# An Interior Inverse Problem for the Diffusion Operator

Authors: A. Dabbaghian; H. Jafari; N. Yosofi
Journal: Abstract and Applied Analysis
 (2013)

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## 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.


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## Body

## 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 problemL(q0(x),q1(x),h,H) of the form
(1)ly(x):=y′′(x)+(ρ2-2ρq1(x)-q0(x))y(x)=0
on the interval 0<x<π with the boundary conditions
(2)U(y):=y′(0)-hy(0)=0,V(y):=y′(π)+Hy(π)=0
and with the jump conditions
(3)y(π2+0)=a1y(π2-0),y′(π2+0)=a1-1y′(π2-0)+a2y(π2-0),
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
(4)∫0π‍q1(x)dx=0.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.Letφ(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(5)Δ(ρ):=-V(φ).The functionΔ(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(6)Q(x):=∫0x‍q1(t)dt.The functions(7)cos(ρx-Q(x)),sin(ρx-Q(x))ρ
form a fundamental system of solutions for the differential equation
(8)y′′+q1′(x)ρ-q1(x)y′+(ρ-q1(x))2y=0.We rewrite (1) in the form
(9)y′′+q1′(x)ρ-q1(x)y′+(ρ-q1(x))2y=(q0(x)+q12(x))y+q1′(x)ρ-q1(x)y′.The functiony=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(10)C(x,ρ)=cos(ρx-Q(x))+∫0x‍sin(ρ(x-t)-Q(x)+Q(t))ρ-q1(t)×(+q1′(t)ρ-q1′(t)(q0(t)+q12(t))C(t,ρ)+q1′(t)ρ-q1′(t)C′(t,ρ))dt.For|ρ|→∞,
(11)C(x,ρ)=cos(ρx-Q(x))+O(1ρexp(|Imρ|x)).Substituting this asymptotic into (10), we calculate
(12)C(x,ρ)=cos(ρx-Q(x))+sin(ρx-Q(x))2ρ∫0x‍(q0(t)+q12(t))dt+12ρ∫0x‍(q0(t)+q12(t))×sin(ρ(x-2t)-Q(x)+2Q(t))dt+o(1ρexp(|Imρ|x)).Differentiating (12) with respect to x, we get
(13)C′(x,ρ)=-ρsin(ρx-Q(x))+cos(ρx-Q(x))2∫0x‍(q0(t)+q12(t))dt+12∫0x‍(q0(t)+q12(t))×cos(ρ(x-2t)-Q(x)+2Q(t))dt+o(exp(|Imρ|x)).Analogously, one can obtain for the functionS(x,ρ)(14)S(x,ρ)=sin(ρx-Q(x))ρ-cos(ρx-Q(x))2ρ2×∫0x‍(q0(t)+q12(t))dt+12ρ2∫0x‍(q0(t)+q12(t))×cos(ρ(x-2t)-Q(x)+2Q(t))dt+o(1ρ2exp(|Imρ|x)),(15)S′(x,ρ)=cos(ρx-Q(x))-sin(ρx-Q(x))2ρ∫0x‍(q0(t)+q12(t))dt-12ρ∫0x‍(q0(t)+q12(t))×sin(ρ(x-2t)-Q(x)+2Q(t))dt+o(1ρexp(|Imρ|x)).Sinceφ(x,ρ)=C(x,ρ)+hS(x,ρ), by similar arguments in [8], one can calculate, for |ρ|→∞,
(16)φ(x,ρ)=cos(ρx-Q(x))+(h+12∫0x‍(q0(t)+q12(t))dt)×sin(ρx-Q(x))ρ+o(1ρexp(|Imρ|x)),x<π2,(17)φ(x,ρ)=b1cos(ρx-Q(x))+b2cos(ρ(π-x)-2Q(π2)+Q(x))+M(x)sin(ρx-Q(x))2ρ+N(x)sin(ρ(π-x)-2Q(π/2)+Q(x))2ρ+o(1ρexp(|Imρ|x)),x>π2,(18)φ′(x,ρ)=-ρsin(ρx-Q(x))+(h+12∫0x‍(q0(t)+q12(t))dt)×cos(ρx-Q(x))+o(exp(|Imρ|x)),x<π2,(19)φ′(x,ρ)=ρ((ρ(π-x)-2Q(π2)+Q(x))-b1sin(ρx-Q(x))+b2sin(ρ(π-x)-2Q(π2)+Q(x)))+M(x)cos(ρx-Q(x))2-N(x)cos(ρ(π-x)-2Q(π/2)+Q(x))2+o(exp(|Imρ|x)),x>π2,
where
(20)M(x)=b1(2h+∫0x‍(q0(t)+q12(t))dt)+a2,N(x)=b2(2h-∫0x‍(q0(t)+q12(t))dt+2∫0π/2‍(q0(t)+q12(t))dt)-a2.It follows from (6), (17), and (19) that
(21)Δ(ρ)=b1[ρ(sinρπ+b2b1sin2Q(π2))-ωcosρπ2+ω12]+o(exp(|Imρ|x)),
where
(22)ω=2H+2h+∫0π‍(q0(t)+q12(t))dt+a2b1,ω=-[-2∫0π/2‍(q0(t)+q12(t))dt)b2b1(2H-2h+∫0π‍(q0(t)+q12(t))dt-2∫0π/2‍(q0(t)+q12(t))dt)+a2b1]cos2Q(π2).Using (21) by the well-known method (see, e.g., [3]), one has that, for n→∞,
(23)ρn=n+ω0+12nπ(ω+(-1)n-1ω1)+o(1n),
where
(24)ω0=(-1)n-1b2b1πsin2Q(π2).


## 3. Main Result

In this section, we will give a uniqueness theorem. It says that the potential functionq0(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 withL, 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(25)φ′′+(ρ2-2ρq1(x)-q0(x))φ=0,
with the initial conditions φ(0)=1, φ′(0)=h and
(26)φ~′′+(ρ2-2ρq1(x)-q~0(x))φ~=0,
with the initial conditions φ~(0)=1, φ~′(0)=h.Forx≤π/2, the following representation holds (see [21]):
(27)φ(x,ρ)=cos(ρx-α(x))+∫0x‍A(x,t)cosρtdt+∫0x‍B(x,t)sinρtdt,(28)φ~(x,ρ)=cos(ρx-α(x))+∫0x‍A~(x,t)cosρtdt+∫0x‍B~(x,t)sinρtdt,
where
(29)α(x)=xp(0)+2∫0x‍[A(ζ,ζ)sinα(ζ)-B(ζ,ζ)cosα(ζ)]dζ.The kernelsA(x,t) and B(x,t) are the solution of the problem
(30)∂2A(x,t)∂x2-2p(x)∂B(x,t)∂t-q(x)A(x,t)=∂2A(x,t)∂t2,∂2B(x,t)∂x2+2p(x)∂A(x,t)∂t-q(x)B(x,t)=∂2B(x,t)∂t2,q(x)=-p2(x)+2ddx[A(x,x)cosα(x)+B(x,x)sinα(x)],A(0,0)=h,B(x,0)=0,∂A(x,t)∂t|t=o=0,α(x)=∫0x‍p(t)dt.Hence,(31)φφ~=12{∫0x‍H(x,τ)cos2[ρτ-α(τ)]1+cos2[ρx-α(x)]φφ~=+∫0x‍H(x,τ)cos2[ρτ-α(τ)]}dτ,
where
(32)H(x,t)=2[∫-xx-2τ‍A(x,x-2τ)+A~(x,x-2τ)∫-xx-2τ‍+B(x,x-2τ)+B~(x,x-2τ)+∫-x+2τx‍A(x,s)A~(x,s-2τ)ds+∫-x+2τx‍B(x,s)B~(x,s-2τ)ds+∫-xx-2τ‍A(x,s)A~(x,s+2τ)ds+∫-xx-2τ‍B(x,s)B~(x,s+2τ)ds].The eigenvalues and the corresponding eigenfunctions of the problemL are denoted by ρn and φn(x), n∈N, respectively.Theorem 1.
If for anyn∈N,
(33)ρn=ρ~n,[φn,φ~n](π2-0)=0,
then q0=q~0 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
(34)D(ρ)≔∫0(π/2)-0‍[q~(x)-q(x)]φ(x,ρ)φ~(x,ρ)dx=[φ~′(x,ρ)φ(x,ρ)-φ~(x,ρ)φ′(x,ρ)]|0(π/2)-0.
By using the properties ofφ 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(ρn)=0, n∈N.
In addition, by (27), (28), and (34), for 0<x<π, we find
(35)|D(ρ)|≤M,
where M>0 is constant. Now, we define an entire function
(36)Ψ(ρ)=D(ρ)Δ(ρ).
From (21) and (35), it follows that
(37)|Ψ(ρ)|=O(1ρ),
for large |ρ|. So, for all ρ, from the Liouville theorem, we get
(38)Ψ(ρ)=0orD(ρ)=0.
Definer(x)=q~(x)-q(x). Further substituting (31) into (34) and (38), we obtain
(39)12∫0(π/2)-0‍r(x){∫0x‍[1+cos2(ρx-α(x))]+∫0x‍H(x,τ)cos2(ρτ-α(τ))dτ}dx=0,
which can be rewritten as
(40)∫0(π/2)-0‍r(x)dx+∫0(π/2)-0‍cos2(ρτ-α(τ))×[r(τ)+∫τ(π/2)-0‍r(x)H(x,τ)dx]dτ=0.
Lettingρ→∞ for real ρ, we conclude from Riemann-Lebesgue lemma that
(41)∫0(π/2)-0‍r(x)dx=0,(42)∫0(π/2)-0‍cos2(ρτ-α(τ))×[r(τ)+∫τ(π/2)-0‍r(x)H(x,τ)dx]dτ=0.
Then, by using the trigonometric expansion ofcos function and the completeness of the functions cos and sin, we obtain
(43)r(τ)+∫τ(π/2)-0‍r(x)H(x,τ)dx=0,0<τ<π2.
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)=q~0(x) almost everywhere on [0,π/2].
To prove thatq0(x)=q~0(x) almost everywhere on [π/2,π], we will consider the supplementary problem L^:
(44)φ′′(x)+(ρ2-2ρq11(x)-q01(x))φ(x)=0,0<x<π,U(φ):=φ′(0)-Hφ(0)=0,V(φ):=φ′(π)+hφ(π)=0,φ(π2+0)=a1-1φ(π2-0),φ′(π2+0)=a1φ′(π2-0)+a1a2φ(π2-0),
where
(45)q11(x)=q1(π-x),q01(x)=q0(π-x).
Note that, ify(x) and z(x) satisfy the matching conditions (3), then a direct calculation yields
(46)[y,z](π2+0)=[y,z](π2-0).
The assumption of Theorem1 and (46) imply that
(47)[φn,φ~n](π2+0)=0.
A direct calculation implies thatφ^n(x):=φn(π-x) is the solution to the supplementary problem L^ and φ^n((π/2)-0):=φn((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 yieldsr(π-x)=0 on 0<x<π/2; that is, q0(x)=q~0(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).



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