Spectral problems on compact graphs Текст научной статьи по специальности «Математика»

MSC 47A10

DOI: 10.14529/ mmp 170314

SPECTRAL PROBLEMS ON COMPACT GRAPHS S.I. Kadchenko1'2, S.N. Kakushkin1, G.A. Zakirova2

1Nosov Magnitogorsk State Technical University, Magnitogorsk, Russian Federation 2

E-mail: kadchenko@masu.ru, kakushkin-sergei@mail.ru, zakirovaga@susu.ru

The method of finding the eigenvalues and eigenfunetions of abstract discrete semi-bounded operators on compact graphs is developed. Linear formulas allowing to calculate the eigenvalues of these operators are obtained. The eigenvalues can be calculates starting from any of their numbers, regardless of whether the eigenvalues with previous numbers are known. Formulas allow us to solve the problem of computing all the necessary points of the spectrum of discrete semibounded operators defined on geometric graphs. The method for finding the eigenfunetions is based on the Galerkin method. The problem of choosing the basis functions underlying the construction of the solution of spectral problems generated by discrete semibounded operators is considered. An algorithm to construct the basis functions is developed. A computational experiment to find the eigenvalues and eigenfunetions of the Sturm - Liouville operator defined on a two-ribbed compact graph with standard gluing conditions is performed. The results of the computational experiment showed the high efficiency of the developed methods.

Keywords: perturbed operators; eigenvalues; eigenfunetions; compact graph; continuity conditions; Kirchhoff conditions.

1. Perturbed Operators on Compact Graphs. Recently, the methods of mathematical modelling began to play an important role in the study of the frequency-resonance characteristics of various technical devices described by linear dynamical systems and computer diagnostics of technical systems based on frequencies of natural oscillations. In this case, usually, mathematical model is direct or inverse spectral problem for Sturm -Liouville's operators on geometric graphs. The methods for finding eigenvalues and eigenfunetions of abstract discrete semibounded operators defined on compact graphs are developed in the article.

Let G = G(V, E) be a finite associated oriented compact graph. Here V — _

set of vertices, E — {Ej}j°=1 - set of edges. Suppose, that each edge Ej has the length of lj > 0 and cross-sectiona 1 area dj > 0 On the edges E of the graph G we consider the operator

T + P — (Ti + Pi,T2 + P2, ...,Tj0 + j acting in the Hilbert space

H = L2(G) = {g = {g1,g2,...,gjo), gj E L2{°,lj), j = l,jo}

with the scalar product [1]

j

(g, h) = dj gjhjdx, g, h E H. j=i 0

Bulletin of the South Ural State University. Ser. Mathematical Modelling, Programming

& Computer Software (Bulletin SUSU MMCS), 2017, vol. 10, no. 3, pp. 156-162

Here Tj are discrete semibounded operators, and Pj are bounded operators, defined in L2(0,lj) (j = 1, jo)- We consider the boundary value problem

(1) (2)

{Tj + Pj)uj = ^Uj, Uj = Uj(Xj), Xj E (0,lj), j = 1, jo

£

Ek&Ea(Vs)

d,

duk

dxk

xk =o

d

dum

Em&E^ (Vs)

xm —l

0,

m —lm

(3)

ui(0) = uk (0) = um(lm) = uh(lh), Ei, Ek e Ea(Vs), Em, Eh G E"(Vs), Vs e N.

Let Ea(Vs) denote a set of edges with origin at the vertex VS, and E"(Vs) - set of edges with an end at the vertex Vs. Conditions (2) mean that the flow through each vertex must be equal to zero, and (3) means that the solution u = (u1,u2, ■■■,uj0) at each vertex must be continuous. Also consider the boundary value problem:

Tj vj

Xvj, Vj = Vj(xj), xj E (0,lj), j = 1,jo,

d

dvk

Ek&Ea(Vs)

dxk

Xk =0

d

dvm

Em&E^ (Vs)

dxm

0,

xm —lm

(4)

(5)

(6)

^(0) = uk (0) = um(lm) = uh(lh), Ei, Ek G Ea(Vs), Em, Eh G E"(Vs), Vs e N.

We denote by [Xkeigenvalues of the problem (4) - (6), numbered in the order of nondecreasing of their magnitudes, and denote by {vk = (v1k,v2k,...,Vj0,k^^ -eigenvector-functions, corresponding to these eigenvalues Xk. Approximate solution of the problem (l)-(3) can be found in the form

u(n)

Vk,

k—1

(7)

where ak, with undetermined coefficients and vector-functions vk = (v1k,v2k, ■■■,Vj0>k), form a countable basis in the space L2(G) with energy norm \\vk||t+p, induced by the energy scalar product

(vk, Vm)T+P

30 n

((T + P)vk, vm) =Y, dj (Tj + Pj j—1 0

+ Pj )VjkVjmdx.

The space L2(G) with energy norm ||vk||T+P is denoted by HT+P.

Theorem 1. IfT + P is a semibounded from, below operator, acting in the Hilbert space L2(G), then solutions of the problem (4)-(6) form a basis in the energy space HT+P.

Доказательство. The system {vk is a basis in the Hilbert space HT+P in that case,

P

llvk IIT+p = (vk, vk)t+p = ((T + P)vk, v^ = (Tvk, vk) + (Pvk, vk) =

= (Xkvk, vk) + (Pvk, vk) = Xk||vk||2 + (Pvk, vk) < Xk||vkII2 + IlPvk|| • llvk|| < (8) < XkHvk||2 + HPH|vk||2 = (Xk + HP||) ||2.

The operator T is positive defined in space L2(G). Denote by HT energy space, which is a replenishment of space L2(G) % the norm \\vk\\T, induced by the energy scalar product, defined by the relation[2]:

JO j.

(Vk, vm)r = (Tvk, Vm) = dj TjVjkvjmdx.

3=1 0

Considering (4), we get, that \\vk\\T = (Tvk,vk) = \k\\vk\\2, from whence

\vk B2 = \v\. (9)

Ak

Let c be the lower bound of operator T + P. Then

\\vk\\2T+P = ((T + P)vk, vk) > c(vk, vk) = c(Tvk, vk) = c\\vk\\2T. (10) V / Ak Ak

Vector-functions vk are eigenfunctions of operator T and form a basis in L2(G) [1]. Therefore, the system of these functions is closed in L2(G), and tence, in HT. Considering (8) - (10), we get

f \\vk\\T <\vk\\T+p < (1 + P) • W\\T. Ak Ak

Therefore, HT+P consists of the same element, that HT, so that, the system of functions

{vkis ^teed in HT+P. □

Corollary 1. Further, by the theorem 1, when solving the problem (1) - (3) in the form (7), we will use the first n solutions of the problem (4) - (6) as coordinate functions vk, (k = 1, to). If necessary, the elements of the system {vk}n=1 should be normalized.

In the articles of S.I. Kadchenko [3, 4] linear formulas for eigenvalues of perturbed discrete operators were obtained. Analogously to these papers, we can prove the theorem.

Theorem 2. If T = (T1,T2, ...,Tj0) is a discrete semibounded operator, and P = (P1, P2,Pj0) - bounded operator, acting in a separable Hilbert space L2(G), and the system of vector-functions {vk = (v1k,v2k,...,Vj0>k^^ forms an orthonormal basis in L2(G), then eigenvalues ¡im of operator T + P can be found from formulas:

j0

Vm = Am + ^ dj PjVjmVjmdx + 8(m). (11)

3=1 0

m— 1

Here 8(m) = \Pjk(m — 1) — fik(m)], and fik(m) - m-th Galerkin's approximation of k-th k=1

eigenvalues.

Following the Galerkin method, coefficients ak (k = 1,n), included in (7), are found from the solution of a system of linear homogeneous equations

n

y^ak[(vk, vm)T+P — v(vk, vm)] = 0. (12)

k=1

I co Bulletin of the South Ural State University. Ser. Mathematical Modelling, Programming

° & Computer Software (Bulletin SUSU MMCS), 2017, vol. 10, no. 3, pp. 156-162

Using the theorem 2, by the formulas (11) we find n eigenvalues (k = l,n) of operator T + P. We substitute some fim into the system (12) instead of parameter Then the determinant of this system is equal to zero, and the system (12) will have nontrivial solutions. We denote the coefficients ak, included in (7) and corresponding to this solutions, via a^ (k = 1, n). We use the normalization condition (um(n), um(n)) = 1. Convert it, taking into account the system orthonormality }1k=v

(um(n), um(n)^j

Т.Т.^аГЫ, vi ) k=i i=i

E

i=i

(13)

Having supplemented the system of equations (12) by equation (13), we find the coefficients am (k = l,n).

2. Double-Edge Graph. As an example, consider a compact graph G, consisting of two edges E1 and E2 on Figure.

Graph G

On each edge Ej (j = 1, 2) we introduce the real parameter Xj, varying from 0 till lj. On graph G define a vector-function u = (u1, u2), which component Uj is a function of parameter Xj E [0,lj], corresponds to an edge Ej (j = 1, 2). On each edge Ej of the G

-u'j + qj (xj )uj

flUj.

(14)

u

standard gluing conditions, including the condition (2), analogous to the Kirchhoff's condition, and continuity condition (3). The continuity condition means, that, since the vertex V2 is an incident to the edges E1 and E2, then the values of the components of the vector-function u on these edges in the ends, corresponding to the vertex V2, are coincide:

ui(li) = u2(0).

(15)

Condition (2) means that the sum of normal derivatives of the components of the vector-function u in the vertices Vj (j = 1, 3) is equal to zero, i.e. if Vj corresponding to Xj = 0, then the derivative of the component Uj in the point, corresponds to the vertex Vj, is taken with a sign " + ", and with a sign " —", if Vj corresponds to Xj = lj-.

du1 dx1

+ du2 xi=li dx2

X2=0

(16)

In the boundary vertices Vi and V3 the conditions (2) transform to the Neumann's conditions:

dU =^ (17)

dx1 £i=o

2

1

0

du2 dx2

0.

Xi=l2

(18)

We use the system of coordinate functions {vk}n=1 to construct the solution of the problem (7), (14) - (18) and while finding the eigenvalues {^k}]n=1 by the formulas (11). To find the system we solve the boundary value problem:

—vj = Av3 , j = 1, 2, V1(h)= v2(0),

+ dvi -

x1=l1 dx2 x2=0

dV1 = 0,

xi=0

dv1 dx1

dx1 dV2

dx2

(19)

0.

Xl=l2

It can be shown that eigenvalues of the spectral problem (19) are

Ak = (^ )■

and eigenfunctions are

V1k (X1) = CkCosVAk xu

V2kx) = Ck cos y/Akh cos \f\kx2 — siny/Xk¡1 siny/Xkx2 Ck

Through vk = (v1k ,v2k) denote the vector-functions corresponding to the eigenvalues Xk. To find eigenvalues fik and vector-functions uk of the boundary problem (14) - (18) a computational experiment was conducted. Verification of the spectral characteristics was performed by substituting them into the equation (14). The Table shows the norms of the left and right side of the equation (14) and the difference between them.

Conclusion. New algorithm for finding eigenvalues of abstract discrete semibounded operators on geometric graph is developed. Numerous computational experiments have shown high computational efficiency of the algorithm.

0

References

1. Bayazitova A.A. The Sturm - Liouville Problem on Geometric Graph. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2010, no. 16 (192), issue 5, pp. 4-10. (in Russian)

2. Vlasova E.A., Zarubin V.S., Kuvyrkin G.N. Priblizhennyye metody matematicheskoy fiziki [Approximate Methods of Mathematical Physics]. Moscow, Bauman MSTU, 2004. 704 p.

3. Kadchenko S.I., Kakushkin S.N. The Numerical Methods of Eigenvalues and Eigenfunctions of Perturbed Self-Adjoin Operator Finding. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2012, no. 27 (286), issue 13, pp. 45-57. (in Russian)

4. Kadchenko S.I. Numerical Method for the Solution of Inverse Problems Generated by Perturbations of Self-Adjoint Operators by Method of Regularized Traces. Vestnik of Samara State University. Natural Science Series, 2013, no. 6 (107), pp. 23-30. (in Russian)

Bulletin of the South Ural State University. Ser. Mathematical Modelling, Programming

& Computer Software (Bulletin SUSU MMCS), 2017, vol. 10, no. 3, pp. 156-162

Table

Values ||(T + P)uk|| and uk|| of the boundary problem (14) - (18) calculated with parameters l1 = 1, l2 = 2, q1(x1) = sin(2x1 + 1) and q2(x2) = x\ + 3x2 + 2

k \\(T + P К || \\^k Uk\\ \\(T + P)uk - HkUk\\

1 10,4227164 9,1403925 1,2823239

2 15,5409132 14,3518941 1,1890191

3 23,0321518 22,0003709 1,0317808

4 32,7097768 31,9689829 0,7407939

5 44,5560760 43,9575040 0,5985720

6 58,7018393 58,1884571 0, 5133822

7 75,1222924 74,7029771 0,4193153

8 93,6696164 93,3045596 0, 3650569

18 400,5064008 400,3463671 0,1600337

19 443,2899290 443,1407540 0,1491750

20 488,2289128 488,0880478 0,1408650

21 535, 3685999 535,2323454 0,1362546

22 584, 731325)3 584,6030267 0,1282986

30 1058,4182713 1058,3240996 0,0941718

31 1127, 5178582 1127,4275559 0, 0903023

32 1198,7865687 1198,6994693 0,0870994

33 1272, 2515155 1272,1661447 0, 0853708

34 1347,9301120 1347, 8479328 0, 0821793

Received Aprile 21, 2017

УДК 519.624.3 Б01: 10.14529/ттр170314

РЕШЕНИЕ СПЕКТРАЛЬНЫХ ЗАДАЧ НА КОМПАКТНЫХ ГРАФАХ

С.И. Кадченко1'2, С.Н. Какушкин1, Г.А. Закирова2

1 Магнитогорский государственный технический университет им. Г.И. Носова,

г. Магнитогорск

2

Разработана методика нахождения собственных чисел и собственных функций абстрактных дискретных полуограниченных операторов, заданных на компактных графах. Получены линейные формулы, позволяющие с высокой вычислительной эффективностью вычислять собственные значения этих операторов, начиная с любого их номера, независимо от того, известны ли собственные значения с предыдущими номерами. Данные формулы решают проблему вычисления всех необходимых точек спектра дискретных полуограниченных операторов, заданных на геометрических графах.

Собственные функции находятся на основе метода Галеркина. Рассмотрен вопрос выбора базисных функций, лежащих в основе построения решения спектральных задач, порожденных дискретными полуограниченными операторами, и приводится алгоритм их построения. Проведен вычислительный эксперимент по нахождению собственных чисел и собственных функций оператора Штурма - Диувилля, заданного на двухре-берном компактном графе со стандартными условиями склейки. Результаты вычислительных экспериментов показали высокую эффективность разработанной методики.

Ключевые слова: возмущенные операторы; собственные числа; собственные функции; компактный граф; условия непрерывности; условия Кирхгофа.

Литература

1. Баязитова, A.A. Задача Штурма - Диувилля на геометрическом графе / A.A. Баязито-ва // Вестник ЮУрГУ. Серия: Математическое моделирование и программирование. -2010. - № 16 (192), вып. 5. - С. 4-10.

2. Власова, Е.А. Приближенные методы математической физики / Е.А. Власова, B.C. Зарубин, Г.Н. Кувыркни. - М.: Изд-во МГТУ им. Н.Э. Баумана, 2004. - 704 с.

3. Кадченко, С.И. Численные методы нахождения собственных чисел и собственных функций возмущенных самосопряженных операторов / С.И. Кадченко, С.Н. Какушкин // Вестник ЮУрГУ. Серия: Математическое моделирование и программирование. - 2012. -№ 27 (286), вып. 13. - С. 45-57.

4. Кадченко, С.И. Численный метод решения обратных задач, порожденных возмущенными самосопряженными операторами, методом регуляризованных следов / С.И. Кадченко // Вестник Самарского университета. Естественнонаучная серия. - 2013. - № 6 (7). -С. 23-30.

Сергей Иванович Кадченко, доктор физико-математических наук, профессор, кафедра «Прикладная математика и информатика:», Магнитогорский государственный технический университет им. Г.И. Носова (г. Магнитогорск, Российская Федерация); кафедра «Уравнения математической физики», Южно-Уральский государственный университет (г. Челябинск, Российская Федерация), kadchenko@masu.ru.

Сергей Николаевич Какушкин, КсШДИДсХТ физико-математических наук, кафедра «Прикладная математика и информатика», Магнитогорский государственный технический университет им. Г.И. Носова (г. Магнитогорск, Российская Федерация), kakushkin-sergei@mail.ru.

Галия Амрулловна Закирова, КсШДИДсХТ физико-математических наук, доцент, «» государственный

университет (г. Челябинск, Российская Федерация), zakirovaga@susu.ru.

Поступила в редакцию 21 а,прем,я 2011 г.

Bulletin of the South Ural State University. Ser. Mathematical Modelling, Programming

& Computer Software (Bulletin SUSU MMCS), 2017, vol. 10, no. 3, pp. 156-162