On an inverse problem for Sturm-Liouville Equation

Transcription

1 EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 1, No. 3, 17, ISSN Published by New York Business Global On an inverse problem for Sturm-Liouville Equation Döne Karahan 1,, Khanlar R. Mamedov 1 Department of Mathematics, Harran University, Turkey Department of Mathematics, Mersin University, Turkey Abstract. In this study, the theorem on necessary and sufficient conditions for the solvability of inverse problem for Sturm-Liouville operator with discontinuous coefficient is proved and the algorithm of reconstruction of potential from spectral data eigenvalues and normalizing numbers is given. 1 Mathematics Subject Classifications: 34A55, 34B4 Key Words and Phrases: Sturm-Liouville operator, inverse problem, necessary and sufficient conditions 1. Introduction We consider the boundary value problem y + qxy = λ ρxy, x π, 1 y =, yπ =, where q x L, π is a real-valued function, ρx is a piecewise continuous function, λ is a complex parameter. This spectral problem appears while solving wave or heat equations for nonhomogeneous density of the material [1], []. Physical applications of discontinuous Sturm-Liouville problem are given in [3]-[8]. For simplicity, we will assume that the density function has only one discontinuity point such that { 1, x a, ρx = α 3, a < x π, where < α 1. Direct problem of spectral analysis for Sturm-Liouville problem is investigated properties of eigenvalues and eigenfunctions, finding normalizing numbers, spectrum set of the boundary value problem, scattering data and some other values. It is important to investigate these properties. Inverse problem of spectral analysis is to final the coefficient of Corresponding author. addresses: dkarahan@harran.edu.tr D. Karahan, hanlar@mersin.edu.tr Kh. R. Mamedov c 17 EJPAM All rights reserved.

2 D. Karahan, Kh. R. Mamedov / Eur. J. Pure Appl. Math, , the equation for given spectral data. This has to be done uniquely, so that it gives the uniqueness of the inverse problem. In the process of the solution of the inverse problem giving an algorithm for constructing the potential is important. For ρx 1, solutions of inverse problem for equation 1 is given by [9]-[16]. For ρx 1, under different boundary conditions similar problem is solved in [17]-[1]. When boundary conditions contain spectral parameter, it is solved by [], [3]. The inverse problem for this equation is to find necessary and sufficient conditions for any data set to be spectral data. The main of this work is to find these conditions for 1, boundary value problem. Firstly spectral data is defined. Characteristic properties of these values are investigated in [] and also uniqueness of the solution of the inverse problem is proved. Consequently, in this work for 1, spectral problem, solution of the inverse problem is given with respect to the spectral data. For 1, boundary value problem in [], it is shown that the real numbers { λ n, α n }n 1 satisfy the following λ n = λ n + d n λ + k n n n, α n = α n + t n n, {k n}, {t n } l, 4 where λ n are zeros of the function λ = α cos λµ+ π α cos λµ π, h + sin λ d n = nµ + π + h sin λ nµ π α µ+ π sin λ nµ + π α µ π sin λ nµ π is a bounded sequence. In [18] it is proved, that the solution ϕx, λ of the equation 1 with initial date ϕ, λ = 1, ϕ, λ = can be represented as ϕx, λ = ϕ x, λ + µ + x Ax, t cos λtdt, 5 where Ax, t belongs to the space L, π for each fixed x [, π] and is related to the coefficient qx of the equation 1 by the formula: d dx Ax, 1 µ+ x = qx, 6 ρx ρx ϕ x, λ = 1 is the solution of 1 when qx, cos λµ + x + 1 ρx µ + x = ±x ρx + a 1 1 ρx cos λµ x 7 1 ρx. 8

3 D. Karahan, Kh. R. Mamedov / Eur. J. Pure Appl. Math, , The characteristic function λ of the problem 1, is λ :=< ϕx, λ, ψx, λ >= ϕx, λψ x, λ ϕ x, λψx, λ where λ is independent from x [, π]. Substituting x = and x = π into above the equation, we get λ = ϕπ, λ = ψ, λ. Theorem 1. For each fixed x [, π] the kernel Ax, t from the representation 5satisfies the following linear functional integral equation 1 + ρt A x, µ + t + 1 ρa t 1 + A x, a t + ρa t where F x, t = 1 +F x, t + F x, t = µ + x Ax, ξf ξ, tdξ =, < t < x 9 ϕ t, λ n cos λ n x ϕ t, λ n cos λ nx n=1 α n F µ + x, t + 1 ρx α n ρx F µ x, t 11 { λ n } are eigenvalues and α n are norming constants of the boundary value problem 1, when qx. Theorem. For each fixed x [, π] main equation 9 has a unique solution Ax,. L,ρ, µ + x. The proof of Theorem 1 and Theorem is given in [1].. Sufficient conditions for solvability of the inverse problem Assume that the real numbers { λ n, α n is given by the formula 4. Now, let s }n 1 construct F x, t and F x, t functions by using the formulas 1, 11 and write the integral equation 9. We determine Ax, t from the main equation 9. We shall construct the function ϕx, λ with the formula 5 i.e. ϕx, λ := ϕ x, λ + and the function qx with formula qx := µ + x Ax, t cos λtdt, 4ρx ρx + 1 d dx A x, µ + x. 1

4 D. Karahan, Kh. R. Mamedov / Eur. J. Pure Appl. Math, , Denote bx := cos λn x n=1 α n λ n cos λ nx αnλ n. Similar to Lemma in [15], it is shown that bx W 1, π. According to 4 and 5 we have F tt x, t = ρtf xx x, t, ρtf xx x, t = ρxf tt x, t, 13 F x, t x= =, F x, t t= =, 14 x F µ ± x, t = ± ρ x ξ F ξ, t ξ=µ ± x. 15 Using the main equation 9 it can be proved that Ax, =, 16 ρx 1 d ρx + 1 dx Ax, µ+ x = d { Ax, µ x + Ax, µ x }. 17 dx.1. Derivation of the Differential Equation Lemma 1. The following relations hold Proof. Assume that bx W, π and ϕ x, λ + qxϕx, λ = λ ρxϕx, λ, 18 ϕ, λ = 1, ϕ, λ =. 19 Jx, λ := 1 + ρt A x, µ + t + 1 ρa t 1 + A x, a t + ρa t +F x, t + µ + x Differentiating twice with respect to x and t we get Ax, ξf ξ, tdξ =, J xxx, t ρxj ttx, t qxjx, λ. Using the formulas 9, 1-15 and 17, we obtain the following homogeneous equation 1 + [ Axx x, µ + t ρxa tt x, µ + t qxa x, µ + t ] + ρt + 1 ρa t 1 + ρa t [A xx x, a t ρxa tt x, a t qxa x, a t] +

5 D. Karahan, Kh. R. Mamedov / Eur. J. Pure Appl. Math, , µ + x [A xx x, ξ ρxa ξξ x, ξ qxax, ξ] F ξ, tdξ =. We know that from [1] this equation has only trivial solution: A xx x, t ρxa tt x, t qxax, t =, < t < x. 1 Differentiating 5 twice, integrating by parts twice and using 16 we obtain ϕ x, λ + λ ρxϕx, λ qxϕx, λ = ϕ x, λ + µ + x A xx x, t cos λtdt+ λρxax, µ + x sin λµ + x + ρxa x x, µ + x cos λµ + x+ +λρx sin λµ x A x, µ x + A x, µ x + + ρx cos λµ x d A x, µ x + A x, µ x + dx + ρx cos λµ + Ax, t x x + t=µ + x + ρx cos λµ x Ax, t x t=µ x+ Ax, t x ϕ x, λ + λρx sin λµ + xax, µ + x + ρx cos λµ + x t=µ x Ax, t t λρx sin λµ x { A x, µ x + A x, µ x } + [ ] +ρx cos λµ Ax, t x Ax, t t t + 1 ρx qx [ 1 µ + x t=µ x 1 1 ρx cos λµ x + A ttx, t cos λtdt ρx cos λµ + x+ µ + x t=µ x+ Ax, t cos λtdt ]. t=µ + x Hence using 1, 17 and 1 we arrive at 18. The relations 19 follow from 5 for x =. Lemma 1 is proved in the case bx W, π. The proof of Lemma 1 in the case bx W 1, π is carried out by a standard method see e.g. [8] p. 4. As in the theory of Sturm-Liouville problems see [15], Lemma and Corollary the following lemmas can be proved.

6 D. Karahan, Kh. R. Mamedov / Eur. J. Pure Appl. Math, , Lemma. For each function gx L,ρ, π, ρxg xdx = 1 α n=1 n Corollary 1. For arbitrary functions fx, gx L,ρ, π, ρxfxgxdx = 1 α n=1 n ρtgtϕt, λ n dt. ρtftϕt, λ n dt ρtgtϕt, λ n dt. 3 Using the below lemmas the following lemma is proved with standard method. Lemma 3. The following relation holds ρxϕt, λ n ϕt, λ k dt = {, n k α n, n = k. 4.. Derivation of Boundary Condition Lemma 4. For all n 1 the equality holds. we get Proof. Since From 5 we have By 4 we get ϕπ, λ n = ϕ x, λ n + qxϕx, λ n = λ nρxϕx, λ n, ϕ x, λ m + qxϕx, λ m = λ mρxϕx, λ m, d ϕx, λn ϕ x, λ m ϕ x, λ n ϕx, λ m = dx = λ n λ m ρxϕx, λn ϕx, λ m 5 λ n λ m ρxϕx, λ n ϕx, λ m dx = = ϕπ, λ n ϕ π, λ m ϕ π, λ n ϕπ, λ m. ϕπ, λ n ϕ π, λ m ϕ π, λ n ϕπ, λ m =. 6 Clearly, ϕ π, λ n, for all n 1. Indeed, if we suppose that ϕ π, λ m = for a certain m, then ϕπ, λ m, and in view of 6 ϕ π, λ n = for all n. On the other hand, ϕ π, λ n = ϕ π, λ n + Oe Imλ µ+ x, λ

7 D. Karahan, Kh. R. Mamedov / Eur. J. Pure Appl. Math, , i.e. for any n, ϕ π, λ n ϕ π, λ n as n, that contradicts the condition ϕ π, λ n =, n m. Thus, ϕ π, λ n, for all n 1 and from 6 we have ϕπ, λ n ϕ π, λ n = ϕπ, λ m ϕ π, λ m = H, i.e. for any n, ϕπ, λ n = Hϕ π, λ n. Since ϕπ, λ n = o1 as n, we have H = i.e. ϕπ, λ n =. Thus, we prove that the numbers { λ n, α n are spectral data of the constructed }n 1 boundary value problem 1,. Then, the following theorem is proved. Theorem 3. For the sequences { λ n, α n }n 1, where λ n λ m for n m, α n > for all n to be spectral date of a problem Lqx of the form 1-3 with qx L, π, it is necessary and sufficient to satisfy conditions Here λ n are the zeros of the function λ = α ϕ x, λ = 1 λ n = λ n + d n λ + k n n n, α n = αn + t n n, {k n}, {t n } l α n = cos λµ + π + 1 ϕ x, λ n ρxdx, cos λµ + x + 1 ρx d n is a bounded sequence; {k n }, {t n } l. 1 1 cos λµ π, α µ ± x = ±x ρx + a 1 ρx, 1 1 ρx cos λµ x, Algorithm of the construction of the function qx by spectral date { λ n, α n } follows from the proof of the Theorem 3: 1 By the given numbers { λ n, α n }n 1 the functions F x, t and F x, t are constructed by the formulas 1 and 11, respectively; The function Ax, t is found from equation 9; 3 qx is calculated by the formula 1. Acknowledgements This work is supported by the Scientific and Technological Research Council of Turkey TUBITAK.

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