Multipoint initial-final problem for one class of Sobolev type models of higher order with additive "white noise" Текст научной статьи по специальности «Математика»

MSC 35G05, 35G16, 47D09, 60H30

DOI: 10.14529/mmp 180308

MULTIPOINT INITIAL-FINAL PROBLEM FOR ONE CLASS OF SOBOLEV TYPE MODELS OF HIGHER ORDER WITH ADDITIVE "WHITE NOISE"

G.A. Sviridyuk1, A.A. Zamyshlyaeva1, S.A. Zagrebina1

^outh Ural State University, Chelyabinsk, Russian Federation E-mail: sviridyuk@susu.ru, zamyshliaevaaa@susu.ru, zagrebinasa@susu.ru

Sobolev type equations theory has been an object of interest in recent years, with much attention being devoted to deterministic equations and systems. Still, there are also mathematical models containing random perturbation, such as white noise. A new concept of "white noise", originally constructed for finite dimensional spaces, is extended here to the case of infinite dimensional spaces. The main purpose is to develop stochastic higherorder Sobolev type equations theory and provide some practical applications. The main idea is to construct "noise" spaces using the Nelson-Gliklikh derivative. Abstract results concerning initial-final problems for higher order Sobolev type equations are applied to the Boussinesq-Love model with additive "white noise". We also use well-known methods in the investigation of Sobolev type equations, such as the phase space method, which reduces a singular equation to a regular one, as defined on some subspace of the initial space.

Keywords: Sobolev type equation; propagator; "white noise"; Wiener K-process; multipoint initial-final problem.

Dedicated, to Professor Jan Kisyrlski on the occasion

of his 85th birthday.

Introduction

Sobolev type equations make up a vast area of nonclassical equations of mathematical physics. Their systematic study started in the middle of the twentieth century with the seminal work of S.L. Sobolev, though many such equations had been studied earlier on; we recall, in particular, the famous Xavicr Siokcs equation system (see the excellent review in [1]). Recently, there has been a major increase in the research of Sobolev type equations. We should mention several monographs about these problems [2-7]. Different aspects of the incomplete higher-order Sobolev type equations

Au(n) = Bu + g, (1)

with the assumption ker A = {0}, have been studied [8-11]. Here the operators A, B E L(U;F) (i-e- linear and continuous), U and F are Banach spaces, absolute term g = g(t) models the external force, and n > 2 is a natural number. One of the prototypes of equation

(1) 1S (A - A)vtt = aAv + g, (2)

which models, among others, the incompressible fluid free surface perturbation under the assumption of motion potentiality and conservation of mass in a layer [12], longitudinal vibrations of an elastic rod [13] and wave processes in smectic and plasma [14].

The shortcoming of the model (2) with the deterministic absolute term is that in natural experiments the system is exposed to random perturbation, for example in the form

of white noise. Stochastic ordinary differential equations with different additive random processes (i.e. not only white noise, but more general Markov and diffusion processes) are now actively studied [15]. The traditional Ito-Stratonovich-Skorohod approach is the most widely followed, although new and very promising avenues of research have recently appeared [11,16].

The first results concerning stochastic Sobolev type equations of the first order can be found in [17]. They are based on the extension of the Ito-Stratonovich-Skorokhod method to partial differential equations (see, for example, [18-20]). In this paper, the stochastic higher order Sobolev type equation

An(n) = Bn + Nw (3)

is considered. Here, w in the right hand side denotes the random process. It is required to find the random process n(t), satisfying (in some sense) equation (3) and the multipoint initial-final conditions

Pj(n[k) (Tj) - j) = 0, j = 0m, k = 0, n — 1, (4)

where Tj E R with Tj < Tj+1, j = 0,m, j k = 0,n — 1, are given random variables, and Pj

w

Wiener process. Later, a new approach to the investigation of equation (3) appeared [15] and is being actively developed [16, 21-23], where "white noise" means the Nelson-Gliklikh [15, 24] derivative of the Wiener process. This "white noise" was first used in optimal measurement theory [25,26], which constructs a special space of "noises". The concept of "white noise" in this theory (that is, only in the finite dimensional spaces) proved to be highly efficient, therefore suggesting to extend the concept to infinite-dimensional spaces [17,27]. The main goal of this extension is to develop a theory of stochastic Sobolev type equations and its applications to nonclassical models of mathematical physics of practical importance [28].

Besides the introduction, the paper consists of three sections. The first one deals with the deterministic inhomogeneous linear Sobolev type equation of higher order. We define a multipoint initial-final problem and state a theorem on the existence of a unique solution. We borrowed results from [10,29] and therefore give them without proofs. The second section extends the deterministic results of the first one to the stochastic setup by analogy with [23]; sketches of proofs complement the results. In the third section we consider the linear stochastic Boussinesq-Love equation. In conclusion, we outline possible directions for further research.

1. A Deterministic Linear Sobolev Type Equation of Higher Order with Relatively p-Bounded Operators

Let U and F be separable Hilbert spaces, operators A,B E L(U;F)- Following [18], A

pA(B) = [v E C : (M — B)-1 E L(F;U)}

and an A-spectrum aA(B) = C \ pA (B) of operator B. The operator-functions (vA — B)-1, Ra(B) = (vA — B)-1A, La(B) = A(^A — B)-1 with the domain pA(B) are called

the A-resolvent, the right and the left A-resolvents of operator B correspondingly. If the set aA(B) is bounded (i.e. there exists a > 0 : \j\ < a for all j E aA(B)) then the operator B is railed (A, a)-bounded.

Let the operator B be (A, a)-bounded, p E {0} U N. Construct the set ) = {j E C : jn E aA (B)}; it is compact in C due to the compactness of the A-spectrum of operator B. Take the contour y = {j E C : \j\ = r,rn > a} that bounds the domain containing the points of aA(B) and construct the projectors

P

1

lin-lRAn (B)d» e L(U),

1

2ni

Q = I (B)d» e L(F).

Here, R^n(B) = (^nA - BL^(B) = A(^nA - B)-1. Set U0(U1) = ker P(imP), F0(F1) = ker Q(imQ). Thus, the spaces U and F can be decomposed into direct sums U = U0 © U1 and F = F0 © F1, where as U0 D ker A. By Ak (Bk ) define the restriction of operator A(B) onto Uk, k = 0,1.

Lemma 1. [10] The operators Ak,Bk E L(Uk;Fk),k operators B-1 E L(F°; U0) and A-1 e L(Fl;U1).

0, 1

Construct the operators H = B0-1Ao E L(U0), S = A~[1B1 E ¿(U1). The (A, a)-bounded operator B is railed (A,p)-bounded, p E {0} U N, if to is a removable singular point (i.e. H = O,p = 0) or a pole of order p E N (i.e. Hp = O, Hp+1 = O) °f the A-resolvent (¡A — B)-1 of operator B. Introduce the following condition:

°A(B) = U af(B), for m e N; moreover, af(B) =

3=0

there exists a closed contour Yj C C, bounding a domain Dj D aA(B), such that Dj n aA(B) = 0 and 0 for all j,k,l = l,m with k = l.

з - -3

Dk n Di

(A)

Then we have

Lemma 2. [10] If the operator B is (A, a)-bounded and condition (A) is fulfilled then (i) there exist relatively spectral projectors

P3

1

fin-1RAn (B)d» e L(U),

Yj

1 , m,

Qj = »

2ni

■n-1lAn (B)d» e L(F),

Yj

Moreover,

(u) PjP = PP3 = P3, Q3Q = QQ3 = Q3;

(Hi) Pk Pi = PiPk = O for a Ilk, l = 1, m with k = l.

j = 1, m.

j

Put Po — P Pj E L(U)-, Qo — Q - ^ Qj E L(F)- Due to Lemma 3 operators

j=i j=i _

P0, Q0 are projectors. Moreover, PjP0 — P0Pj — O, QjQ0 — Q0Qj — O for j — 1,m.

Thus, assume that condition (A) is fulfilled. Fix Tj E R with Tj < Tj+1, vectors Uj E U for j — 0,m, and vector-function f E F) Consider the linear inhomogeneous

Sobolev type equation

Au(n) — Bu + f. (5)

Refer to a vector-function u E C^(R; U) satisfying (5) as a solution to (5). Refer to a solution u — u(t) to (5) satisfying the conditions

Pj(u(k)(rj) - uk) — 0, j — 0m, k — 0,n - 1, (6)

as a solution to the multipoint initial-final value problem (6) for (5). Introduce the following operator families

1

2ni

Uk(t) — — I ¡in-k-1(^nA - B)-1Ae^tdfj,,

Yj

k — 0,1,... ,n - 1, j — 1,... ,m.

Lemma 3. [10] If the operator B is (A,, p)-bounded, and condition (A) is fulfilled then

(i) Uk(t) k — 0,1,... ,n - 1, j — 1,... ,m are propagators of homogeneous (f = 0) equation (5);

(ii){Uk (t))® — Uk-1 (t) for k — 0,1,...,n - 1 j — 0,1,...,m,l — 0,1,...,k;

(Hi) (Uk(t))(° — O for k — I; J tj t=o

(w) {Uk(t))

t

(k)

(t)

—P

t=o

Introduce the subspaces U1j — im Pj and F1j — imQ j for j — 0,m. By construction,

m m

U1 — 0 U1j and F1 — 0 F1j.

=o =o

Denote by A1 j the restriction of A to U1j and by B1j the restriction of B to U1j for j — 0,m.

Theorem 1. [29] (generalized spectral theorem) Suppose that A,B E L(U; F); the operator B is (A, a)-bounded, and condition (A) is satisfied then (i) Aj E L(U1j; F1j) an d Bj E L(U1j; F1j) for j — 0m; (H) the operators A-j E L(F1j; U1j) exist, for j — 0,m.

Theorem 2. [10] If the operator B is (A,p)-bounded for p E {0} U N and condition (A) holds then for all f E Cpn+n(R; F) and uk E U, for j — 0,m, k — 0,n - 1, there exists a unique solution to (5), (6) given by

u(t) — - £ HqB-1(I - Q)f (qn)(t) +

q=0

m n— 1 m t (7)

Y,Y,Uk (t - Tj )uk + £ / Un—1(t - s)A—jQj f (s)ds. j=0 k=0 j=0 ¿,

2. A Stochastic Linear Sobolev Type Equation of Higher Order with Relatively p-Bounded Operators

For a real separable Hilbert space U = (U, (•, •)), take an operator K G L(U) whose spectrum a(K) is nonnegative, discrete, finite, and accumulates only to zero. Denote by [Xj} the sequence of eigenvalues of K enumerated in the non-increasing order taking into account the multiplicities. The linear span of the set [^j} of corresponding orthonormal K U K

<x

trace Tr K = E Xj < j=i

Take a sequence [nj} of independent stochastic processes nj ■ Q xI ^ R, a complete probability space and an interval I C R. Equip R with the Borel a-algebra. Assume that the random variables nj(u,t) G L2 are Gaussian for all u G A and t G I, where A is a a-algebra on Q. In addition, the sample trajectory nj(u, •) is almost surely continuous, that is, nj G CL2 (for a detailed description oft he spaces Cl L2 for l G [0} U N, see [23]). UK

<x

®K(t) = Y. vXnj(t)<Pj (8)

j=i

I

if [nj} C CL2 then the existence of a stochastic K-process 6K implies that its trajectories are almost surely (a.s.) continuous. Introduce the Nelson-Gliklikh derivatives

<x

o(kt) = Y, VXj (9) j=i

K

lI

a detailed description of the Nelson-Gliklikh derivative, see [15,23]). Introduce [23] the space of differentiable "noises" ClKL2 of stochastic K-processes whose trajectories are a.s. continuously differentiable on I in the sense of Nelson-Gliklikh up to order l G [0} U N.

K

a.s. coincide with zero (that is, absolute silence), as well as "white noise"

Wk (t) = ^, (10)

K

wk(t) = yl vxь, t g r+

j=i

Here fij = (3j (t) is the Brownian motion of the form

n , ч • n(2k + 1) -

в (t) = Y. j 2- t, t G R+,

k=l

where j are pairwise independent Gaussian random variables such that Ej = 0 and

D£

jk

n(2k + 1) 2

-2

that is, Cjk £ L2.

Having considered the deterministic equation (5) in the previous section, we now proceed to the stochastic equation (3). Assume that the operator B is (A,p)-bounded, with p E {0} U N, and condition (A) is satisfied. Consider the linear stochastic Sobolev type equation

„(n)

(11)

O (n)

An = Bn + Nw,

°(n)

where n = n(t) is the required stochastic K-process, n is its Nelson-Gliklikh derivative of the n-th OTder, w = w(t) is a known stochastic K-process, and the operator N is defined below.

Take t0 = 0 and Tj E R+ with Tj-\ < Tj for j = 1,m. Complement (11) with the multipoint initial-final conditions

(k)

Pj(n (Tj) - Ck) = 0, j = 0,m,k = 0,n - 1,

(12)

Pj

have to consider the weak (in the sense of S. Krein) multipoint initial-final conditions

lim Po(n(k) (t) - Ck) = 0, Pj(n(k)(Tj) - j) = 0, j = im,k = M-I. (13)

Here

ck = E v^kc

%Vk, j

0, m, k = 0, n — 1,

(14)

i=1

j E L2 is a Gaussian random variable such that series (14) converges. (For instance D& ^ Cj, i E N j = 0,m, k = 0,n — 1). Call a stoch astic K-process n E CK L2

a (classical) solution to (11) whenever a.s. all its trajectories satisfy (11) for some stochastic K-process w E CK L2, some operator N E L(U; F), and all t El. (Here and henceforth I = (0, +to)). Call a solution n = n(t) to (11) a (classical) solution to problem (11), (12) (problem (11), (13)) whenever in addition condition (12) (condition (13)) is satisfied.

Theorem 3. For p E {0} U N take an (A,p)-bounded operator B and assume that condition (A) holds. Given Tj E for j = 1,m, an operator N E L(U; F) a nuclear-operator K E L(U) with real spectrum a(K), a stochastic K-process w = w(t) such that (I — Q)Nw E CKn+nL 2 mid QNw E CkL2, and random variables j E L2; for j = 0,m k = 0,n — 1, such that (14) is fulfilled, there exists a unique solution n E C'KL2 to problem (11), (12) given by

j=o

n(t) = HqBo_1(i - Q) w(qn)(t)+

q=0

n— 1

£ Uk (t - Tj )j + Ujn—1(t - s)A—1Qj w(s)ds

k=o

t.

(15)

t

Let us sketch the proof. It is straight forward to verify that (17) is a solution to problem (11), (12). To establish the uniqueness, reduce the problem to the equivalent system

o (n) ◦ (k) ___

An = Bn, Pj n (j ) = 0,j = 0,m, k = 0,n — 1. By Theorem 1 the first equation here is equivalent to the system

H(n0)(n) = n0, (n1)(n) = Sn\ (16)

where n° = (I — P)n and n1 = P'l- Taking now the n-th Nelson-Gliklikh derivative of the

H

0 = hp+1( n 0)(np+n) =... = h2( n 0)(2n) =... = h (n0 )(n) = n0 ■

By Theorem 2 and the initial-final conditions (12), the second equation of (16) yields

m n— 1

n1 = ££ Uk(t — Tj)0 = 0.

j=0 k=0

In view of (10), problem (11), (12) is not solvable when the right-hand side of (11) is

o

the "white noise" w(t) =WK (t). In this case instead of conditions (12) we should consider conditions (13).

o

Corollary 1. If all conditions of Theorem 3 hold and w(t) =WK (t) then, for random variables ^k G L2 given by (14) there exists a unique solution to problem (11), (13) given by

p

n(t) = —J2 HqB—1(I — Q) WK{qn+1)(t) +

q=0

n— 1

j (t — Tj )j — UV-1'

k=0

+ I Un—2(t — s)A—jQjNWk(s)ds

+

j=0

J2Ukk(t - Tj)j - Un-l(t - Tj)A-jjQjNWk(j)+ (17)

t e x.

The proof of Corollary 1 is similar to that of Theorem 3. The difference in the additive terms is caused by an application of integration "by parts",

t

j Un-(t - s)AjQj N Wk (s)ds =

Tj

t

-Ujn-1 (t - Tj )AjQj NWk (Tj) + / Uj^-2(t - s)A-j Qj NWk (s)ds,

Tj

j

which follows from the properties of the Nelson-Gliklikh derivative.

3. The Multipoint Initial-Final Problem for the Stochastic Boussinesq-Love Equation with Additive "White Noise"

Let D C Rd be a bounded domain with the boundary dD of class CFix l E {0} U N and set F = W2(D), U = {u E Wl2+2(D) : u(x) = 0,x E dD}. Obviously, U is a real separable Hilbert space densely and continuously embedded in F-

Let {vj} be the sequence of eigenvalues of the Laplace operator with homogenous Dirichlet boundary conditions, numbered in nondecreasing order according to multiplicity, and by {Vj} denote the set of corresponding eigenfunctions, orthonormal in the sense of

F

UK

A = (—1)m-1 Am with domain domA = {w2+2(m+1)(D) : Aku(x) = 0,x E dD,k = 0,1, ...,m — 1},m E N Note that the operator A has the same eigenfunctions {Vj}, as the Laplace operator, but its spectrum consists of eigenvalues \vj\m. Since their asymptotic

2m

\vj\m ~ j^ ^,j ^ we consider such number m E N for a fixed d E N that the

<x

series \vj \-m converges (in part icular m = d). Then the ope rator A is continuously j=i

U

consisting of eigenvalues Xj = \vj\-m. We take that very operator as the nuclear operator KK

<x

Wk(t) = £ y/XjPj(t)vj-

j:vj=X

In the cylinder D x [0,T], T E R+ consider the Dirichlet problem

f(x, t) = 0, (x, t) E dD x [0, T] (18)

for the equation

o o

(X — Ax) ftt= aAxf+ Wk - (19)

Put A = X — Ax, B = a Ax, N = I.

Lemma 4. [10] For all a E R+; X E R the opera tor B is (A, 0)-bounded.

In order to state initial-final conditions, we need relatively spectral projectors. In this example for the sake of simplicity we confine to three initial-final conditions. First of all present the projectors

p (Q)

IU (IF) if X = Vj for all j E N, IU — Vj)U Vj ( IF — Vj>F Vj

Furthermore, choose h1,h2 E R+ such that h1 < h2 and the sets ) = {^j E ja(B) : \ < hj, <jA(B) = {^ E ja(B) : h1 < \ < h2}, and a£(B) = {^ E ja(B) : \^3 \ > h2} are not empty; hence, j^A(B) n ja(B) n ja(B) = 0 and condition (A) holds.

\=Uj

\=Uj

Construct the projectors

Pi = IU - Y1 )u ф, p2 = IU - Y1 )U ф'

hi<\^j\<h2 h2<\^j \

Qi = IF - Y^ (•'Vj^ ф' Q2 = IF - Y k'j ф' hi<\^j\<h2 h2<\^j \ Po = P - Pi - P2,QO = Q - Qi - Q2.

(20)

Finally, choose t1 G (0,T) as well as random variables ^ and C2 independent of each

o

K WK

Urn Po(V(t) - Co) = 0, Pi(n(Ti)- ti) = 0,P2(V(T) - & = 0,

(21)

where

ro ro ro

Co = Y y/KZoiVi, C1 = Y y/K&iVh 6 = Y ^/Xi&i(22)

i=1 i=1 i=1

Applying the results of Section 2 to problem (18), (19), (21), we obtain the following theorem.

Theorem 4. For all numbers X G R, a G R \ [0} mid t1 G (0,T), as well as random variables C0i, Cu an d C2i such th at Dj < Cj fo r i G N, j = 0,1, 2 for som e Cj G R+ there exists a unique solution n = n(t)> f°r t G R+; to problem (18), (19), (21) given by

n(t) = —Bo—1 (I — Q) Wk (t) + U0(t)c0 + u0(t)C0+

t

+ i U0(t — s)A——o1QoWK(s)ds + U0(t — T1)C0 + U1(t — n)C1 —

Here

u0(t) =

-Ul(t - Ti)A-lQiWK(Ti)+ U0(t - s)A-IQiWk(s)ds+

Tl

+U0(t - T)$ + Ui(t - T)£ - Ul(t - T)A-21Q2Wk(T)+ t

+j US (t - s^M G R+.

T

B-i = ^ ^'Фк )Фк

Vk=A

e

Ilk G aA(B), X< Vk

фк )'£k ch

(23)

avk X - Vk

t+

e

lk G oA(B), X> Vk

•,фk)фk cos J aVk t, Vk - X

Ul(t)= y,

lk G oA(B), X< Vk

x X-Vu aVk v-^ Vk-X . aVk

•,фk)Фk\ -sh\ -t + v^k)Pk\ -sinW--t,

V aVk V X-V^^ V aVk \j Vk-X

lk G a£(B),

X> Vk

t

Aw = Y^ (A -Vk) '

t + £ (-,<£k)Pk cos W v-^t,

ßk e oA(B), Uk ßk e aAA(B), k

\<Vk A > Vk

Ui1(t)= £ )Vk\ ^^s\ ^^t+ V )Vk\ sWt, ^ V avk V A-v^^ \ avk V Vk-A

ßk e aA(B), ßk e aA(B),

A < Vk A > Vk

A- = £ (A - Vk)- 1 (•,Pk)Pk,

tik eaA(B)

U20(t) = )Pk Aa-k uk t + ^ )Pk ^^jV^—A1,

ßk e aA(B), ßk e aA(B),

A< k A> k

U21(t)= V )Pk\ ^^sM^^t+ V (^k)Vk\ V— sinjt,

^ V a Vk V A- Vk ' V a Vk y Vk-A

ßk e aA(B), ßk e aA(B),

A< k A> k

A-2 = £ (A - Vk)_1(^k^k•

k •

Conclusion

The next stage of our studies is to spread the ideas and the developed methods of the theory of multipoint initial-final problems for linear Sobolev type equations of higher order from the relatively p-bounded to the relatively (n,p)-sectorial case and to the case of complete higher order Sobolev type equations with initial-final conditions. In addition, it would be interesting to apply these ideas to inverse problems for Sobolev type equations of higher order.

Acknowledgement. The work was supported by Act 211 Government of the Russian Federation, contract no. 02.AOS.21.0011.

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21. Favini A., Sviridyuk G., Manakova N. Linear Sobolev Type Equations with Relatively p-Sectorial Operators in Space of "Noises". Abstract and Applied Analysis, 2015, vol. 2015, Article ID 697410, 8 p.

22. Favini A., Sviridyuk G., Sagadeeva M., Linear Sobolev Type Equations with Relatively p-Radial Operators in Space of "Noises". Mediterranean Journal of Mathematics, 2016, vol. 13, no. 6, pp. 4607-4621. DOI: 10.1007/s00009-016-0765-x

23. Favini A., Sviridyuk G.A., Zamyshlyaeva A.A. One Class of Sobolev Type Equations of Higher Order with Additive "White Noise". Communications on Pure and Applied Analysis, 2014, vol. 15, no. 1, pp. 185-196. DOI: 10.3934/cpaa.2016.15.185

24. Nelson E. Dynamical Theories of Brownian Motion. Princeton, Princeton University Press, 1967.

25. Shestakov A.L., Keller A.V., Nazarova E.I. Numerical Solution of the Optimal Measurement Problem. Automation and Remote Control, 2012, vol. 73, no. 1, pp. 97-104.

26. Keller A.V., Sagadeeva M.A. The Optimal Measurement Problem for the Measurement Transducer Model with a Deterministic Multiplicative Effect and Inertia. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2014, vol. 7, no. 1, pp. 134-138. DOI: 10.14529/mmpl40111

27. Konkina A.S. Multipoint Initial-Final Value Problem for the Model of Devis with Additive White Noise. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2017, vol. 10, no. 2, pp. 144-149. DOI: 10.14529/mmpl70212

28. Manakova N.A. Mathematical Models and Optimal Control of the Filtration and Deformation Processes. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2015, vol. 8, no. 3, pp. 5-24. DOI: 10.14529/mmpl50301

29. Zagrebina S.A. A Multipoint Initial-Final Value Problem for a Linear Model of Plane-Parallel Thermal Convection in Viscoelastic Incompressible Fluid. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2014, vol. 7, no. 3, pp. 5-22. DOI: 10.14529/mmp 140301

Received February 8, 2018

УДК 517.9 Б01: 10.14529/ттр180308

МНОГОТОЧЕЧНАЯ НАЧАЛЬНО-КОНЕЧНАЯ ЗАДАЧА ДЛЯ ОДНОГО КЛАССА МОДЕЛЕЙ СОБОЛЕВСКОГО ТИПА ВЫСОКОГО ПОРЯДКА С АДДИТИВНЫМ <БЕЛЫМ ШУМОМ>

Г.А. Свиридюк1, А.А. Замышляева1, С.А. Загребина1

1 Южно-Уральский государственный университет, г. Челябинск, Российская Федерация

Теория уравнений Соболева была предметом интереса многих исследователей последние годы, при этом много внимания уделялось детерминированным уравнениям и системам. Тем не менее, существуют также математические модели, содержащие случайные возмущения, такие как белый шум. Новая концепция «белого шума», первоначально построенная для конечномерных пространств, в данной работе распространяется на случай бесконечномерных пространств. Основная цель заключается в

разработке стохастической теории уравнений соболевского типа высокого порядка и

предоставлении некоторых практических приложений. Основная идея состоит в том,

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лиха. Абстрактные результаты, касающиеся начально-конечных задач для уравнений соболевского типа высокого порядка, применяются к математической модели Бусси-неска - Лява с аддити в н ы м «»

рии уравнений соболевского типа, как метод фазового пространства, заключающийся в редукции сингулярного уравнения к регулярному, определенному на некотором подпространстве исходного пространства, понимаемом как фазовое пространство.

«»

ровский K-процесс; многоточечная начально-конечная задача.

Статья выполнена при поддержке Правительства РФ (Постановление №211 от 16.03.2013 г.), соглашение № 02.А03.21.0011.

Литература

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A.G. Sveshnikov. - De Gruyter, 2011.

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5. Showalter, R.E. Hilbert Space Methods for Partial Differential Equations / R.E. Showalter. - Pitman; London; San Francisco; Melbourne, 1977.

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B. Loginov, A. Sinithyn, M. Falaleev. - Dordrecht; Boston; London: Kluwer Academic Publishers, 2002.

7. Sviridyuk, G.A. Linear Sobolev Type Equations and Degenerate Semigroups of Operators / G.A. Sviridyuk, V.E. Fedorov. - Utrecht; Boston; Koln; Tokyo: VSP, 2003.

8. Zamyshlyaeva, A.A. The Linearized Benney - Luke Mathematical Model with Additive White Noise / A.A. Zamyshlyaeva, G.A. Sviridyuk // Springer Proceedings in Mathematics and Statistics. - 2015. - V. 113. - P. 327-337.

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16. Gliklikh, Yu.E. Stochastic Leontieff Type Equations and Mean Derivatives of Stochastic Processes / Yu.E. Gliklikh, E.Yu. Mashkov // Вестник ЮУрГУ. Серия: Математическое моделирование и программирование. - 2013. - V. 6, № 2. - Р. 25-39.

17. Zagrebina, S.A. The Stochastic Linear Oskolkov Model of the Oil Transportation by the Pipeline / S.A. Zagrebina, E.A. Soldatova, G.A. Sviridyuk // Springer Proceedings in Mathematics and Statistics. - 2015. - V. 113. - P. 317-325.

18. Da Prato, G. Stochastic Equations in Infinite Dimensions of Encyclopedia of Mathematics and Its Applications / G. Da Prato, J. Zabczyk. - Cambridge: Cambridge University Press, 1992.

19. Arato, M. Linear Stochastic Systems with Constant Coefficients / M. Arato. - Berlin: Springer, 1982.

20. Kovacs, M. Introduction to Stochastic Partial Differential Equations / M. Kovacs, S. Larsson // Proceedings of «New Directions in the Mathematical and Computer Sciences:», National Universities Commission, Abuja, Nigeria, October 8-12, 2007. V. 4. - Lagos: Publications of the ICMCS, 2008. - P. 159-232.

21. Favini, A. Linear Sobolev Type Equations with Relatively p-Sectorial Operators in Space of «Noises» //A. Favini, G.A. Sviridyuk, N.A. Manakova / / Abstract and Applied Analysis. -2015. - V. 2015. - Article ID 697410. - 8 p.

22. Favini, A. Linear Sobolev Type Equations with Relatively p-Radial Operators in Space of «Noises» / A. Favini, G. Sviridyuk, M. Sagadeeva // Mediterranean Journal of Mathematics. - 2016. - V. 13, № 6. - P. 4607-4621.

«

»

Applied Analysis. - 2014. - V. 15, № 1. - P. 185-196.

24. Nelson, E. Dynamical Theories of Brownian Motion / E. Nelson. - Princeton: Princeton University Press, 1967.

25. Shestakov, A.L. Numerical Solution of the Optimal Measurement Problem / A.L. Shestakov, A.V. Keller, E.I. Nazarova // Automation and Remote Control. - 2012. - V. 73, № 1. -P. 97-104.

26. Keller, A.V. The Optimal Measurement Problem for the Measurement Transducer Model with a Deterministic Multiplicative Effect and Inertia / A.V. Keller, M.A. Sagadeeva // Вестник ЮУрГУ. Серия: Математическое моделирование и программировние. - 2014. -V. 7, № 1. - Р. 134-138.

27. Konkina, A.S. Multipoint Initial-Final Value Problem for the Model of Devis with Additive White Noise / A.S. Konkina // Вестник ЮУрГУ. Серия: Математическое моделирование и программировние. - 2017. - V. 10, № 2. - Р. 144-149.

28. Манакова, Н.А. Математические модели и оптимальное управление процессами фильтрации и деформации / Н.А. Манакова // Вестник ЮУрГУ. Серия: Математическое моделирование и программировние. - 2015. - Т. 8, № 3. - С. 5-24.

29. Zagrebina, S.A. A Multipoint Initial-Final Value Problem for a Linear Model of Plane-Parallel Thermal Convection in Viscoelastic Incompressible Fluid /S.A. Zagrebina / / Вестник ЮУр-ГУ. Серия: Математическое моделирование и программировние. - 2014. - V. 7, № 3. -Р. 5-22.

Георгий Анатольевич Свиридюк, доктор физико-математических наук, профессор, кафедра «Уравнения математической физики», Южно-Уральский государствен-ныи университет (г. Челябинск, Российская Федерация), sviridyuk@susu.ru.

Алена Александровна Замышляева, доктор физико-математических наук, доцент, кафедра «Прикладная математика и программирование:», ЮжноУральский государственный университет (г. Челябинск, Российская Федерация), zamyshliaevaaa@susu. ru.

Софья Александровна Загребина, доктор физико-математических наук,

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Уральский государственный университет (г. Челябинск, Российская Федерация), zagrebinasa@susu.ru.

Поступила в редакцию 8 февраля 2018 г.