CC BY
0
MSC 35Q99
DOI: 10.14529/ mmp240207
NUMERICAL ALGORITHM AND COMPUTATIONAL EXPERIMENTS FOR ONE LINEAR STOCHASTIC HOFF MODEL
E.A. Soldatova1, A.V. Keller2
1 South Ural State University, Chelyabinsk, Russian Federation 2Voronezh State Technical University, Voronezh, Russian Federation E-mail: soldatovaea@susu.ru, alevtinak@inbox.ru
Investigated is a model of deformation in a structure composed of I-beams with random external effect; it is based on stochastic Hoff equations with an initial-final condition. The article describes an algorithm for a numerical solution of the initial-final problem for stochastic Hoff equations; the algorithm is based on the Galerkin method. Provided is a numerical investigation algorithm providing for numerical solutions for both degenerate and non-degenerate equations. The main theoretical results that enabled this numerical investigation are the methods of the theory of degenerate groups of operators and of the theory of the Sobolev type equations. The algorithms are represented by schemes enabling building flowcharts of programs for computational experiments. Results of computational experiments. In addition, numerical investigation of the stochastic model involves further obtaining and processing the results of n experiments at various values of a random variable, including those related to rare events.
Keywords: Hoff model; geometric graph; initial-final condition; numerical investigation; algorithm; Sobolev type stochastic equations; computational experiment.
Introduction
Theoretical and applied research associated with the tasks of information processing and analysis, identification and management uses stochastic models [1-3] to assess the state and parameters of complex physical and financial systems. Of note, Sobolev type stochastic equations (1) have been used for several decades to describe and simulate a large number of physical, technical and technological processes
Analytical and numerical investigations of non-classical stochastic models are developing in two directions. One of them uses the concept of "white noise" as the Nelson-Gliklikh derivative of the Wiener K-process [4,5]. This approach has been widely used in recent years for Sobolev type stochastic equations in the works of G.A. Sviridyuk, A. Favini, A.A. Zamyshlyaeva, S.A. Zagrebina, N.A. Manakova, M.A. Sagadeeva, T.G. Sukacheva [6-10] and for Leontief type stochastic systems in the works of Yu.E. Gliklikh, E.Yu. Mashkov [5], G.A. Sviridyuk, A.L. Shestakov, A.A. Zamyshlyaeva, A.V. Keller [11].
The other direction relates to development of the Ito-Stratonovich-Skorokhod approach [5]. It is one of the very first areas of investigation of differential equations and is currently considered classical. In the works of G. Da Prato, J. Zabczyk [12] the Ito-Stratonovich-Skorokhod approach is applied in the infinite-dimensional case. The works of I.V. Melnikova [13] investigate stochastic equations in Schwarz spaces, while using the traditional approach to the concept of white noise as a generalized derivative of the
Ld( = M(dt + NdW.
(1)
Wiener process. M. Kovacs and S. Larsson [14] investigated non-degenerate models of mathematical physics using the Ito-Stratonovich-Skorokhod approach. In the last decade, the works of G.A. Sviridyuk, A.A. Zamyshlyaeva [15], S.A. Zagrebina [16] investigate non-classical stochastic models within this approach. Therefore, new results for the theory of Sobolev type stochastic equations enabling the investigation of various mathematical models with the development of numerical methods and algorithms are relevant.
Suppose that G = G(V; E) is a geometric graph, V = {Vi} denotes a set of vertices and E = {Ej} denotes a set of edges. On the edges Ej of the graph G let us consider the linear stochastic Hoff equations
AdQ + dujxx = 3j j dt + NdWj (2)
with an initial-final condition
Po(Z(to) - £o) = Pi(Z(ti) - 6) = 0, (3)
where W = (W1, W2,..., Wn) is an F-digit nuclear K-Wiener process, operator K G L(Z) is nuclear, P0,P1 are relatively spectral projectors. At the vertices Vi of the graph G let us set continuity conditions
Zj (0,t) = Zk (0 , t) = Zm(lm,t) = (n(L,t),
Ej,Ek G Ea(Vi),Em,En G (V)
(4)
and flow balance conditions
J] djZjx(0,t) - ^ dkZkx(lk,t) = 0, (5)
Ej&Ea(Vi) EfceE" (Vi)
where Ea (Vi) (Ew (Vi)) is the set of edges having, at the vertices Vi which is the beginning (end), lj > 0 and dj > 0 which are the length of the edge and the diameter of its cross section. Equations (2) describe the buckling dynamics of I-beams in a structure under constant load with a random external action. Here, the A G R+ parameter characterizes the load on the jth beam, the ¡3 G R parameter, in turn, describes the properties of the jth beam material, the random (j = (j(x,t), (x,t) G (0, lj) x R process characterizes the deviation of the jth beam from the equilibrium position.
Let us turn to linear mathematical Hoff model [17]. Initial boundary value problems for the Hoff equation in a bounded domain Q were first studied by N.A. Sidorov [18] and his students [19,20]. Hoff equations on a geometrical graph with the Cauchy condition were first studied by G.A. Sviridyuk together with V.V. Shemetova [21]. They managed to describe completely the phase space on a geometrical graph. Later, the inverse problem for the Hoff equation was solved on geometric graphs [22]. Furthermore, solution stability for the Cauchy problem of the Hoff equations was investigated, and sufficient conditions for stability and asymptotic stability of solutions to the Cauchy problem for the Hoff equations in the domain and on a geometric graph were obtained [23]. Optimal control of solutions to the Hoff equation was studied by N.A. Manakova and her students (see, for example, [24]). Numerical investigation of the non-autonomous Hoff equation on a geometric graph with the Showalter-Sidorov condition was carried out in [25].
The linear stochastic Hoff model for deformation in an I-beam structure with a multipoint initial-final condition was analytically studied in [26] in the sense of Nelson-Gliklich derivative to the "white noise" concept, and in [27] in the sense of the traditional
approach. [27] proves the solvability of the initial-final problem (3) for an abstract linear stochastic Sobolev type equation (2).
Theorem 1. Suppose the process W is the F1 -digit K-Wiener process, the operator N £ ¿(F1) and for each fixed t random variables £0,£1 £ L2(Q, Z1) and the W process are independent. Then for any £0,£1 £ L2(Q, Z1) the problem (2), (3) has a unique solution defined by expression
Z (t) = Z-T0 Co + Z0-s L-01Q0 NW (s)ds +
T0 (6)
t
+ Z1-T1 Ci + Zl-sL-11 Q1NW (s)ds,
where Zb0 и Z1
1
Zl = — {sI - S,)~lestds, Z\ = — / (si - S^e^ds.
2ni J 2ni J
Y0 Y1
1. The Algorithm of the Numerical Method
We will look for an approximate solution of the problem (2) - (5) in the form of
C(x,t) = {(1(x,t),(2(x,t),...,(n(x,t)^ , (7)
where Zj (x,t) is the approximate solution on the jth beam of the graph of the form
N
Cj(x,t) = ZN(x,t) = Y, ak(t)<fi(x). (8)
k=1
Here, } = {(p\ ,<^k} refer to the corresponding orthonormal eigenfunctions relative to the scalar product of L2 (G).
Next, we apply the representation (8) to the Hoff equation (2), resulting in a system of equations by scalarly multiplying by eigenfunctions. Each system's equation will contain only one unknown Galerkin coefficient; therefore, the expression (8) is a partial sum of the series, whose convergence provides for the convergence of the approximate solution to the exact one.
Note that the linearized Hoff model is considered on the graph. Let us build a description of the algorithm of the numerical solution.
Step 1. Given: N is the number of summands of the Galerkin sum; j, dj are the length and cross-sectional area of the edges of the graph, respectively (equal for all edges), u is the parameter of external action, a random variable A. Xj, ftj are the coefficients of the linear Hoff equation. The coefficients are taken equal for all edges of the graph.
Step 2. Initial conditions are set. £0(x), ^(x) are functions of the initial-final condition, whose coefficients are normally distributed random variables.
t
Step 3. Generation of approximate solutions
N
Ô (t,x) = £ ak (t)<p{ (x), (9)
k=i
and application of (7) to the equation.
Step 4■ Using the differential equation from the previous step relative to the unknown variables ak(t), we will multiply it scalarly by functions ^jk(x), k = 1,...,N, to obtain a system of differential equations.
Step 5. Random variables £0k, £1k are generated.
Step 6. For numbers k0, for which the A parameter coincides with the eigenvalue vko of the A operator, a system of corresponding algebraic equations is made and solved.
Step 7. For numbers k0, for which the A parameter does not coincide with the eigenvalue vko of the A operator, a system of corresponding differential equations is made and solved. One (the other one) includes differential equations whose numbers coincide with the numbers of the eigenvalues related to af0(M) (ofl(M)).
Step 8. The first (second) system of differential equations is solved with initial (final) conditions.
2. Computational Experiments
Consider a five-edge graph with six vertices, shown in Fig. 1, the lengths of all edges are different: l 1 = n, l2 = 2n, = 3n, l4 = 4n, l5 = 5n, and the diameter of the sections is the same for all edges dj = 1, j = 1, 2,..., 5.
Fig. 1. The graph for computational experiment
For such a graph, let us write down the continuity conditions (4)
Ci(ii) - (2(0) = 0,
(2(I2) - Ca(0) = 0, Z2(I3) - Ca(0) = 0, Z4(I4) - (5(0) = 0
and the flow balance conditions (5)
Zix (0) = 0
Zix(ii) - C2*(0)
C2*(I2) - C3x(0) C3x(I3) - Z4X(0)
Z4x (14) - Z5x (0) Z5x(15) = 0
= 0,
= 0,
= 0,
= 0,
(11)
With the given coefficients ^ = 0,15, A = 1, 44, the Hoff equations will be set on the edges of the graph
1,44d(j + dZjxx = 0,15 Zj dt + NdWj, j = 1, 2,..., 5. (12)
We will look for solutions Zj (t,x), j = 1, 2,..., 5 of this problem in the form of Galerkin sums, by taking 7 summands.
6
o(t,x) = ak(t) j(x)' j =1,2,•••,5 k=i
Having solved the Sturm-Liouville problem, we get
Afc = <pltk(x) = ^ + cos (£x) ,
¥>2,fc(z) = COS (¿(x + 7r)) ,
fa,k(x) = cos (TE(x + 37r)) '
= COS (£(x + 671-)) ,
= cos (te(x + 107r)) •
Note that A = A6. Thus, A6 G ^^(M). According to the analytical investigation of the Hoff model of deformation in a structure of I-beams with an initial-final condition, let us assume that A 1, A2, A3, G af0(M) and A4, A5, A7 G afl(M).
We will assume that the system is exposed to the same effect, therefore all random variables here are normally distributed Gaussian quantities ~ N(0; 0, 6). Using the initialfinal conditions, let us obtain a representation for the initial values considered as follows
Po(Z(0) - £0) = X] (Z(0,x) - £o(x),^k= 0, (13)
k:^k €ctl10(M )
Pi(Z(0) - £1) = E (Z(T,x) - Ci(x),^k)^k = 0, (14)
k:\k€aLl (m)
further £r = (£1,0, £2,0, £3,0, £4,0, £5,0) and £1 = (£1,1,£2,1,£3,1,£4,1,£5,1) according to the number of edges j = 5.
As a result of the generation of random variables included in the decomposition for the functions of the initial condition, we obtained
£1,01 = -1, 57203253, £1,0, = -0, 6534010832, £1,03 = -0, 01797861729,
£2,0l = 0, 769351807, £2,02 = 0,1959283956, £2,03 = 0,1276538831,
£3,0l = 0, 4600965754, £3,0, = 0, 05775170542, £3,03 = 0, 4919564389,
£4,0l = -0, 2374773235, £4,02 = 0, 00273696281, £4,03 = 0, 491218275,
£5,01 = 0, 470553412, £5,0, = 0, 5003177325, £5^ = -0, 4645654522.
By generating random variables that are part of the decomposition for the initial functions of the final condition, we obtained
£1,14 = -0, 3283899452, £Mb = 0, 8185883182, £M? = 0, 7807642225,
£2,14 = 0, 206083644203, £2,1b = -1, 36940970662, £2,1y = -0,187482338909,
£3,14 = 0, 4579221402, £3,1b = 0, 4316586561, £3,1y = -0, 2344698016,
£4^ = -0, 05175526728, £4,1, = 0, 2550668598, £4,1, = -0, 9918095115,
£5,14 = 0,11864946, £5,1, = 1, 749752786; £5,1, = -0, 2830376578.
The external effect on each edge may be represented by decomposition
NdWj = A sin(wt)01^1(x) + A sin(wt)02^2(x) + A sin(wt)03^3(x) + +A sin(wt)0^4(x) + A sin(wt)^5(x) + A sin(wt)^7(x)
with A = 0,765405620436, w = 4>k = (0ifc> 02fc, 03fc, 04fc, 05fc) in accordance with the number of edges j = 5.
The generation of random variables included in the decomposition for random external action gave the following results:
011 = 0, 2246948044, 012 = 1,36697483468, 013 = -0, 02737836647,
014 = 0, 8498896526, 015 = -0, 3518854196, 017 = 0, 07826279773,
02i = 0, 08735263553, 022 = -0, 931286038, 023 = -0,1017266735,
024 = -0, 2982196884, 025 = -0, 5344867959, 027 = -0, 03159215838,
03i = -0, 2212717272, 032 = 0,7903122257, 033 = 0,6887758545,
034 = 0,03068167176, 035 = -0, 7041800119, 027 = -0, 6699585179,
04i = 0,4722770182, 042 = -0, 2433091863, 043 = 0,04289751467,
044 = -0, 5012154257, 045 = 0,297270174, 047 = 0, 9044232645,
05i = 0, 0272506223, 052 = 0, 2489458583, 053 = -0, 04979900558,
054 = -0, 2764538139, 055 = 0,5463800986, 057 = -0, 3528926737.
The first system of differential equations with initial conditions contains three equations and three initial conditions. The second system of differential equations contains three equations and three final conditions for t = 10. Obtained was one algebraic equation, the sixth one (since the sixth eigenvalue equals X). The first system
- 0,lhaAt) - 0, 063296358 sinO, 166667i = 0, dt 1
da2(t) . .
0, 9955555556—- 0,15a2(i)-dt
-8,17168916 ■ 10-16 cos(0,166667t) - 0, 27851468 sin(0,166667t) = 0,
da3(t) , ,
0, 982222222—- 0,15a3(i) + dt
-2, 04292229 ■ 10-15 cos(0,166667t) - 0, 044257363 sin(0,166667t) = 0, with conditions
a1(0) = -0, 376410562, a2(0) = 0, 0164114362, a3(0) = 0, 0862050486.
The second system
dt
-1, 09101285 ■ 10-15 cos 0,166667t + 0, 248536829 sin0,166667t = 0,
0, 928888888^^ - 0,15a5(i) + dt
+4,11751779 ■ 10-16 cos 0,166667t - 0, 30034281 sin0,166667t = 0,
0, 8888888^^ - 0,15a7(i)-dt
-1,10136402 ■ 10-16 cos 0,166667t - 0,197981101 sin0,166667t = 0,
with conditions
a4(10) = -0, 748694532, a5(10) = 0, 263697458, a7(10) = -0, 069132153. Algebraic equation has the form
a6 (t) = 0, a6(0) = 0.
Next, we combine and solve the two systems, after which an approximate solution to the initial-final problem is found on the graph in question. The analytical result is rather cumbersome; therefore, the results of the computational experiment are presented graphically (Figs. 2 - 4) in the form of two-dimensional graphs at various points in time t*. The abscissa axis reflects the values of the variable x and reflects the length of the edges of the graph. The ordinate axis reflects the values of the (j (x,t*) function: the dynamics of buckling of I-beams in a structure under constant load with random external effect.
The colors show solutions on different edges of the graph: (1(x,t*) - blue, (2(x,t*) -green, <3(x,t*) - black, (4(x,t*) - red, (5(x,t*) - magenta.
The sequence of graphs reflects the development of the process over time, taking into account the structure of the graph. Of note, the curve of the graph characterizes the deviation of the I-beam from the vertical on the corresponding edge of the graph.
2 -2
Fig. 2. Results of computational experiment
2 -2
Fig. 3. Results of computational experiment
Fig. 4. Results of computational experiment
References
1. Frolov A.V. Dinamiko-stokhasticheskie modeli mnogoletnikh kolebanii urovnya protochnykh ozer [Dynamic-Stochastic Models of Long-Term Level Fluctuations in Non-Terminal Lakes]. Moscow, Nauka, 1985. (in Russian)
2. Breer V.V., Novikov D.A., Rogatkin A.D. Stochastic Models of Mob Control. Automation and Remote Control, 2016, vol. 77, pp. 895-913. DOI: 10.1134/S000511791605012X
3. Kibzun A.I., Ivanov S.V., Stepanova A.S. Construction of Confidence Absorbing Set for Analysis of Static Stochastic Systems. Automation and Remote Control, 2020, vol. 81, no. 4, pp. 589-601. DOI: 10.1134/S0005117920040025
4. Nelson E. Dynamical Theories of Brownian Motion. Princeton, Princeton University Press, 1967.
5. Gliklikh Yu.E. Global and Stochastic Analisys with Applications to Mathematical Physicas. London, Dordrecht, Heidelberg, New York, Springer, 2011.
6. Sviridyuk G.A., Manakova N.A. The Dynamical Models of Sobolev Type with Showolter-Sidorov Condition and Additive "Noise". Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2014, vol. 7, no. 1, pp. 90-103. DOI: 10.14529/mmp140108
7. Favini A., Sviridyuk G.A., Manakova N.A. 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. DOI: 10.1155/2015/697410
8. Favini A., Sviridyuk G.A., Zamishlyaeva A.A. One Class of Sobolev Type Equations of Higher Order with Additive "White Noise". Communications on Pure and Applied Analysis, 2016, vol. 15, issue 1, pp. 185-196. DOI: 10.3934/cpaa.2016.15.185
9. 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, issue 6, pp. 4607-4621. DOI: 10.1007/s00009-016-0765-x
10. Zagrebina S., Sukacheva T., Sviridyuk G. The Multipoint Initial-Final Value Problems for Linear Sobolev-Type Equations with Relatively p-Sectorial Operator and Additive "Noise". Global and Stochastic Analysis, 2018, vol. 5, no. 2, pp. 129-143.
11. Shestakov A.L., Keller A.V., Sviridyuk G.A. The Theory of Optimal Measurements. Journal of Computational and Engineering Mathematics, 2014, vol. 1, issue 1, pp. 3-16.
12. Da Prato G., Zabczyk J. Stochastic Equations in Infinite Dimensions. Cambridge, Cambridge University Press, 1992.
13. Melnikova I.V., Filinkov A.I., Anufrieva U.A. Abstract Stochastic Equations. I. Classical and Distributional Solutions. Journal of Mathematical Sciences, 2002, vol. 111, no. 2, pp. 3430-3475.
14. Kovacs M., Larsson S. Introduction to Stochastic Partial Differential Equations. Proceedings of New Directions in the Mathematical and Computer Sciences, Abuja, 2008, vol. 4, pp. 159-232.
15. Zamyshlyaeva A.A. Stochastic Incomplete Linear Sobolev Type High-Ordered Equations with Additive White Noise. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2012, no. 40 (299), issue 14, pp. 73-82.
16. Zagrebina S.A., Soldatova E.A. The Linear Sobolev-Type Equations With Relatively p-Bounded Operators and Additive White Noise. Bulletin of Irkutsk State University. Series: Mathematics, 2013, vol. 6, no. 1, pp. 20-34.
17. Hoff N.J. The Analysis of Structures. New York, John Wiley, 1956.
18. Sidorov N.A. Obshchie voprosy regulyarizacii v zadachah teorii vetvleniya [General Questions of Regularization in Problems of Bifurcation Theory]. Irkutsk, Irkutsk State University Publisher, 1982. (in Russian)
19. Sidorov N.A., Romanova O.A. On Application of Some Results of the Branching Theory in the Process of Solving Differential Equations with Degeneracy. Differential Equations, 1983, vol. 19, no. 9, pp. 1516-1526.
20. Sidorov N.A., Falaleev M.V. Generalized Solution of Differential Equations with a Fredholm Operator at the Derivative. Differential Equations, 1987, vol. 23, no. 4, pp. 726-728.
21. Sviridyuk G.A., Shemetova V.V. Hoff Equations on Graphs. Differential Equations, 2006, vol. 42, no. 1, pp. 139-145. DOI: 10.1134/S0012266106010125
22. Sviridyuk G.A., Bayazitova A.A. On Direct and Inverse Problems for the Hoff Equations on Graph. Journal of Samara State Technical University. Series: Physical and Mathematical Sciences, 2009, no. 1 (18), pp. 6-17.
23. Zagrebina S.A., Moskvicheva P.O. Stability in Hoff Models. Saarbrücken, LAMBERT Academic Publishing, 2012. (in Russian)
24. Manakova N.A., Dylkov A.G. Optimal Control of Solutions of Initial-Finish Problem for the Linear Sobolev Type Equations. Bulletin of the South Ural State University. Series: Mathematical Modeling, Programming and Computer Software, 2011, no. 17 (234), issue. 8, pp. 113-114.
25. Sagadeeva M.A., Generalov A.V. Numerical Solution for Non-Stationary Linearized Hoff Equation Defined on Geometrical Graph. Journal of Computational and Engineering Mathematics, 2018, vol. 5, no. 3, pp. 61-74.
26. Favini A., Zagrebina S.A., Sviridyuk G.A. The Multipoint Initial-Final Value Condition for the Hoff Equations on Geometrical Graph in Spaces of K-"Noises". Mediterranean Journal of Mathematics, 2022, vol. 19, article ID: 53. DOI: 10.1007/s00009-021-01940-0
27. Soldatova E.A. The Initial-Final Problem for the linear Stochastic Hoff Model. Bulletin of the South Ural State University. Series: Mathematical Modeling and Programming and Computer Software, 2014, vol. 7, no. 2, pp. 124-128. DOI: 10.14529/mmp140212
Received March 14, 2024
УДК 517.9 DOI: 10.14529/mmp240207
ЧИСЛЕННЫЙ АЛГОРИТМ И ВЫЧИСЛИТЕЛЬНЫЕ ЭКСПЕРИМЕНТЫ ДЛЯ ОДНОЙ ЛИНЕЙНОЙ СТОХАСТИЧЕСКОЙ МОДЕЛИ ХОФФА
E.A. Солдатова1, Л.В. Келлер2
1 Южно-Уральский государственный университет, г. Челябинск, Российская Федерация
2Воронежский государственный технический университет, г. Воронеж, Российская Федерация
Исследуется модель деформации под действием высокой температуры в конструкции из двутавровых балок со случайным внешним воздействием, в ее основе лежат
стохастические уравнения Хоффа на геометрическом графе с начально-конечным условием. В статье приводится описание алгоритма численного исследования рассматриваемой модели, в основе которого лежит метод Галеркина. Представленный алгоритм предусматривает получение численного решения в случае вырожденности, так и невырожденности уравнений. Основными теоретическими результатами, позволившими провести данное численное исследование, являются методы теории вырожденных групп операторов и теории уравнений соболевского типа. Алгоритмы представлены схемами, позволяющими построить на их основе блок-схемы программ для проведения вычислительных экспериментов. Кроме того, численное исследование стохастической модели предполагает в дальнейшем получение и обработку результатов n экспериментов при различных значениях случайной величины, в том числе, относящихся к редким событиям.
Ключевые слова: модель Хоффа; геометрический граф; начально-конечное условие; численное исследование; алгоритм; стохастическое уравнение соболевского типа; вычислительный эксперимент
Литература
1. Фролов, А.В. Динамико-стохастические модели многолетних колебаний уровня проточных озер / А.В. Фролов. - М.: Наука, 1985.
2. Бреер, В.В. Стохастические модели управления толпой / В.В. Бреер, Д.А. Новиков, А.Д. Рогаткин // Управление большими системами. - 2014. - № 52. - С. 85-117.
3. Кибзун, А.И. Построение доверительного множества поглощения в задачах анализа статических стохастических систем / А.И. Кибзун, С.В. Иванов, А.С. Степанова // Автоматика и телемеханика. - 2020. - Т. 81, № 4. - С. 21-36.
4. Nelson, E. Dynamical Theories of Brownian Motion / E. Nelson. - Princeton: Princeton University Press, 1967.
5. Gliklikh, Yu.E. Global and Stochastic Analisys with Applications to Mathematical Physicas / Yu.E. Gliklikh. - London; Dordrecht; Heidelberg; New York: Springer, 2011.
6. Свиридюк, Г.А. Динамические модели соболевского типа с условием Шоуолтера-Сидорова и аддитивными <шумами>/ Г.А. Свиридюк, Н.А. Манакова // Вестник ЮУр-ГУ. Серия: Математическое моделирование и программирование. - 2014. - Т. 7, № 1. -С. 90-103.
7. 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.
8. Favini, A. One Class of Sobolev Type Equations of Higher Order with Additive «White Noise> / A. Favini, G.A. Sviridyuk, A.A. Zamishlyaeva // Communications on Pure and Applied Analysis. - 2016. - V. 15, № 1. - P. 185-196.
9. 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.
10. Zagrebina, S. The Multipoint Initial-Final Value Problems for Linear Sobolev-Type Equations with Relatively p-Sectorial Operator and Additive «Noise> / S. Zagrebina, T. Sukacheva, G. Sviridyuk // Global and Stochastic Analysis. - 2018. - V. 5, № 2. - P. 129-143.
11. Shestakov, A.L. The Theory of Optimal Measurements / A.L. Shestakov, A.V. Keller, G.A. Sviridyuk // Journal of Computational and Engineering Mathematics. - 2014. - V. 1, № 1. - P. 3-16.
12. Da Prato, G. Stochastic Equations in infinite dimensions / G. Da Prato, J. Zabczyk. -Cambridge: Cambridge University Press, 1992.
13. Melnikova, I.V. Abstract Stochastic Equations. I. Classical and Distributional Solutions / I.V. Melnikova, A.I. Filinkov, U.A. Anufrieva // Journal of Mathematical Sciences. - 2002. -V. 111, № 2. - P. 3430-3475.
14. Kovacs, M. Introduction to Stochastic Partial Differential Equations / M. Kovacs, S. Larsson // Proceedings of «New Directions in the Mathematical and Computer Sciences:». - Abuja, 2008. - V. 4. - P. 159-232.
15. Замышляева, А.А. Стохастические неполные линейные уравнения соболевского типа высокого порядка с аддитивным белым шумом / А.А. Замышляева // Вестник ЮУр-ГУ. Серия: Математическое моделирование и программирование. - 2012. - № 40 (299), вып. 14. - С. 73-82.
16. Загребина, С.А. Линейные уравнения соболевского типа с относительно p-ограниченными операторами и аддитивным белым шумом / С.А. Загребина, Е.А. Сол-датова // Известия Иркутского государственного университета. Серия: Математика. -2013. - Т. 6, № 1. - С. 20-34.
17. Hoff, N.J. The Analysis of Structures / N.J. Hoff. - New York: John Wiley, 1956.
18. Сидоров, Н.А. Общие вопросы регуляризации в задачах теории ветвления / Н.А. Сидоров. - Иркутск: Издательство Иркутского государственного университета, 1982.
19. Сидоров, Н.А. О применении некоторых результатов теории ветвления при решении дифференциальных уравнений / Н.А. Сидоров, О.А. Романова // Дифференциальные уравнения. - 1983. - Т. 19, № 9. - С. 1516-1526.
20. Сидоров, Н.А. Обобщенные решения дифференциальных уравнений с фредгольмовым оператором при производной / Н.А. Сидоров, М.В. Фалалеев // Дифференциальные уравнения. - 1987. - Т. 23, № 4. - С. 726-728.
21. Свиридюк, Г.А. Уравнения Хоффа на графах / Г.А. Свиридюк, В.В. Шеметова // Дифференциальные уравнения. - 2006. - Т. 42, № 1. - С. 126-131.
22. Свиридюк, Г.А. О прямой и обратной задачах для уравнений Хоффа на графе / Г.А. Свиридюк, А.А. Баязитова // Вестник Самарского государственного технического университета. Серия: Физ.-мат. науки. - 2009. - № 1 (18). - С. 6-17.
23. Загребина, С.А. Устойчивость в моделях Хоффа / С.А. Загребина, П.О. Москвичева. -Saarbrücken: LAMBERT Academic Publishing, 2012.
24. Манакова, Н.А. Оптимальное управление решениями начально-конечной задачи для линейных уравнений соболевского типа / Н.А. Манакова, А.Г. Дыльков //Вестник ЮУр-ГУ. Серия: Математическое моделирование и программирование. - 2011. - № 17 (234), вып. 8. - С. 113-114.
25. Sagadeeva, M.A. Numerical Solution for Non-Stationary Linearized Hoff Equation Defined on Geometrical Graph / M.A. Sagadeeva, A.V. Generalov // Journal of Computational and Engineering Mathematics. - 2018. - V. 5, № 3. - P. 61-74.
26. Favini, A. The Multipoint Initial-Final Value Condition for the Hoff Equations on Geometrical Graph in Spaces of K-«Noises» / A. Favini, S.A. Zagrebina, G.A. Sviridyuk // Mediterranean Journal of Mathematics. - 2022. - V. 19. - Article ID: 53.
27. Солдатова Е.А. Начально-конечная задача для линейной стохастической модели Хоффа / Е.А. Солдатова // Вестник ЮУрГУ. Серия: Математическое моделирование и программирование. - 2014. - Т. 7, № 2. - C. 124-128.
Екатерина Александровна Солдатова, кандидат физико-математических наук, кафедра «Математическое и компьютерное моделирование:», ЮжноУральский государственный университет (г. Челябинск, Российская Федерация), soldatovaea@susu.ru.
Алевтина Викторовна Келлер, доктор физико-математических наук, профессор кафедры «Прикладная математика и механика>, Воронежский государственный технический университет, (г. Воронеж, Российская Федерация), alevtinak@inbox.ru.
Поступила в редакцию 14 марта 2024 г-
