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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 320456, 6 pages
http://dx.doi.org/10.1155/2013/320456
Research Article

An Interior Inverse Problem for the Diffusion Operator

1Islamic Azad University, Neka Branch, P.O. Box 48411-86114, Neka, Iran
2Department of Mathematics, University of Mazandaran, P.O. Box 47416-95447, Babolsar, Iran
3Department of Mathematics, Islamic Azad University, Sari Branch, Sari, Iran

Received 27 April 2013; Accepted 4 June 2013

Academic Editor: Dumitru Baleanu

Copyright © 2013 A. Dabbaghian et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

An inverse problem for the diffusion operator on a finite interval with discontinuities conditions inside the interval is studied. We have shown that the potential function of the diffusion operator can be established uniquely by a set of values of eigenfunctions at the midpoint of the interval and one spectrum.

1. Introduction

In an inverse spectral problem, one seeks to determine coefficients in a differential operator from information about the spectrum of the operator, subject to specific side conditions. These kinds of problems arise in a remarkable variety of applications, for example, geophysics, seismology, seismic tomography, optics, and graph theory (see [17]).

We consider the boundary value problem of the form on the interval with the boundary conditions and with the jump conditions where is the spectral parameter, and are real functions in , and the numbers , , , and are real and . Without loss of generality, we assume that

Boundary value problems with discontinuities inside the interval are extensively studied [8, 9]. These kinds of problems are often appear in mathematics, mechanics, physics, and other branches of natural sciences. For example, discontinuous inverse problems appear in electronics for constructing parameters of heterogeneous electronic lines with desirable technical characteristics [1012]. Also, boundary value problems with discontinuities in an interior point appear in geophysical models for oscillations of the Earth (see [13, 14]). Discontinuous inverse problems (in various formulations) have been considered in [1517] and other works.

The inverse problem for interior spectral data of the differential operator consists in reconstruction of this operator from the known eigenvalues and some information on eigenfunctions at some internal point.

In the later years, interior inverse problems were studied by several authors [1820]. In particular, research in [20] discussed the inverse problem for Sturm-Liouville operators with discontinuous boundary conditions and proved that the spectral data of parts of two spectra and some information on eigenfunctions at some interior point of the interval are sufficient to determine the potential.

The aim of this paper is to study the inverse problem of reconstructing the diffusion operator with discontinuous conditions on the basis of spectral data of a kind: one spectrum and some information on eigenfunctions at the midpoint of the interval .

2. Auxiliary Assertions

Before giving the main results of this work, we will mention some results which will be needed later.

Let , , and be solutions of (1) under the initial conditions , , and under the jump conditions (3). For each fixed , the functions , , and together with their derivatives with respect to are entire in .

Denote

The function is called the characteristic function of . The function is entire in of order , and its zeros coincide with the eigenvalues of .

Denote

The functions form a fundamental system of solutions for the differential equation

We rewrite (1) in the form

The function is a solution of the Cauchy problem for (9) with the initial conditions , .

By the method of variation of parameters, we deduce that

For ,

Substituting this asymptotic into (10), we calculate

Differentiating (12) with respect to , we get

Analogously, one can obtain for the function

Since , by similar arguments in [8], one can calculate, for , where

It follows from (6), (17), and (19) that where

Using (21) by the well-known method (see, e.g., [3]), one has that, for , where

3. Main Result

In this section, we will give a uniqueness theorem. It says that the potential function for a diffusion operator is uniquely determined by one spectrum and some information on eigenfunctions at the midpoint of the interval . The technique we used is similar to those used in [6, 9].

Together with , we consider a boundary value problem of the same form but with a different coefficient . We agree that, if a certain symbol denotes an object related to , then will denote an analogous object related to .

Consider the problems with the initial conditions , and with the initial conditions , .

For , the following representation holds (see [21]): where

The kernels and are the solution of the problem

Hence, where

The eigenvalues and the corresponding eigenfunctions of the problem are denoted by and , , respectively.

Theorem 1. If for any , then almost everywhere on .

Proof. If we multiply (25) by and (26) by , and then subtract, after integrating on , we obtain
By using the properties of and , we conclude that the function is an entire function. From condition of the theorem, together with the initial-value condition at , it follows that , .
In addition, by (27), (28), and (34), for , we find where is constant. Now, we define an entire function
From (21) and (35), it follows that for large . So, for all , from the Liouville theorem, we get
Define . Further substituting (31) into (34) and (38), we obtain which can be rewritten as
Letting for real , we conclude from Riemann-Lebesgue lemma that
Then, by using the trigonometric expansion of function and the completeness of the functions and , we obtain
Since (43) is a Volterra integral equation, it has only trivial solution. Hence, we have obtained our result on ; that is, almost everywhere on .
To prove that almost everywhere on , we will consider the supplementary problem : where
Note that, if and satisfy the matching conditions (3), then a direct calculation yields
The assumption of Theorem 1 and (46) imply that
A direct calculation implies that is the solution to the supplementary problem and . Thus, for the supplementary problem , the assumption conditions in the theorem are still satisfied.
If we repeat the previous arguments, then this yields on ; that is, almost everywhere on . The proof of the theorem is finished.

We suggest to extend this work for fractional differential equations and local fractional differential equations [2224] when the order of is noninteger in (1).

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