## Advanced Theoretical and Applied Studies of Fractional Differential Equations 2013

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A. Dabbaghian, H. Jafari, N. Yosofi, "An Interior Inverse Problem for the Diffusion Operator", *Abstract and Applied Analysis*, vol. 2013, Article ID 320456, 6 pages, 2013. https://doi.org/10.1155/2013/320456

# An Interior Inverse Problem for the Diffusion Operator

**Academic Editor:**Dumitru Baleanu

#### 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 [1–7]).

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 [10–12]. 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 [15–17] 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 [18–20]. 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 [22–24] when the order of is noninteger in (1).

#### References

- K. Aki and P. G. Richards, in
*Quantitative Seismology: Theory and Methods*, vol. 1, chapter 8, pp. 337–381, W. H. Freeman, New York, NY, USA, 1980. - W. Rundell and P. E. Sacks, “Reconstruction of a radially symmetric potential from two spectral sequences,”
*Journal of Mathematical Analysis and Applications*, vol. 264, no. 2, pp. 354–381, 1991. View at: Publisher Site | Google Scholar - H. P. Baltes,
*Inverse Scattering Problems in Optics*, vol. 20 of*Topics in Current Physics*, Springer, Berlin, Germany, 1980. View at: MathSciNet - L. Hogben, “Spectral graph theory and inverse eigenvalue problem of a graph,”
*Chamchuri Journal of Mathematics*, vol. 1, no. 1, pp. 51–72, 2009. View at: Google Scholar - C. R. Johnson, A. Leal-Duarte, and C. M. Saiago, “Inverse eigenvalue problems and lists of multiplicities of eigengvalues for matrices whose graph is a tree: the case of generalized stars and double generalized stars,”
*Linear Algebra and Its Applications*, vol. 373, pp. 311–330, 2003. View at: Publisher Site | Google Scholar | MathSciNet - R. L. Parker and K. A. Whaler, “Numerical methods for establishing solutions to theinverse problem of electromagnetic induction,”
*Journal of Geophysical Research*, vol. 86, no. 10, pp. 9574–9584, 1981. View at: Publisher Site | Google Scholar - V. Yurko, “Uniqueness of recovering differential operators on hedgehog-type graphs,”
*Advances in Dynamical Systems and Applications*, vol. 4, no. 2, pp. 231–241, 2009. View at: Google Scholar | MathSciNet - G. Freiling and V. A. Yurko,
*Inverse Sturm-Liouville Problems and Their Applications*, Nova Science, New York, NY, USA, 2001. View at: MathSciNet - O. H. Hald, “Discontinuous inverse eigenvalue problems,”
*Communications on Pure and Applied Mathematics*, vol. 37, no. 5, pp. 539–577, 1984. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - O. N. Livinenko and V. I. Soshnikov,
*The Theory of Heterogeneous Lines and Their Applications in Radio Engineering*, Radio, Moscow, Russia, 1964 (Russian). - J. R. McLaughlin and P. L. Polyakov, “On the uniqueness of a spherically symmetric speed of sound from transmission eigenvalues,”
*Journal of Differential Equations*, vol. 107, no. 2, pp. 351–382, 1994. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - V. P. Meschanov and A. L. Feldstein,
*Automatic Design of Directional Couplers*, Sviaz, Moscow, Russia, 1980. - R. S. Anderssen, “The effect of discontinuities in density and shear velocity onthe asymptotic overtone structure of torsional eigenfrequencies of the Earth,”
*Geophysical Journal of the Royal Astronomical Society*, vol. 50, pp. 303–309, 1997. View at: Google Scholar - F. R. Lapwood and T. Usami,
*Free Oscillations of the Earth*, Cambridge University Press, Cambridge, UK, 1981. - G. Freiling and V. A. Yurko, “Inverse spectral problems for singular non-selfadjoint differential operators with discontinuities in an interior point,”
*Inverse Problems*, vol. 18, no. 3, pp. 757–773, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - R. J. Krueger, “Inverse problems for nonabsorbing media with discontinuous material properties,”
*Journal of Mathematical Physics*, vol. 23, no. 3, pp. 396–404, 1982. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - V. A. Yurko, “On boundary value problems with discontinuity conditions inside an interval,”
*Differentsial'nye Uravneniya*, vol. 36, no. 8, pp. 1139–1140, 2000 (Russian), English Translation in*Differential Equations*, vol. 8, no. 8, pp. 1266–1269, 2000. View at: Publisher Site | Google Scholar | MathSciNet - K. Mochizuki and I. Trooshin, “Inverse problem for interior spectral data of the Sturm-Liouville operator,”
*Journal of Inverse and Ill-Posed Problems*, vol. 9, no. 4, pp. 425–433, 2001. View at: Google Scholar | Zentralblatt MATH | MathSciNet - K. Mochizuki and I. Trooshin, “Inverse problem for interior spectral data of the Dirac operator on a finite interval,”
*Publicationsof the Research Institute for Mathematical Sciences, Kyoto University*, vol. 38, no. 2, pp. 387–395, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - C. F. Yang and X. P. Yang, “An interior inverse problem for the Sturm-Liouville operator with discontinuous conditions,”
*Applied Mathematics Letters*, vol. 22, no. 9, pp. 1315–1319, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - M. G. Gasymov and G. Š. Guseĭnov, “Determination of a diffusion operator from spectral data,”
*Akademiya Nauk Azerbaĭdzhanskoĭ SSR. Doklady*, vol. 37, no. 2, pp. 19–23, 1981. View at: Google Scholar | MathSciNet - D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo,
*Fractional calculus: models and numerical methods*, vol. 3 of*Series on Complexity, Nonlinearity and Chaos*, World Scientific, Boston, Mass, USA, 2012. View at: Publisher Site | Zentralblatt MATH | MathSciNet - X. J. Yang,
*Advanced Local Fractional Calculus and Its Applications*, World Science, New York, NY, USA, 2012. - X. J. Yang,
*Local Fractional Functional Analysis and Its Applications*, Asian Academic, Hong Kong, China, 2011.

#### Copyright

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.