On Completeness of the System of Eigenfunctions of the Dirichlet Problem for Lame Equations in the Strip

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1 Advanced Studies in Teoretical Pysics Vol. 9, 2015, no. 16, HIKARI Ltd, ttp://dx.doi.org/ /astp On Completeness of te System of Eigenfunctions of te Diriclet Problem for Lame Equations in te Strip Nikolai Plescinskii, Kristina Stekina and Dmitrii Tumakov Kazan Federal University 18 Kremlyovskaya St., Kazan, , Russian Federation Copyrigt c 2015 Nikolai Plescinskii, Kristina Stekina and Dmitrii Tumakov. Tis article is distributed under te Creative Commons Attribution License, wic permits unrestricted use, distribution, and reproduction in any medium, provided te original work is properly cited. Abstract In te present work, te Diriclet problem for Lame equations in te strip is considered. A caracteristic equation for te problem is obtained. It is sown tat te caracteristic equation as a denumerable number of simple roots. Te conclusion is drawn tat any solution to te considered problem can be represented in te form of an eigenfunction series. Keywords: Completeness, System of eigenfunctions, Diriclet problem, Lame equations. 1 Introduction One of key questions in matematical pysics is te completeness of eigenfunctions of specific problems. Terefore te question is often raised in researces. For example, te completeness of te generalized eigenfunctions of te differential operators of fractional orders is investigated in [1]. In [2] is studied te completeness of te set of te eigenfunctions corresponding to a system of two simultaneous Sturm-Liouville problems coupled by means of two different

2 812 Nikolai Plescinskii, Kristina Stekina and Dmitrii Tumakov spectral parameters. Te completeness of generalized eigenfunctions of general ordinary differential operators is investigated in [3]. Te completeness of generalized eigenfunctions of a polynomial nonselfadjoint operator pencil as been addressed in [4]. Te question of te completeness of root vectors generated by systems of coupled yperbolic equations is considered in [5]. Various approaces are used in representation of required functions at te solution to problems of te teory of elasticity [6]. Te classic presentation is te form of a vector and of a scalar displacement function, based upon te decomposition of a vector into te sum of a gradient and a curl [7]. Different views of solutions of Lame equations can also received [8 11]. Te completeness of te representations are proved in [12, 13]. Wen solving various problems, it often becomes necessary to represent a desired solution in te form of a basis-function series. For example, in te cylindrical system of coordinates, Bessel functions are used for te basis functions, wereas in te sperical system of coordinates, sperical functions are cosen for te basis-functions instead, etc. Completeness of suc classical systems of basis-functions is so obvious tat no one would ever raise any doubts on tis matter. However, wen it comes to solving various applied problems of matematical pysics, it often becomes necessary to expand a solution into an eigenfunction series [14]. For example, in problems of diffraction by waveguide structures, it is convenient to seek a field excited in te waveguide in te form of a combination of eigenwaves [15]. In te present work, completeness of te system of eigenfunctions of te Diriclet problem for Lame equations in te strip is investigated. Te initial boundary value problem (BVP) is reduced to te system of ordinary differential equations (ODE). For te obtained system, te caracteristic equation is developed. It is sown tat roots of te caracteristic equation and eigenvalues of a matrix of te equivalent system of differential equations are connected wit eac oter. In tis regard, a conclusion is drawn tat te caracteristic equation as a denumerable set of simple roots. As te main result, a teorem is formulated tat any solution of an initial BVP can be presented in te form of an eigenfunction series.

3 On completeness of te system of eigenfunctions Problem statement Let us consider te following system of differential equations (Lame equations) (λ + 2µ) 2 u x x 2 + µ 2 u x y 2 (λ + µ) 2 u x x y + u y µ 2 x 2 + ρω2 u x + (λ + µ) 2 u y x y = 0, + (λ + 2µ) 2 u y y 2 + ρω2 u y = 0 (1) in te strip 0 < y <. Let te following boundary conditions be applied to te two sides of te strip: u x (x, 0) = 0, u y (x, 0) = 0, (2) u x (x, ) = 0, u y (x, ) = 0. (3) Te purpose of te present work is to sow tat any solution to te BVP (1) (3) can be expanded into a series in terms of a set of particular solutions. 3 Infinite system of linear differential equations Let us seek a solution to te BVP (1) (3) in te form were u x (x, y) = a n (x)s n (y), u y (x, y) = b n (x)s n (y), (4) s n (y) = 2 πn sin y, n = 1, 2,... is an ortonormal system of functions at te interval [0, ] going to zero at te ends of te interval. Te following denotation is utilized ere: I mn = 0 s m (y)s n(y)dy, m, n = 1, 2,... Lemma 3.1 Te BVP (1) (3) is equivalent to te system of differential

4 814 Nikolai Plescinskii, Kristina Stekina and Dmitrii Tumakov equations [ ( ) ] πm 2 (λ + 2µ) c m(x) + ρω 2 µ a m (x) + (λ + µ) d n (x) I mn = 0, [ ( ) ] πm 2 (λ + µ) c n (x) I mn + ρω 2 (λ + 2µ) b m (x) + µ d m(x) = 0, a m(x) = c m (x), b m(x) = d m (x), m = 1, 2,... (5) Proof. Let us substitute expansion (4) into equation (1) and obtain te following: [ ( ) ] πn 2 (λ + 2µ) a n(x)s n (y) + ρω 2 µ a n (x)s n (y)+ +(λ + µ) b n(x)s n(y) = 0, [ ( ) ] πn 2 (λ + µ) a n(x)s n(y) + ρω 2 (λ + 2µ) b n (x)s n (y)+ (6) +µ ( iα j ) 2 b n(x)s n (y) = 0. By multiplying bot equations (6) by s m (y) and integrating wit respect to y from 0 to, one obtains [ ( ) ] πm 2 (λ + 2µ) a m(x) + ρω 2 µ a m (x) + (λ + µ) b n(x) I mn = 0, [ ( ) ] πm 2 (λ + µ) a n(x) I mn + ρω 2 (λ + 2µ) b m (x) + µ b m(x) = 0. (7) Next, let us introduce te following desired functions c m (x) = a m(x), d m (x) = b m(x), m = 1, 2,.... (8) and reduce te order of te system of equations (7). In doing tis, one obtains te lemma s statement. Te system of equations (5) represents an infinite system of linear differential equations of te first order wit constant coefficients of te type w = Mw,

5 On completeness of te system of eigenfunctions were M = ρω2 µ ( ) 2 π λ + 2µ 0 ρω2 (λ + 2µ) ( ) 2 π µ 0 0 (λ + µ)i 11 λ + 2µ (λ + µ)i 11 µ is a coefficient matrix and w = w(x) = (a 1 (x), b 1 (x), c 1 (x), d 1 (x), a 2 (x), b 2 (x), c 2 (x), d 2 (x),...) is an infinite vector of te desired functions. In order to find eigenvalues and eigenfunctions of te matrix M, we use a different metod of constructing te BVP (1) (3). 4 Caracteristic equation Let us seek a solution to te BVP (1) (3) in te form u x (x, y) = u x (y)e iαx, u y (x, y) = u y (y)e iαx, (9) were α is some number. In te teory of elasticity, te spectral parameter α is called a longitudinal constant of propagation of an elastic eigenwave. It follows from equations (1) tat te functions u x (y) and u y (y) must satisfy te system of ODE of te second order (λ + 2µ)( α 2 )u x + µu x + ρω 2 u x + (λ + µ)( iα)u y = 0, (λ + µ)( iα)u x + µ( α 2 )u y + (λ + 2µ)u y + ρω 2 u y = 0 (10) (all te derivatives ere are assumed to be wit respect to argument y). Let us reduce te order of te system of equations (10) by using te substitution of te desired functions v 1 = u x, v 2 = u x, v 3 = u y, v 4 = u y and turn to te system of equations of te first order of te type v = K v, were te

6 816 Nikolai Plescinskii, Kristina Stekina and Dmitrii Tumakov desired vector functions are v = (v 1, v 2, v 3, v 4 ) and te constant matrix is K = α 2 (λ + 2µ) ρω 2 Assume tat te following is true: µ 0 0 iα(λ + µ) µ γ j = γ j (α) = iα(λ + µ) λ + 2µ α 2 µ ρω 2 (λ + 2µ) 0. k 2 j α 2, j = 1, 2. (11) We coose single-value continuous brances of functions γ j (α), j = 1, 2 suc tat in one of te points of te positive imaginary semi-axis in te complex plane, wic is cut off at te interval of te real axis [ k j, k j ], te values are positive (e.g., γ j (i) = k 2 j + 1). Ten for te transformation γ j (α), te upper and lower alf-planes of te complex plane turn out to be te rigt and left alf-planes, respectively [16]. Te matrix K as eigenvalues ±iγ 1 and ±iγ 2 ; eigenvectors ( α, iγ 1 α, γ 1, ±iγ 2 1) and (γ 2, ±iγ 2 2, ±α, iγ 2 α) are consistent wit tese eigenvalues (ereinafter, dependence on parameter α will not be accounted for). Terefore, te general solution to te system of equations (10) takes te form: u x (y) = αa e iγ 1y + αb e iγ 1y + γ 2 C e iγ 2y + γ 2 D e iγ 2y, u y (y) = γ 1 A e iγ 1y + γ 1 B e iγ 1y + αc e iγ 2y αd e iγ 2y, (12) were A, B, C and D are some arbitrary constants. Let us coose te constants in equation (12) suc tat te boundary conditions (2) and (3) are satisfied. From te boundary condition at y = 0, it follows tat C = (A + B)α2 (A B)γ 1 γ 2 2αγ 2, (13) D = (A + B)α2 + (A B)γ 1 γ 2 2αγ 2. (14)

7 On completeness of te system of eigenfunctions Te constant B can be expressed via te constant A in te following manner: B = A (α2 + γ 1 γ 2 )e iγ 1 (α 2 γ 1 γ 2 )e i(γ 1+2γ 2 ) 2γ 1 γ 2 e i(2γ 1+γ 2 ) (α 2 γ 1 γ 2 )e iγ 1 2e iγ 2γ1 γ 2 + (α 2 + γ 1 γ 2 )e i(γ 1+2γ 2 ). (15) Substitution of expressions (13), (14) and (15) into te remaining condition (3) gives te following equation (α 4 + γ 2 1(α)γ 2 1(α))(1 e 2iγ 1(α) )(1 e 2iγ 2(α) )+ +2α 2 γ 1 (α)γ 2 (α)(1 + e 2iγ 1(α) )(1 + e 2iγ 2(α) ) (16) 8α 2 γ 1 (α)γ 2 (α)e i(γ 1(α)+γ 2 (α)) = 0. All possible values of α must satisfy te equation (16). Tus, te following statement is formulated. Lemma 4.1 Functions (9) are solutions to te BVP (1) (3), if and only if α is a root of te caracteristic equation (16). Tus, te functions u x (y), u y (y) ave te form (12); tey are determined wit accuracy up to a constant multiplier. 5 Completeness of te system of particular solutions Let us formulate one more auxiliary statement. Lemma 5.1 Te number iα j is an eigenvalue of te matrix M, if and only if α j is a root of te caracteristic equation (16). Proof. To every root α j of te caracteristic equation, tere corresponds a solution to te BVP (1) (3). Let us expand te functions u j x(y) and u j y(y) into a Fourier series in terms of functions s n (y): u j x(x, y) = u j xns n (y) e iαjx, u j y(x, y) = u j yns n (y) e iαjx. (17) Ten, for te given solution to te BVP, we ave a m (x) = u j xm e iα jx, b m (x) = u j ym e iα jx, m = 1, 2,... (18) and te vector function w j = (u j x1, u j y1, iα j u j x1, iα j u j y1, u j x2, u j y2, iα j u j x2, iα j u j y2,...)e iα jx (19)

8 818 Nikolai Plescinskii, Kristina Stekina and Dmitrii Tumakov is a solution to te system of equations (5). Terefore, iα j is an eigenvalue of te matrix M and (u j x1, u j y1, iα j u j x1, iα j u j y1, u j x2, u j y2, iα j u j x2, iα j u j y2,...) is an eigenvector corresponding to it. Te inverse statements also old. Using any solution to te system of equations (5), corresponding to a simple eigenvalue of te matrix M, it is easy to construct a solution to te BVP. Taking into account tat to every value of parameter α, tere corresponds a unique solution (wit accuracy up to a constant multiplier) of te BVP, te matrix of M as a simple spectrum, i.e. all of its eigenvalues are simple. It follows from tis tat, first, te caracteristic equation (16) as a denumerable set of simple roots. Second, te final result can be formulated as follows. Teorem 5.2 If α j, j = 1, 2,... are roots of te caracteristic equation (16), and u j x(x, y), u j y(x, y) are particular solutions to te BVP (1) (3), corresponding to tose roots, ten any solution to te problem can be represented in te form u x (x, y) = c j u j x(x, y), u y (x, y) = c j u j y(x, y), (20) j=1 j=1 were c j are some constants. Note tat if te number α is a root of te caracteristic equation, ten α is also a root. Tis statement follows from te fact tat α is introduced to te caracteristic equation only in even powers. Pysical interpretation of solutions to te BVP (1) (3) is as follows: every solution determines an elastic eigenwave of an elastic strip, te sides of wic are motionless. To te values of te spectral parameter α and α, tere correspond some waves transferring energy in te opposite directions. Te statement of teorem 1 is reduced to te statement tat any wave in an elastic strip wit te fixed walls is a result of superposition of a denumerable set of eigenwaves. 6 Conclusion Completeness of te system of eigenfunctions of te Diriclet problem for Lame equations in te strip is proven. As te result of te teorem, any solution of

9 On completeness of te system of eigenfunctions an initial boundary value problem can be presented in te form of an infinite eigenfunction series. Acknowledgements. Tis work was supported by te subsidy of te Russian Government to support te Program of competitive growt of Kazan Federal University among world class academic centers and universities References [1] A.V. Agibalova, On te completeness of system of eigenfunctions and associated functions of differential operators of orders 2 ε and 1 ε, J. Mat. Sciences, 174 (2011), no. 4, ttp://dx.doi.org/ /s [2] M. Faierman, M. Möller and B.A. Watson, Completeness teorems for a non-standard two-parameter eigenvalue problem, Int. Eqs. Oper. Teory, 60 (2007), ttp://dx.doi.org/ /s [3] J. Locker, Spectral teory of Non-self-adjoint Two-Point Differential Operators, Matematical Surveys and Monograps, AMS, 73 (1999). ttp://dx.doi.org/ /surv/073 [4] C. Tretter, Linear operator pencils A λb wit discrete spectrum, Integr. Eqs. Oper. Teory, 37 (2000), no. 3, ttp://dx.doi.org/ /bf [5] M.A. Subov, On te completeness of root vectors generated by systems of coupled yperbolic equations, Mat. Nacr., 287 (2014), no. 13, ttp://dx.doi.org/ /mana [6] P.P. Teodorescu, Treatise on Classical Elasticity: Teory and Related Problems, Springer, ttp://dx.doi.org/ / [7] B.G. Galerkin, Contribution à la solution générale du problème de la téorie de l élasticité dans le cas de trois dimensions, C. Rend. Hebd. des séances de l Acad. des Sci., 190 (1930), [8] H. Neuber, Ein neuer Ansatz zűr Lösung räumlicer Probleme der Elastizitätsteorie. Der Holkegel unter Einzellast als Beispiel, Z. Angew. Mat. Mec., 14 (1934), ttp://dx.doi.org/ /zamm

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