On the Solvability of Stochastic Helmholtz Problem

Tleubergenov, M. I.; Azhymbaev, D. T.

doi:10.1007/s10958-021-05229-1

Cited by 7 publications

(3 citation statements)

References 5 publications

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“…Suppose that the Lagrange function is of the form L = In [26], the method of additional variables was applied to the problems in Examples 1 and 2 to construct Lagrangians in the class of stochastic differential equations equivalent a.s. by the given equations ( 29) and (38).…”

Section: Examplesmentioning

confidence: 99%

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Stochastic Helmholtz problem and convergence in distribution

Tleubergenov

Vassilina

Azhymbaev

2022

Filomat

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In the present paper, the solvability of the stochastic Helmholtz problem is investigated in the class of stochastic differential equations equivalent in distribution. Earlier, by additional variables method the Helmholtz problem was investigated in the class of stochastic differential equations equivalent almost surely (a.s.). The study of the stochastic Helmholtz problem in the class of equations equivalent in distribution allows us to significantly expand the region of its solvability. This is due to the possibility of using well-known methods of the theory of stochastic processes, such as the method of the phase space transformation, the method of absolutely continuous change of measure, and the method of random change of time. In that paper stochastic equations of the Lagrangian structure equivalent in distribution are constructed by the given second order Ito stochastic equations using the methods of phase space transformation, absolutely continuous measure transformation and randomtime substitution. The obtained results are illustrated by specific examples.

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Section: Examplesmentioning

confidence: 99%

“…In [20,21,25], the authors present their studies on the Helmholtz problem, mainly for ODEs and PDEs, as well as a historical overview of the development and generalization of the problem. In contrast to this paper, the stochastic Helmholtz problem is solved in the class of stochastic differential equations equivalent a.s. [26].…”

Section: Introductionmentioning

confidence: 99%

Stochastic Helmholtz problem and convergence in distribution

Tleubergenov

Vassilina

Azhymbaev

2022

Filomat

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“…These issues are closely related to the inverse problem of the calculus of variations (IPCV) in the following statement: for a given equation, one needs to construct a functional such that its set of stationary points coincides with the set of solutions to this equation. There is a large number of works devoted to inverse problems of the calculus of variations: for ordinary differential equations and partial differential equations [3], [4], [7], [9], [19], [20], [26], [27], operator equations [6], [21], [22], differential-difference equations [8], [17], [18], stochastic differential equations [23], [24], [25], fractional differential equations [1], [10], [14], [28]. In these works, nonlocal bilinear forms were mainly used to solve the IPCV.…”

Section: Introductionmentioning

confidence: 99%

On indirect representability of fourth order ordinary differential equation in form of Hamilton - Ostrogradsky equations

Budochkina,

Luu,

Shokarev

2023

Ufimsk. Mat. Zh.

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In the paper we solve the problem on the representability of a fourth order ordinary differential equation in the form of Hamilton-Ostrogradsky equations. Local bilinear forms play an essential role in the investigation of the potentiality property of the considered equation. It is well known that the problem of representing differential equations in the form of Hamilton-Ostrogradsky equations is closely related to the existence of a solution to the inverse problem of the calculus of variations, that is, for a given equation one needs to construct a functional-variational principle. To solve this problem, we first obtain necessary and sufficient conditions for the given equation to admit an indirect variational formulation relative to a local bilinear form and then construct the corresponding Hamilton-Ostrogradsky action. Note that the found conditions are analogous to the Helmholtz potentiality conditions for the given ordinary differential equation. We also define the structure of the considered equation with the potential operator and use the Ostrogradsky scheme to reduce the given equation to the form of Hamilton-Ostrogradsky equations.It should be noted that applications and extensions of the work are the possibility to establish connections between invariance of the functional and first integrals of the given equation and to extend the proposed scheme to partial differential equations and systems of such equations.

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