^{2}$ is a second-order differential operator, its domain is ${D}_{B}=\{u\in {H}^{2,2}(0,1)\cap {H}^{1,2}[0,1]:{u}_{x}(0)=u(1)=0\}$. Since the initial value function $g(x)$ belongs to ${H}^{2,2}[0,1]$, it is obvious that $g(x)\in {D}_{B}$.

Denote by $S(t)$ the semigroup of linear operators generated by the operator −*A* [5, 6]. As in the previous section, we can easily find the eigenvalues and eigenfunctions of the differential operator *B*. Furthermore, the semigroup $S(t)$ can be easily constructed by using the eigenvalues and eigenfunctions of the differential operator *B*. For this reason, we first consider the following eigenvalue problem:

This problem is called a Sturm-Liouville problem. We can easily determine that the eigenvalues are ${\lambda}_{n}=k(1){(2n-1)}^{2}{\pi}^{2}/4$ for all $n=0,1,\dots $ and the corresponding eigenfunctions become ${\varphi}_{n}(x)=\sqrt{2}cos((2n-1)\pi /2x)$. Hence the semigroup $S(t)$ can be represented in the following form:

The null space of the semigroup $S(t)$ of the linear operators can be defined as follows:

Since the Sturm-Liouville problem generates a complete orthogonal family of eigenfunctions, we can say that the null space of the semigroup $S(t)$ is an empty set, *i.e.*, $N(S)=\mathrm{\varnothing}$. This result is very important for the uniqueness of the unknown coefficient $k(x)$.

The unique solution of the initial value problem (16) in terms of a semigroup $S(t)$ can be represented in the following form:

Hence, by using identity (6), the solution $u(x,t)$ of the parabolic problem (15) in terms of a semigroup can be written in the following form:

Defining the following:

The solution representation of the parabolic problem (17) can be rewritten in the following form:

Differentiating both sides of the above identity with respect to *x* and substituting $x=1$ yield

Taking into account the overmeasured data $k(1){u}_{x}(1,t)=h(t)$, we get

Now we can determine the value $k(1)$. From the overmeasured data $k(1){u}_{x}(1,t)=h(t)$, the identity $k(1)=h(t)/{u}_{x}(1,t)$ for all $t>0$ can be rewritten in terms of a semigroup in the following form:

Taking limit as $t\to 0$ in the above identity yields

The right-hand side of the above identity defines *the semigroup representation of the input-output mapping* $\mathrm{\Psi}[k]$ on the set of admissible unknown diffusion coefficient $\mathcal{K}$:

Differentiating both sides of identity (17) with respect to *t*, we get

Using semigroup properties, we obtain

Taking $x=1$ in the above identity, we get

Since $u(1,t)={\psi}_{1}$, we have ${u}_{t}(1,t)=0$. Taking into account this and substituting $t=0$, we get

Solving this equation for ${k}^{\prime}(1)$ and substituting ${u}_{x}(1,0)=h(0)/k(1)$, we reach the following result:

Then we can define the admissible set of diffusion coefficients as follows:

The following lemma implies the relation between the coefficients ${k}_{1}(x),{k}_{2}(x)\in \mathcal{K}$ at $x=1$ and the corresponding outputs ${h}_{j}(t):={k}_{j}(1){u}_{x}(1,t;{k}_{j})$, $j=1,2$.

**Lemma 3.1** *Let* ${u}_{1}(x,t)=u(x,t;{k}_{1})$ *and* ${u}_{2}(x,t)=u(x,t;{k}_{2})$ *be solutions of the direct problem* (16) *corresponding to the admissible coefficients* ${k}_{1}(x),{k}_{2}(x)\in \mathcal{K}$. *Suppose that* ${h}_{j}(t)=u(1,t;{k}_{j})$, $j=1,2$, *are the corresponding outputs and denote by* $\mathrm{\Delta}h(t)={h}_{1}(t)-{h}_{2}(t)$, $\mathrm{\Delta}{w}_{1}(x,t,s)={w}_{1}^{1}(x,t,s)-{w}_{1}^{2}(x,t,s)$. *If the condition*

*holds*, *then the outputs* ${h}_{j}(t)$, $j=1,2$, *satisfy the following integral identity*:

*for each* $\tau \in (0,T]$.

*Proof* The solutions of the direct problem (15) corresponding to the admissible coefficients ${k}_{1}(x),{k}_{2}(x)\in \mathcal{K}$ can be written at $x=1$ as follows:

respectively, by using formula (20). From definition (18), it is obvious that ${z}_{1}^{1}(1,\tau )={z}_{1}^{2}(1,\tau )$ for each $\tau \in (0,T]$. Hence the difference of these formulas implies the desired result. □

This lemma with identity (23) implies the following conclusion.

**Corollary 3.1** *Let the conditions of Lemma* 3.1 *hold*. *Then* ${h}_{1}(t)={h}_{2}(t)$, $\mathrm{\forall}t\in [0,T]$, *if and only if*

*hold*.

Since the null space of it consists of only zero function, *i.e.*, $N(S)=\{0\}$, Corollary 3.1 states that ${h}_{1}\equiv {h}_{2}$ if and only if ${\chi}^{1}(x,t)-{\chi}^{2}(x,t)=0$ for all $(x,t)\in {\mathrm{\Omega}}_{T}$. The definition of $\chi (x,t)$ implies that ${k}_{1}(x)={k}_{2}(x)$ for all $x\in (0,1]$.

**Theorem 3.1** *Let conditions* (C1) *and* (C2) *hold*. *Assume that* $\mathrm{\Psi}[\cdot ]:{\mathcal{K}}_{1}\to {C}^{1}[0,T]$ *is the input*-*output mapping defined by* (3) *and corresponding to the measured output* $h(t):=k(1){u}_{x}(1,t)$. *Then the mapping* $\mathrm{\Psi}[k]$ *has the distinguishability property in the class of admissible coefficients* ${\mathcal{K}}_{1}$, *i*.*e*.,

## 4 The inverse problem with mixed output data

Consider now the inverse problem (1)-(2) with two measured output data $f(t)$ and $h(t)$. As shown before, having these two data, the values $k(0)$ as well as $k(1)$ can be defined by the above explicit formulas. Based on this result, let us define now the set of admissible coefficients ${\mathcal{K}}_{2}$ as an intersection:

On this set, both input-output mappings $\mathrm{\Phi}[k]$ and $\mathrm{\Psi}[k]$ have distinguishability property.

**Corollary 4.1** *The input*-*output mappings* $\mathrm{\Phi}[\cdot ]:{\mathcal{K}}_{2}\to {H}^{1,2}[0,T]$ *and* $\mathrm{\Psi}[\cdot ]:{\mathcal{K}}_{2}\to {H}^{1,2}[0,T]$ *distinguish any two functions* ${k}_{1}(x)\ne {k}_{2}(x)$ *from the set* ${\mathcal{K}}_{2}$, *i*.*e*.,

## 5 Conclusion

The aim of this study was to analyze distinguishability properties of the input-output mappings $\mathrm{\Phi}[\cdot ]:{\mathcal{K}}_{2}\to {H}^{1,2}[0,T]$ and $\mathrm{\Psi}[\cdot ]:{\mathcal{K}}_{2}\to {H}^{1,2}[0,T]$ which are naturally determined by the measured output data. In this paper we show that if the null spaces of the semigroups $T(t)$ and $S(t)$ include only zero function then the corresponding input-output mappings $\mathrm{\Phi}[\cdot ]$ and $\mathrm{\Psi}[\cdot ]$ have distinguishability property.

This study shows that boundary conditions and the region on which the problem is defined have a significant impact on the distinguishability of the input-output mappings $\mathrm{\Phi}[\cdot ]$ and $\mathrm{\Psi}[\cdot ]$ since these key elements determine the structure of the semigroups $T(t)$ and $S(t)$ of linear operators and their null spaces.

## References

- 1.
DuChateau P: Monotonicity and invertibility of coefficient-to-data mappings for parabolic inverse problems.

*SIAM J. Math. Anal.*1995, 26: 1473-1487. 10.1137/S0036141093259257 - 2.
Isakov V: On uniqueness in inverse problems for quasilinear parabolic equations.

*Arch. Ration. Mech. Anal.*1993, 124: 1-13. 10.1007/BF00392201 - 3.
Pilant MS, Rundell W: A uniqueness theorem for conductivity from overspecified boundary data.

*J. Math. Anal. Appl.*1988, 136: 20-28. 10.1016/0022-247X(88)90112-6 - 4.
Renardy M, Rogers R:

*An Introduction to Partial Differential Equations*. Springer, New York; 2004. - 5.
Cannon JR:

*The One-Dimensional Heat Equation*. Addison-Wesley, Reading; 1984. - 6.
DuChateau P, Thelwell R, Butters G: Analysis of an adjoint problem approach to the identification of an unknown diffusion coefficient.

*Inverse Probl.*2004, 20: 601-625. 10.1088/0266-5611/20/2/019 - 7.
Showalter R:

*Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations*. Am. Math. Soc., Providence; 1997. - 8.
Hasanov A, Demir A, Erdem A: Monotonicity of input-output mappings in inverse coefficient and source problem for parabolic equations.

*J. Math. Anal. Appl.*2007, 335: 1434-1451. 10.1016/j.jmaa.2007.01.097 - 9.
Hasanov A, DuChateau P, Pektas B: An adjoint approach and coarse-fine mesh method for identification of the diffusion coefficient in a linear parabolic equations.

*J. Inverse Ill-Posed Probl.*2006, 14: 435-463. 10.1515/156939406778247615 - 10.
DuChateau P, Gottlieb J:

*Introduction to Inverse Problems in Partial Differential Equations for Engineers, Physicists and Mathematicians*. Kluwer Academic, Dordrecht; 1996. - 11.
Ashyralyev A, San ME: An approximation of semigroup method for stochastic parabolic equations.

*Abstr. Appl. Anal.*2012., 2012: Article ID 684248. doi:10.1155/2012/684248 - 12.
Demir A, Ozbilge E: Semigroup approach for identification of the unknown diffusion coefficient in a quasi-linear parabolic equation.

*Math. Methods Appl. Sci.*2007, 30: 1283-1294. 10.1002/mma.837 - 13.
Ozbilge E: Identification of the unknown diffusion coefficient in a quasi-linear parabolic equation by semigroup approach with mixed boundary conditions.

*Math. Methods Appl. Sci.*2008, 31: 1333-1344. 10.1002/mma.974 - 14.
Demir A, Ozbilge E: Analysis of a semigroup approach in the inverse problem of identifying an unknown coefficient.

*Math. Methods Appl. Sci.*2008, 31: 1635-1645. 10.1002/mma.989 - 15.
Demir A, Hasanov A: Identification of the unknown diffusion coefficient in a linear parabolic equation by the semigroup approach.

*J. Math. Anal. Appl.*2008, 340: 5-15. 10.1016/j.jmaa.2007.08.004

## Acknowledgements

Dedicated to Professor Hari M Srivastava.

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

## Author information

## Additional information

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

## Rights and permissions

## About this article

#### Received

#### Accepted

#### Published

#### DOI

### Keywords

- Inverse Problem
- Null Space
- Distinguishability Property
- Initial Boundary
- Parabolic Problem