An Interior Inverse Problem for the Diffusion Operator

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 (0,𝜋) 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 𝐿(𝑞₀(𝑥),𝑞₁(𝑥),ℎ,𝐻) of the form 𝑙𝑦(𝑥)∶=𝑦^′′(𝑥)+𝜌²−2𝜌𝑞₁(𝑥)−𝑞₀(𝑥)𝑦(𝑥)=0 (1) on the interval 0<𝑥<𝜋 with the boundary conditions 𝑈𝑦∶=𝑦^′(0)(0)𝑉𝑦−ℎ𝑦=0,∶=𝑦^′(𝜋)+𝐻𝑦(𝜋)=0 (2) and with the jump conditions 𝑦𝜋2+0=𝑎₁𝑦𝜋2,𝑦−0_′𝜋2+0=𝑎₁⁻¹𝑦^′𝜋2−0+𝑎₂𝑦𝜋2−0, (3) where 𝜌 is the spectral parameter, 𝑞₀(𝑥) and 𝑞₁(𝑥) are real functions in 𝐿²[0,𝜋], and the numbers ℎ, 𝐻, 𝑎₁, and 𝑎₂ are real and 𝑎₁>0. Without loss of generality, we assume that ^𝜋₀𝑞₁(𝑥)𝑑𝑥=0. (4)

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 (0,𝜋) 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 (0,𝜋).

2. Auxiliary Assertions

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

Let 𝜑(𝑥,𝜌), 𝐶(𝑥,𝜌), and 𝑆(𝑥,𝜌) be solutions of (1) under the initial conditions 𝜑(0,𝜌)=𝐶(0,𝜌)=𝑆^′(0,𝜌)=1, 𝜑^′(0,𝜌)=ℎ, 𝐶^′(0,𝜌)=𝑆(0,𝜌)=0 and under the jump conditions (3). For each fixed 𝑥∈[0,𝜋], the functions 𝜑(𝑥,𝜌), 𝐶(𝑥,𝜌), and 𝑆(𝑥,𝜌) together with their derivatives with respect to 𝑥 are entire in 𝜌.

Denote Δ𝜌∶=−𝑉𝜑. (5)

The function Δ(𝜌) is called the characteristic function of 𝐿. The function Δ(𝜌) is entire in 𝜌 of order 1/2, and its zeros coincide with the eigenvalues of 𝐿.

Denote 𝑄(𝑥)∶=^𝑥₀𝑞₁(𝑡)𝑑𝑡. (6)

The functions (𝑥),(𝑥)cos𝜌𝑥−𝑄sin𝜌𝑥−𝑄𝜌 (7) form a fundamental system of solutions for the differential equation 𝑦^′′+𝑞^′₁(𝑥)𝜌−𝑞₁(𝑥)𝑦^′+𝜌−𝑞₁(𝑥)²𝑦=0. (8)

We rewrite (1) in the form 𝑦^′′+𝑞^′₁(𝑥)𝜌−𝑞₁(𝑥)𝑦^′+𝜌−𝑞₁(𝑥)²𝑦=𝑞₀(𝑥)+𝑞²₁(𝑥𝑞)𝑦+^′1(𝑥)𝜌−𝑞₁(𝑥)𝑦^′. (9)

The function 𝑦=𝐶(𝑥,𝜌) is a solution of the Cauchy problem for (9) with the initial conditions 𝑦(0)=1, 𝑦^′(0)=0.

By the method of variation of parameters, we deduce that 𝐶(𝑥)𝑥,𝜌=cos𝜌𝑥−𝑄+^𝑥₀sin𝜌(𝑥−𝑡)−𝑄(𝑥)+𝑄(𝑡)𝜌−𝑞₁(𝑡)×𝑞₀(𝑡)+𝑞²₁(𝑡)𝐶+𝑞𝑡,𝜌_′1(𝑡)𝜌−𝑞^′₁(𝑡)𝐶^′𝑡,𝜌𝑑𝑡. (10)

For |𝜌|→∞, 𝐶(𝑥)1𝑥,𝜌=cos𝜌𝑥−𝑄+𝑂𝜌expIm𝜌𝑥. (11)

Substituting this asymptotic into (10), we calculate 𝐶(𝑥)+(𝑥)𝑥,𝜌=cos𝜌𝑥−𝑄sin𝜌𝑥−𝑄2𝜌^𝑥₀𝑞₀(𝑡)+𝑞²₁(𝑡)+1𝑑𝑡2𝜌^𝑥₀𝑞₀(𝑡)+𝑞²₁(𝑡𝜌()(𝑥)(𝑡)1)×sin𝑥−2𝑡−𝑄+2𝑄𝑑𝑡+𝑜𝜌expIm𝜌𝑥. (12)

Differentiating (12) with respect to 𝑥, we get 𝐶^′(𝑥)+(𝑥)𝑥,𝜌=−𝜌sin𝜌𝑥−𝑄cos𝜌𝑥−𝑄2^𝑥₀𝑞₀(𝑡)+𝑞²₁(𝑡+1)𝑑𝑡2^𝑥₀𝑞₀(𝑡)+𝑞²₁(𝑡)𝜌()(𝑥)(𝑡)𝑥.×cos𝑥−2𝑡−𝑄+2𝑄𝑑𝑡+𝑜expIm𝜌 (13)

Analogously, one can obtain for the function 𝑆(𝑥,𝜌)𝑆=(𝑥)𝑥,𝜌sin𝜌𝑥−𝑄𝜌−cos𝜌𝑥−𝑄(𝑥)2𝜌²×^𝑥₀𝑞₀(𝑡)+𝑞²₁(𝑡)+1𝑑𝑡2𝜌^2𝑥₀𝑞₀(𝑡)+𝑞²₁(𝑡)𝜌()(𝑥)(𝑡)1×cos𝑥−2𝑡−𝑄+2𝑄𝑑𝑡+𝑜𝜌²expIm𝜌𝑥, (14)𝑆^′(𝑥)−(𝑥)𝑥,𝜌=cos𝜌𝑥−𝑄sin𝜌𝑥−𝑄2𝜌^𝑥₀𝑞₀(𝑡)+𝑞²₁(𝑡)−1𝑑𝑡2𝜌^𝑥₀𝑞₀(𝑡)+𝑞²₁(𝑡𝜌()(𝑥)(𝑡)1)×sin𝑥−2𝑡−𝑄+2𝑄𝑑𝑡+𝑜𝜌expIm𝜌𝑥. (15)

Since 𝜑(𝑥,𝜌)=𝐶(𝑥,𝜌)+ℎ𝑆(𝑥,𝜌), by similar arguments in [8], one can calculate, for |𝜌|→∞, 𝜑(𝑥)+1𝑥,𝜌=cos𝜌𝑥−𝑄ℎ+2^𝑥₀𝑞₀(𝑡)+𝑞²₁(𝑡)×(𝑥)𝑑𝑡sin𝜌𝑥−𝑄𝜌1+𝑜𝜌𝑥𝜋expIm𝜌,𝑥<2,𝜑𝑥,𝜌=𝑏₁cos𝜌𝑥−𝑄(𝑥)+𝑏₂𝜌()𝜋cos𝜋−𝑥−2𝑄2(𝑥)(𝑥)(𝑥)+𝑄+𝑀sin𝜌𝑥−𝑄(𝑥)𝜌()()(𝑥)2𝜌+𝑁sin𝜋−𝑥−2𝑄𝜋/2+𝑄12𝜌+𝑜𝜌𝑥𝜋expIm𝜌,𝑥>2,𝜑^′(𝑥)+1𝑥,𝜌=−𝜌sin𝜌𝑥−𝑄ℎ+2^𝑥₀𝑞₀(𝑡)+𝑞²₁(𝑡)(𝑥)𝑥𝜋𝑑𝑡×cos𝜌𝑥−𝑄+𝑜expIm𝜌,𝑥<2,𝜑^′𝑥,𝜌=𝜌−𝑏₁sin𝜌𝑥−𝑄(𝑥)+𝑏₂𝜌()𝜋sin𝜋−𝑥−2𝑄2(𝑥)(𝑥)(𝑥)+𝑄+𝑀cos𝜌𝑥−𝑄2(𝑥)𝜌()()(𝑥)−𝑁cos𝜋−𝑥−2𝑄𝜋/2+𝑄2𝑥𝜋+𝑜expIm𝜌,𝑥>2, (16)(17)(18)(19) where 𝑀(𝑥)=𝑏₁2ℎ+^𝑥₀𝑞₀(𝑡)+𝑞²₁(𝑡)𝑑𝑡+𝑎₂,𝑁(𝑥)=𝑏₂2ℎ−^𝑥₀𝑞₀(𝑡)+𝑞²₁(𝑡)𝑑𝑡+2₀^𝜋/2𝑞₀(𝑡)+𝑞²₁(𝑡)𝑑𝑡−𝑎₂. (20)

It follows from (6), (17), and (19) that Δ𝜌=𝑏₁𝜌𝑏sin𝜌𝜋+²𝑏₁𝜋sin2𝑄2−𝜔cos𝜌𝜋2+𝜔₁2+𝑜expIm𝜌𝑥, (21) where 𝜔=2𝐻+2ℎ+^𝜋₀𝑞₀(𝑡)+𝑞²₁(𝑡)𝑎𝑑𝑡+²𝑏₁,𝑏𝜔=−²𝑏₁2𝐻−2ℎ+^𝜋₀𝑞₀(𝑡)+𝑞²₁(𝑡)𝑑𝑡−2₀^𝜋/2𝑞₀(𝑡)+𝑞²₁(𝑡+𝑎)𝑑𝑡²𝑏₁𝜋cos2𝑄2. (22)

Using (21) by the well-known method (see, e.g., [3]), one has that, for 𝑛→∞, 𝜌_𝑛=𝑛+𝜔₀+1()2𝑛𝜋𝜔+−1^𝑛−1𝜔₁1+𝑜𝑛, (23) where 𝜔₀=(−1)^𝑛−1𝑏₂𝑏₁𝜋𝜋sin2𝑄2. (24)

3. Main Result

In this section, we will give a uniqueness theorem. It says that the potential function 𝑞₀(𝑥) for a diffusion operator is uniquely determined by one spectrum and some information on eigenfunctions at the midpoint of the interval (0,𝜋). The technique we used is similar to those used in [6, 9].

Together with 𝐿, we consider a boundary value problem 𝐿=𝐿(𝑞₀,𝑞₁,ℎ,𝐻) of the same form but with a different coefficient 𝑞₀. We agree that, if a certain symbol 𝛼 denotes an object related to 𝐿, then 𝛼 will denote an analogous object related to 𝐿.

Consider the problems 𝜑^′′+𝜌²−2𝜌𝑞₁(𝑥)−𝑞₀(𝑥)𝜑=0, (25) with the initial conditions 𝜑(0)=1, 𝜑^′(0)=ℎ and 𝜑^′′+𝜌²−2𝜌𝑞₁(𝑥)−𝑞₀(𝑥)𝜑=0, (26) with the initial conditions 𝜑(0)=1, 𝜑^′(0)=ℎ.

For 𝑥≤𝜋/2, the following representation holds (see [21]): 𝜑𝑥,𝜌=cos𝜌𝑥−𝛼(𝑥)+^𝑥₀𝐴()+𝑥,𝑡cos𝜌𝑡𝑑𝑡_𝑥0𝐵()𝜑(𝑥)+𝑥,𝑡sin𝜌𝑡𝑑𝑡,𝑥,𝜌=cos𝜌𝑥−𝛼^𝑥₀𝐴()+𝑥,𝑡cos𝜌𝑡𝑑𝑡_𝑥0𝐵(𝑥,𝑡)sin𝜌𝑡𝑑𝑡, (27)(28) where 𝛼(𝑥)=𝑥𝑝(0)+2^𝑥₀𝐴(𝜁,𝜁)sin𝛼(𝜁)−𝐵(𝜁,𝜁)cos𝛼(𝜁)𝑑𝜁. (29)

The kernels 𝐴(𝑥,𝑡) and 𝐵(𝑥,𝑡) are the solution of the problem 𝜕²𝐴(𝑥,𝑡)𝜕𝑥²(𝑥)()−2𝑝𝜕𝐵𝑥,𝑡(𝑥)𝐴()=𝜕𝜕𝑡−𝑞𝑥,𝑡²𝐴(𝑥,𝑡)𝜕𝑡²,𝜕²𝐵(𝑥,𝑡)𝜕𝑥²(𝑥)()+2𝑝𝜕𝐴𝑥,𝑡(𝑥)𝐵()=𝜕𝜕𝑡−𝑞𝑥,𝑡²𝐵(𝑥,𝑡)𝜕𝑡²,𝑞(𝑥)=−𝑝²(𝑥)𝑑+2[𝐴()(𝑥)()(𝑥,𝐴()()()𝑑𝑥𝑥,𝑥cos𝛼+𝐵𝑥,𝑥sin𝛼)]0,0=ℎ,𝐵𝑥,0=0,𝜕𝐴𝑥,𝑡𝜕𝑡_𝑡=𝑜=0,𝛼(𝑥)=^𝑥₀𝑝(𝑡)𝑑𝑡. (30)

Hence, 𝜑1𝜑=2(𝑥+1+cos2𝜌𝑥−𝛼)_𝑥0𝐻(𝑥,𝜏)cos2𝜌𝜏−𝛼(𝜏)𝑑𝜏, (31) where 𝐻()𝐴()+𝐴()()+𝐵()+𝑥,𝑡=2𝑥,𝑥−2𝜏𝑥,𝑥−2𝜏+𝐵𝑥,𝑥−2𝜏𝑥,𝑥−2𝜏_{𝑥−𝑥+2𝜏}𝐴()𝐴()+𝑥,𝑠𝑥,𝑠−2𝜏𝑑𝑠_{𝑥−𝑥+2𝜏}𝐵()𝐵()+𝑥,𝑠𝑥,𝑠−2𝜏𝑑𝑠_{𝑥−2𝜏−𝑥}𝐴()𝐴()+𝑥,𝑠𝑥,𝑠+2𝜏𝑑𝑠_{𝑥−2𝜏−𝑥}𝐵(𝑥,𝑠)𝐵(𝑥,𝑠+2𝜏)𝑑𝑠. (32)

The eigenvalues and the corresponding eigenfunctions of the problem 𝐿 are denoted by 𝜌_𝑛 and 𝜑_𝑛(𝑥), 𝑛∈𝑁, respectively.

Theorem 1. If for any 𝑛∈𝑁, 𝜌_𝑛=𝜌_𝑛,𝜑_𝑛,𝜑_𝑛𝜋2−0=0, (33) then 𝑞₀=𝑞₀ almost everywhere on [0,𝜋].

Proof. If we multiply (25) by 𝜑 and (26) by 𝜑, and then subtract, after integrating on [0,(𝜋/2)−0], we obtain 𝐷𝜌∶=₀^(𝜋/2)−0𝑞(𝑥)(𝑥𝜑𝜑=𝜑−𝑞)𝑥,𝜌𝑥,𝜌𝑑𝑥_′𝑥,𝜌𝜑𝑥,𝜌−𝜑𝑥,𝜌𝜑^′𝑥,𝜌₀^(𝜋/2)−0. (34)
By using the properties of 𝜑 and 𝜑, we conclude that the function 𝐷(𝜌) is an entire function. From condition of the theorem, together with the initial-value condition at 0, it follows that 𝐷(𝜌_𝑛)=0, 𝑛∈𝑁.
In addition, by (27), (28), and (34), for 0<𝑥<𝜋, we find 𝐷𝜌≤𝑀, (35) where 𝑀>0 is constant. Now, we define an entire function Ψ𝜌=𝐷𝜌Δ𝜌. (36)
From (21) and (35), it follows that Ψ𝜌1=𝑂𝜌, (37) for large |𝜌|. So, for all 𝜌, from the Liouville theorem, we get Ψ𝜌=0or𝐷𝜌=0. (38)
Define 𝑟(𝑥)=𝑞(𝑥)−𝑞(𝑥). Further substituting (31) into (34) and (38), we obtain 12₀^(𝜋/2)−0𝑟(𝑥(𝑥)+)1+cos2𝜌𝑥−𝛼_𝑥0𝐻(𝑥,𝜏)cos2𝜌𝜏−𝛼(𝜏)𝑑𝜏𝑑𝑥=0, (39) which can be rewritten as ₀^(𝜋/2)−0𝑟(𝑥)+𝑑𝑥_{0(𝜋/2)−0}(𝜏)×𝑟(𝜏)+cos2𝜌𝜏−𝛼_{𝜏(𝜋/2)−0}𝑟(𝑥)𝐻(𝑥,𝜏)𝑑𝑥𝑑𝜏=0. (40)
Letting 𝜌→∞ for real 𝜌, we conclude from Riemann-Lebesgue lemma that ₀^(𝜋/2)−0𝑟(𝑥)𝑑𝑥=0,_{0(𝜋/2)−0}(𝜏)×𝑟(𝜏)+cos2𝜌𝜏−𝛼_{𝜏(𝜋/2)−0}𝑟(𝑥)𝐻(𝑥,𝜏)𝑑𝑥𝑑𝜏=0. (41)(42)
Then, by using the trigonometric expansion of cos function and the completeness of the functions cos and sin, we obtain 𝑟(𝜏)+_𝜏^(𝜋/2)−0𝑟(𝑥)𝐻()𝜋𝑥,𝜏𝑑𝑥=0,0<𝜏<2. (43)
Since (43) is a Volterra integral equation, it has only trivial solution. Hence, we have obtained our result 𝑟(𝑥)=0 on 0<𝑥<𝜋/2; that is, 𝑞₀(𝑥)=𝑞₀(𝑥) almost everywhere on [0,𝜋/2].
To prove that 𝑞₀(𝑥)=𝑞₀(𝑥) almost everywhere on [𝜋/2,𝜋], we will consider the supplementary problem 𝐿: 𝜑^′′(𝑥)+𝜌²−2𝜌𝑞₁₁(𝑥)−𝑞₀₁(𝑥)𝜑(𝑥)𝑈𝜑=0,0<𝑥<𝜋,∶=𝜑^′(0)(0)𝑉𝜑−𝐻𝜑=0,∶=𝜑^′(𝜋)(𝜋)𝜑𝜋+ℎ𝜑=0,2+0=𝑎₁⁻¹𝜑𝜋2,𝜑−0_′𝜋2+0=𝑎₁𝜑^′𝜋2−0+𝑎₁𝑎₂𝜑𝜋2−0, (44) where 𝑞₁₁(𝑥)=𝑞₁(𝜋−𝑥),𝑞₀₁(𝑥)=𝑞₀(𝜋−𝑥). (45)
Note that, if 𝑦(𝑥) and 𝑧(𝑥) satisfy the matching conditions (3), then a direct calculation yields 𝜋𝑦,𝑧2=𝜋+0𝑦,𝑧2−0. (46)
The assumption of Theorem 1 and (46) imply that 𝜑_𝑛,𝜑_𝑛𝜋2+0=0. (47)
A direct calculation implies that 𝜑_𝑛(𝑥)∶=𝜑_𝑛(𝜋−𝑥) is the solution to the supplementary problem 𝐿 and 𝜑_𝑛((𝜋/2)−0)∶=𝜑_𝑛((𝜋/2)+0). Thus, for the supplementary problem 𝐿, the assumption conditions in the theorem are still satisfied.
If we repeat the previous arguments, then this yields 𝑟(𝜋−𝑥)=0 on 0<𝑥<𝜋/2; that is, 𝑞₀(𝑥)=𝑞₀(𝑥) almost everywhere on [𝜋/2,𝜋]. 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 𝑦(𝑥) is noninteger in (1).