CC BY
7
MSC 47A10
DOI: 10.14529/ mmp210205
ALGORITHM FOR NUMERICAL SOLUTION OF INVERSE SPECTRAL PROBLEMS GENERATED BY STURM-LIOUVILLE OPERATORS OF AN ARBITRARY EVEN ORDER
S.I. Kadchenko1, L.S. Ryazanova1, Yu.R. Dzhiganchina2
1Nosov Magnitogorsk State Technical University, Magnitogorsk, Russian Federation 2Bauman Moscow State Technical University, Moscow, Russian Federation E-mails: [email protected], [email protected], [email protected]
The article is devoted to the construction of algorithm for solving inverse spectral problems generated by Sturm-Liouville differential operators of an arbitrary even order. The goal of solving inverse spectral problems is to recover operators from their spectral characteristics and spectral characteristics of auxiliary problems. In the scientific literature, we can not find examples of the numerical solution of inverse spectral problems for the Sturm-Liouville operator of higher than the second order. However, their solution is caused by the need to construct mathematical models of many processes arising in science and technology. Therefore, the development of computationally efficient algorithm for the numerical solution of inverse spectral problems generated by the Sturm-Liouville operators of an arbitrary even order is of great scientific interest.
In this article, we use linear formulas obtained earlier in order to find the eigenvalues of discrete semi-bounded operators and develop algorithm for solving inverse spectral problems for Sturm-Liouville operators of an arbitrary even order.
The results of the performed computational experiments show that the use of the algorithm developed in the article makes it possible to recover the values of the potentials in the Sturm-Liouville operators of any necessary even order.
Keywords: eigenvalues and eigenfunctions; discrete, self-adjoint and semi-bounded operators; Galerkin method; ill-posed problems; Fredholm integral equations of the first kind; asymptotic formulas.
Introduction
The first substantiations of linear formulas for calculating the eigenvalues of discrete semi-bounded operators were made on the basis of the method of regularized traces in the papers [1-5]. Following them, consider a discrete semi-bounded below operator T and a bounded operator P defined in a separable Hilbert space H. Let the eigenvalues {An and the orthonormal eigenfunctions {vn}c^=1 of the operator T be known and enumerated in the non-decreasing order of their values. Denote by vn the multiplicity of the eigenvalue An. Let n0 be the number of all unequal eigenvalues An that belong to the inner part of
the circle Tno of radius pno = I centered at the origin of the complex plane. If,
2||PII
for all n G N, the inequalities qn = —-—- < 1 hold, then the approximate first mo
^ |An+V„ — An 1
eigenvalues |^n}m= 1 of the operator T + P are calculated by the formulas [2]
l^n = Ara + (Pvn,vn) + 8n, n = l,rrio• (1)
Moreover, the numbers 8n satisfy the estimates
2 n о
|<g < (2n - 1 )pn~,-, Q = maxg„, m0 = V];
1 — q nf N —'
n=1
1 — q n€N
Formulas (1) were obtained under the condition
||P|| < 0,5|An+v„ — An|, Vn G N. (2)
Further investigations of the problem on calculating the eigenvalues of discrete semi-bounded operators showed that the restrictions (2) on the norm of the operator P can be removed if the Galerkin method is used to find the eigenvalues of discrete semi-bounded below operators.
In a separable Hilbert space H, consider a discrete semi-bounded operator L with the domain DL C H. The eigenvalues p of the differential operator L are determined by finding solutions to the equation
Lu = pu, (3)
satisfying homogeneous boundary conditions
Gu|r = 0, (4)
where r is the boundary of the domain DL. To calculate them, we use the Galerkin method. In H, we consider the complete sequence (Hn}^=1 of the finite-dimensional subspaces Hn C H. Let the systems of functions }n=1 be orthonormal bases of the spaces Hn. Moreover, suppose that all basis functions satisfy boundary conditions (4). Let us find an approximate solution to spectral problem (3), (4) in the form
Un = ak (n)ifk.
k=i
Theorem 1. Galerkin method, which is applied to the problem on finding eigenvalues of spectral problem (3), (4) and constructed on the system of functions }fc=1; converges [6].
Theorem 2. There exists a unique solution to the problem on finding the eigenvalues and eigenfunctions of the operator L. The approximate values of the eigenvalues can be found by the Galerkin method [6].
Theorem 3. The approximate eigenvalues pn of the operator L are found by the formulas [7]
ptn(n) = (L^n, <£n) + ^n, (5)
n— 1
where 5n = J2 [pfc(n — 1) — pfc(n)]. k= 1
From the theorems, we have
lim 5n = 0. (6)
n
n^-rx
Therefore, as the ordinal number of the eigenvalue calculated according to formulas (5) increases, the accuracy of its calculation increases, and this is confirmed by numerous calculations. Using the example of spectral problems generated by Sturm-Liouville operators of an arbitrary even order, which are considered in the article, it is shown
that linear formulas (5) differ from the known asymptotic formulas only by the order of errors [4]. Compared to classical methods, the formulas drastically reduce the amount of computation and allow to calculate the eigenvalues of discrete semi-bounded operators with any ordinal numbers. The formulas can be used by specialists with no special knowledge in the field of spectral operator theory.
In the articles [3,7], algorithm for solving inverse spectral problems for discrete semi-bounded operators are developed. In this case, numerical examples were considered for differential operators of the second order. Below, using the example of Sturm-Liouville operators of an arbitrary even order, we investigate the possibility of using the developed algorithm for cases where the order of operators is greater than two.
1. Sturm—Liouville Operators of Arbitrary Even Order
Consider a method for solving inverse spectral problems based on formulas (5), using the example of problems generated by Sturm-Liouville operators of an arbitrary even order of the form
Lm«2m(s) = ^2m«2m(s), m G N, 0 < S < n, (7)
4m"1)(0) = 42r1)W = 0, u=l(8)
where L2mu2m{s) = (r2m + p2m(s)^ju2m(s), T2mu2m(s) = (-l)"7^ Recover the
potentials p2m(s) by the eigenvalues ^2mn of boundary value problems (7), (8) that belong to the segments [c2m,d2m], eigenvalues {A2mn and eigenfunctions {v2mn of the corresponding unperturbed boundary value problems
T2mV2m(s) = A2m^2m(s), m G N, 0 < S < n, (9)
^2r1)(0) = ^2r1)W = 0, v = T^i. (10)
In the works [8-10], it is shown that (9), (10) are self-adjoint problems, the eigenvalues A2mn and the corresponding eigenfunctions v2mn are of the form
J-, ra = 0,
A2mn = n2m, v2mn = ancos(ns), an={ ^ ; Vm e N. (11)
\ —, n> 0. V n
It is known that the system of functions {v2mn = an cos(ns)}£=0 is an orthonormal basis of the space L2[0,n]. Moreover, all functions v2mn satisfy boundary conditions (10). Let us write formulas (5) for this system
I^2mn (n) = (LV2mn , ^2mn ) + &2mn = A2mn + (P2m^2mn , ^2mn ) + <W
n " 211 On / 1 211 On 211 On ' \± 2110 211 On ' 211 On J
77 ~
n2m + an/ P2m(s) cos2 (ns)ds + ¿2mn •
(12)
0
Using (12), we construct the Fredholm integral equations of the first kind
Ap2m = J K(x2,„,SK„(S)<iS = f2m(x2m), x2m G [c2m_d2m]. <13)
0
Here K(x2mn, s) = o2n cos2(ns). The segments [c2m,d2m] are chosen such that they contain the required number of known eigenvalues of spectral problems (7), (8). The exact values of the right-hand sides f2m of equations (13) are unknown, but we know their approximate values /2m (X2mn ) = F2mn (n) - n2m - ¿2mn , for which 11 f2m - /2m|||2[c2m,d2m] < ^2m. Let the kernels K(x2m, s) of integral equations (13) be continuous in n2m = [c2m, d2m] x [0,n] and
P2m € W22m[0 , n], /2m (x) € ¿2[c2m ,^2m].
The problems of solving Fredholm integral equations of the first kind (13) are ill-posed. Their approximate numerical solutions F2m are found by the quadrature method, passing to finite subspaces. Let us introduce discretization grids along the s and x2m axes, which are uniform along the s and non-uniform along the x2m axes with an equal number Ns of nodal points.
si = 0, sra+i = sn + hs, n = 1,N - 1, hs
n
Ns - 1
c2m x2mi < x2m2 < ... < x2mNs d2m> h2mi h2m2,
h2mk = x2mk — x2mk-1, k = 2, Ns — 1.
To calculate the definite integrals in equations (13), we interpolate the functions p2m(s) by the Lagrange polynomials
Ns
Ns
P2m(s) = ^ ln(s)'P2mn , ln(s) = Д
s-s
j
n=1
• sn sj j=1 ./
j=n
(14)
Substituting (14) into integral equations (13), we find systems of linear algebraic equations for the values of the functions p2m at the discretization nodes p2mn = p2m(sn)
Ns
Hkn'P2mn = /2m (x2mk ),
(15)
n=1
where iifcra = J ln(s)K(x2mk, s)ds, k,n=l,Ns. 0
Let us write the constructed system of equations (15) in the matrix form
AP2m = Fm. (16)
Here A is a square matrix of the order Ns x Ns, and the column matrices P2m, F2m have the form
/ Hn ... Hins \
A = H21 . . . H2Ns
\ HNS1 • • • HNSNS J
( P2mi \ / •F2m (x2mi)
p2m2
V P2mNs )
2m =
2m =
/*2m (x2m2 ) )
Since the condition number cond(A) of the matrix A of systems of equations (16) is relatively large, and the components f2m(x2mk) of the vectors F2m in the right-hand sides of these systems are approximate, then relative errors can lead to rather large errors of the vectors P2m. In this regard, in order to partially eliminate the undesirable effects of the influence of errors, it is necessary to apply various regularization methods in order to replace the inadmissible solution vector with some "pseudosolution" vector, which is the best for the problem under consideration.
To find "pseudosolutions" to systems of linear algebraic equations (16), we use the Tikhonov regularization method with the choice of the residual regularization parameters. The regularization method of A.N. Tikhonov [12-15] is reduced to minimizing parametric functionals of the form
||AP2m — -^mll + a2m||P2m|| , (17)
where a2m > 0 is the regularization parameter, ||P2m|| is the Euclidean norm of the vector P2m. The problem of minimizing functionals (17) with respect to P2m is equivalent to solving the system of linear algebraic equations
(AT A + a2m/)P2amm = AT F2m, (18)
in this case, the parameters a2m are found by the conditions
HAPar - Fm || = E ||F2m||. (19)
Here POT are vectors of approximate solutions to integral equations (13). The superscript T in (18) means transposition of the matrix A, while E is the specified residual level and I is the identity matrix.
For fixed a2m, the unique solutions P2mm to the problem of minimizing functionals (17) are explicitly expressed by the formulas:
D«2m _
P
2m = (AT A + a 2m/) V F2m • (20)
For a2m ^ 0, the solutions P2mm converge to normal solutions to systems (16) [12]. In (16), inverse matrices exist for any rank of the matrix A and any a2m > 0 [15].
Another method for finding approximate solutions to Fredholm integral equations of the first kind (13), which we use, is based on the construction of A.N. Tikhonov [12,16] smoothing functionals of the form
d2m n
tem./2mi= / [/K<x2m.«w«^-/u*)]^ + ^fej. (2i)
C2m 0
7T
where H2m[p2m] = /[p2m(s) + qp'2m(s)]ds are stabilizing functionals, q > 0. 0
At the discretization nodes of the domain [0, n], the approximate values of the functions P02T are found by the condition that functionals (21) reach the minimum values
&, /2m] = mi i k] (22)
under the boundary conditions
Äm (0)= Äm (п) = о
(23)
Formula (22) implies the Tikhonov equations, which, in these cases, have the form:
n d2m 1
(t) - qfiVm (t)] + / I K(X2m,t)K(X2m, s)dx2m S" (s)ds =
0 LC2m J (24)
(¿2эт
/ K(X2m, t) f2m(%2m)dX2m, 0 < t < П.
С2и
To find the regularization parameters a2m in equations (24), we use the generalized residual method, which, as applied to Fredholm integral equations of the first kind (13), is reduced to solution of the equations
d.2n
C2m 0
K (X2m, s)P2mm (s)ds - /2m(x2m)
2
2m
0.
(25)
In discrete form, equations (25) can be written as
Ns Ns
"Yh Yfc "Yh ( HknPÏmZ - /2mfc
fc=1 n=1
0.
(26)
With this approach, instead of incorrect integral equations (13), we solve either integral equations of the second kind (24) for q = 0 or integro-differential equations for q > 0 taking into account boundary conditions (23).
Discrete versions of Tikhonov equations (24) take the form
Ns £
n=1
®2m
In (t) - q/ra (t) + Gn (t) pa2m = F2mk (t), t G [0,п].
(27)
Ns _ Ns _
Here Gra(t) = E YkK(x mk , t)Hkn F?2m(t) = E IkK(x2mfc , t)/2mfc , Ik are weighting factors k=1 k=1 in the quadrature formula of trapezoids with a variable step. Introducing discretization on
the segment [0,n] with an equal number Ns of nodal points
7r
ti = 0, tfc+i = tk + hs, k = 1,N - 1, hs
Ns - 1
we obtain a system of linear algebraic equations for finding the values of the functions pa2m at the discretization nodes
Ns s
n=1
®2m
Utk) - qln{tk) + Gn{tk) = F2mk(tk), к = 1, Ns.
(28)
Summarizing the above, we formulate an algorithm for the numerical solution of inverse spectral problems (7), (8) using the developed method.
Algorithm. For specific values of m, consider the given Ns eigenvalues G [c2m, d2m] of spectral problems (7), (8), eigenvalues {A2mn}N= 1 and eigenfunctions {v2mn}N= 1 of the corresponding perturbed problems (9), (10).
1. Construct integral equation (13).
2. Using the quadrature method, write system of linear equations (16) or (28) depending
П
2
on the applied method of recovering the values of the functions p^T (s) at the sampling nodes.
3. Using the generalized residual method to select the regularization parameters a2m and conditions (19) or (26), find a2m.
4. Using the found regularization parameter a2m and formulas (20) or solving systems (28), find PZm.
5. To estimate the accuracy of the obtained approximate solution (pm(sn)}%= i, find
and the residual £2m
Dn) J n=
1 Ns _
the average absolute error (2m = — E Hknp^^ds - f2mk
Ns k,n=l
Ns Ns / _ N 2
2m
2 m„
k=l n=l
Ns Ns / ^ \ 2
EY^ ( Hknffimn - f2mkj of equation (13). u—1 « —1 V /
2. Computational Experiment
Using the algorithm described above for solving inverse spectral problems (7), (8) generated by Sturm-Liouville operators of an arbitrary even order, in the Maple mathematical environment, computational experiments were carried out to recover the values of the potentials p2m using the eigenvalues of spectral problems (7), (8) that belong to the segments [c2m ,d2m], the spectra and eigenfunctions of the corresponding unperturbed problems (9), (10). In Maple software environment, using the reserved constant Digits, the operations with real numbers can be performed with a given mantissa, which allows to find numerical solutions to Fredholm integral equations of the first kind (13). All the calculation results given below were obtained with Digits=541.
Table 1
Reconstructed values of potentials at discretization nodes for Sturm-Liouville operators of various even orders
n Sn P(sn) P?(Sn) pT(Sn) Pe6(sn) P166 (sn) P322(s")
1 0,0000 2, 0000 1,8647 1,8596 1,8596 1,8596 1,8596
2 0, 2244 3,1724 4, 0700 4, 0645 4, 0645 4, 0645 4,0645
3 0,4488 4, 4454 5, 8664 5, 8630 5, 8630 5, 8630 5, 8630
4 0, 6732 5,8192 7, 3732 7, 3737 7, 3737 7, 3737 7, 3737
5 0, 8976 7, 2937 8, 6902 8, 6954 8, 6954 8, 6954 8, 6954
6 1,1220 8, 8689 9, 9029 9,9124 9,9123 9,9123 9,9123
7 1,3464 10, 5448 11,0864 11,0986 11,0986 11,0986 11,0986
8 1,5708 12,3214 12,3087 12,3214 12,3214 12,3214 12,3214
9 1,7952 14,1987 13,6343 13,6450 13,6449 13, 6449 13, 6449
10 2,0196 16,1767 15,1269 15,1333 15,1333 15,1333 15,1333
11 2, 2440 18,2555 16,8533 16,8538 16,8538 16,8538 16,8538
12 2,4684 20,4349 18,8866 18,8804 18, 8404 18, 8804 18, 8804
13 2,6928 22,7151 21,3099 21,2974 21,2975 21,2975 21,2975
14 2,9172 24,0960 24,2211 24, 2038 24, 2039 24, 2039 24, 2039
15 3,1416 27, 5776 27, 7382 27,7179 27,7180 27,7180 27,7180
C2 m 1,48- 10~27 3,08•IO-31 1,14 • 10~34 1,10- 10-32 3,35- IO-79
£2 m 3,71 • IO-51 7, 63 • IO-55 3, 67 • 10~5S 3,68 • IO-77 4,24 • 10-1U3
To check the developed algorithm, using Tikhonov equations (24), the values of the same potentials p2m(s) = s2 + 5s + 2 were recovered at the discretization nodes for different orders 2m (m = 1, 2, 3, 4, 8) of the differential operators in inverse spectral problems (7), (8). Some of the results of these calculations are shown in Table 1.
In this case, the number of eigenvalues of spectral problems (7), (8) that belong to the segment [c2m, d2m] used in the algorithm must coincide with the number of discretization nodes of the segment [0,n] in which the values of the potentials pOT of spectral problems (7), (8) are found. The calculations were carried out using systems (28).
The calculation results given in Table 1 show that the developed algorithm for solving inverse spectral problems (7), (8) are computationally efficient and allow to recover the values of the potentials p2m for different orders of differential operators with good accuracy.
The algorithm developed above assume that any segments [c2m, d2m] containing their eigenvalues can be taken to recover the potentials p2m in problems (7), (8). Table 2 shows the results of calculations that allow to restore the values at discretization nodes using different segments [c6,d6] for the potential p6 = sin(s) of spectral problem (7), (8) with m = 3. The calculations were carried out using formulas (20).
Table 2
The reconstructed values of the potential at the nodes of discretization using various
segments containing the known eigenvalues of problem (7), (8)
п P6(s„) P66(sn) 102 P66(sn) IO10 РбЧвп) 1012
CQ = 7, 296184 CQ = 6, 252350 • CQ = 2,081952 •
d6 = 2,413757 - 107 d6 = 2,084224 1011 d6 = 4,195873 • 1012
a6 = 1,985200 • io-34 a6 = 2, 288800 • 10-75 a6 = 1,500000- 10-85
1 0 000000 0,000000 0,000021 0, 000026 0, 000026
2 0 224399 0,222521 0, 222540 0, 222544 0, 222544
3 0 448799 0,433884 0,433897 0, 433900 0, 433900
4 0 673198 0, 623490 0, 623494 0, 623496 0, 623496
5 0 897598 0,781831 0,781827 0,781826 0,781826
6 1 121997 0, 900969 0, 900956 0, 900953 0, 900953
7 1 346397 0, 974928 0, 974909 0, 974905 0, 974905
8 1 570796 1,000000 0, 999979 0, 999974 0, 999974
9 1 795196 0, 974928 0, 974909 0, 974905 0, 974905
10 2 019595 0, 900969 0, 900956 0, 900953 0, 900953
И 2 243995 0,781831 0,781827 0,781826 0,781826
12 2 468394 0, 623490 0, 623494 0, 623496 0, 623496
13 2 692794 0,433884 0,433897 0, 433900 0, 433900
14 2 917193 0,222521 0, 222540 0, 222544 0, 222544
15 3 141593 0,000000 0,000021 0, 000026 0, 000026
Се 6,518733-10 -21 4,160036 • 10 -43 8,241057-10- -4У
£6 1,055714-10 -■¿■A 3,581863-10 -74 2,001959-10- -84
The results of calculations showed that the recovering approximate values of the potential p6 = sin(s) using the developed algorithm practically does not depend on the choice of the segments [c6,d6].
Conclusion
Using the example of spectral problems generated by Sturm-Liouville operators of an arbitrary even order (7), (8), we developed efficient computational algorithm that allow to find the values of the potentials p2m(s) by the eigenvalues ^2mn of these problems that belong to the segments [c2m, d2m], eigenvalues {A2mn and eigenfunctions {v2mn of the corresponding unperturbed boundary value problems (9), (10).
The results of the performed computational experiments confirm the correctness of the idea to develop a new method for solving inverse spectral problems (7), (8) based on the
use of formulas (5). It should be noted that the algorithm for recovering the values of the potentials are quite simple and computationally efficient in their numerical implementation for any m. The order of the differential operators is 2m and, in inverse spectral problems (7), (8), does not lead to additional computational difficulties when finding their numerical solutions.
References
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Received March 3, 2021
УДК 519.624.3 Б01: 10.14529/шшр210205
АЛГОРИТМЫ ЧИСЛЕННОГО РЕШЕНИЯ ОБРАТНЫХ СПЕКТРАЛЬНЫХ ЗАДАЧ, ПОРОЖДЕННЫХ ОПЕРАТОРАМИ ШТУРМА - ЛИУВИЛЛЯ ПРОИЗВОЛЬНОГО ЧЕТНОГО ПОРЯДКА
С.И. Кадченко1, Л.С. Рязанова1, Ю.Р. Джиганчина2
1 Магнитогорский государственный технический университет им. Г.И. Носова, г. Магнитогорск, Российская Федерация
2Московский государственный технический университет имени Н.Э. Баумана, г. Москва, Российская Федерация
Статья посвящена построению алгоритмов решения обратных спектральных задач, порожденных дифференциальными операторами Штурма - Лиувилля произвольного четного порядка. Целью решения обратных спектральных задач является восстановление операторов по их спектральным характеристикам и спектральным характеристикам вспомогательных задач. В научной литературе примеров численного решения обратных спектральных задач для оператора Штурма-Лиувилля выше второго порядка, мы не встречали, хотя их решение вызвано необходимостью построения математических моделей многих процессов возникающих в науке и технике. Поэтому разработка вычислительно эффективных алгоритмов численного решения обратных спектральных задач, порожденных операторами Штурма - Лиувилля произвольного четного порядка, представляет большой научный интерес.
В статье, используя линейные формулы, полученные ранее, для нахождения собственных значений дискретных полуограниченных операторов, разработаны алгоритмы решения обратных спектральных задач для операторов Штурма - Лиувилля произвольного четного порядка.
Результаты проведенных вычислительных экспериментов показали, что используя разработанные в статье алгоритмы можно восстанавливать значения потенциалов в операторах Штурма - Лиувилля любого необходимого четного порядка.
Ключевые слова: собственные значения и собственные функции; дискретные, самосопряженные и полуограниченные операторы; метод Галеркина; некорректно поставленные задачи; интегральные уравнения Фредгольма первого рода; асимптотические формулы.
Литература
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Сергей Иванович Кадченко, доктор физико-математических наук, профессор, профессор кафедры «Прикладная математика и информатика:», Магнитогорский государственный технический университет имени Г.И. Носова (г. Магнитогорск, Российская Федерация), [email protected].
Любовь Сергеевна Рязанова, кандидат педагогических наук, доцент, доцент кафедры «Прикладная математика и информатика», Магнитогорский государственный технический университет имени Г.И. Носова (г. Магнитогорск, Российская Федерация), [email protected].
Юлия Рустамовна Джиганчина, студентка, Московский государственный технический университет имени Н.Э. Баумана (г. Москва, Российская Федерация), [email protected].
Поступила в редакцию 3 марта 2021 г.
