- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 320456, 6 pages
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.
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.
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  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 .
The function is called the characteristic function of . The function is entire in of order , and its zeros coincide with the eigenvalues of .
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
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 , one can calculate, 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 ): where
The kernels and are the solution of the problem
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.
- 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.
- H. P. Baltes, Inverse Scattering Problems in Optics, vol. 20 of Topics in Current Physics, Springer, Berlin, Germany, 1980.
- L. Hogben, “Spectral graph theory and inverse eigenvalue problem of a graph,” Chamchuri Journal of Mathematics, vol. 1, no. 1, pp. 51–72, 2009.
- 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.
- 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.
- 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.
- G. Freiling and V. A. Yurko, Inverse Sturm-Liouville Problems and Their Applications, Nova Science, New York, NY, USA, 2001.
- O. H. Hald, “Discontinuous inverse eigenvalue problems,” Communications on Pure and Applied Mathematics, vol. 37, no. 5, pp. 539–577, 1984.
- 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.
- 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.
- 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.
- R. J. Krueger, “Inverse problems for nonabsorbing media with discontinuous material properties,” Journal of Mathematical Physics, vol. 23, no. 3, pp. 396–404, 1982.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.