Pseudoperiodic solution of a system of integrodifferential equations

Sartabanov, Zh.A.

doi:10.1007/bf01060661

Cited by 5 publications

(4 citation statements)

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“…The foundations of the method used in this note were laid in [1,2], which were further developed in [3][4][5][6][7][8][9][10] and applied to the study of solutions different problems in the partial differential equations [11,12]. These methods with simple modifications extend to the study solutions of problems of the differential and integro-differential equations of different types [1][2][3][4][5][6][7][8][9][10][11][12], in particular, problems on multi-frequency solutions of equations from control theory [13]. The methods of research for multiperiodic solutions are successfully combined by methods for studying solutions of boundary value problems for equations of mathematical physics.…”

Section: Introductionmentioning

confidence: 99%

Research of Multiperiodic Solutions of Perturbed Linear Autonomous Systems With Differentiation Operator on the Vector Field

Sartabanov¹,

Omarova²,

Kerimbekov³

2020

OF THE NATIONAL ACADEMY OF SCIENCES OF THE REPUBLIC OF KAZAKHS

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News of the National Academy of sciences of the Republic of Kazakhstan 2 Б а с р е д а к т о р ы ф.-м.ғ.д., проф., ҚР ҰҒА академигі Ғ.М. Мұтанов Р е д а к ц и я а л қ а с ы: Жұмаділдаев А.С. проф., академик (Қазақстан) Кальменов Т.Ш. проф., академик (Қазақстан) Жантаев Ж.Ш. проф., корр.-мүшесі (Қазақстан) Өмірбаев У.У. проф. корр.-мүшесі (Қазақстан) Жүсіпов М.А. проф. (Қазақстан) Жұмабаев Д.С. проф. (Қазақстан) Асанова А.Т. проф. (Қазақстан) Бошкаев К.А. PhD докторы (Қазақстан) Сұраған Д. корр.-мүшесі (Қазақстан) Quevedo Hernando проф. (Мексика), Джунушалиев В.Д. проф. (Қырғыстан) Вишневский И.Н. проф., академик (Украина) Ковалев А.М. проф., академик (Украина) Михалевич А.А. проф., академик (Белорусь) Пашаев А. проф., академик (Әзірбайжан) Такибаев Н.Ж. проф., академик (Қазақстан), бас ред. орынбасары Тигиняну И. проф., академик (Молдова) «ҚР ҰҒА Хабарлары. Физика-математикалық сериясы».

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Section: Introductionmentioning

confidence: 99%

Research of Multiperiodic Solutions of Perturbed Linear Autonomous Systems With Differentiation Operator on the Vector Field

Sartabanov¹,

Omarova²,

Kerimbekov³

2020

OF THE NATIONAL ACADEMY OF SCIENCES OF THE REPUBLIC OF KAZAKHS

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“…The foundations of the method used in this note were laid in [1,2], which were further developed in [3][4][5][6][7][8][9][10][11][12][13][14] and applied to the study of solutions different problems in the partial differential equations [15,16]. These methods with simple modifications extend to the study solutions of problems of the differential and integro-differential equations of different types [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16], in particular, problems on multi-frequency solutions of equations from control theory [17]. Many oscillatory phenomena are described by systems with a differentiation operator with respect to toroidal vector fields, and new methods based on the ideas of the Fourier [18], Poincaré-Lyapunov and Hamilton-Jacobi methods [19,20] appear to establish their periodic oscillatory solutions.…”

Section: Introductionmentioning

confidence: 99%

Research of Multiperiodic Solutions of Perturbed Linear Autonomous Systems With Differentiation Operator on the Vector Field

Sartabanov¹,

Omarova²,

Kerimbekov³

2019

OF THENATIONAL ACADEMY OF SCIENCES OF THE REPUBLIC OF KAZAKHSTA

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A linear system with a differentiation operator D with respect to the directions of vector fields of the form of the Lyapunov's system with respect to space independent variables and a multiperiodic toroidal form with respect to time variables is considered. All input data of the system multiperiodic depend on time variables or do not depend on them. The autonomous case of the system was considered in our early work. In this case, some input data received perturbations depending on time variables. We study the question of representing the required motion described by the system in the form of a superposition of individual periodic motions of rationally incommensurable frequencies. The initial problems and the problems of multiperiodicity of motions are studied. It is known that when determining solutions to problems, the system integrates along the characteristics outgoing from the initial points, and then, the initial data is replaced by the first integrals of the characteristic systems. Thus, the required solution consists of the following components: characteristics and first integrals of the characteristic systems of operator D, the matricant and the free term of the system itself. These components, in turn, have periodic and non-periodic structural components, which are essential in revealing the multiperiodic nature of the movements described by the system under study. The representation of a solution with the selected multiperiodic components is called the multiperiodic structure of the solution. It is realized on the basis of the well-known Bohr's theorem on the connection of a periodic function of many variables and a quasiperiodic function of one variable. Thus, more specifically, the multiperiodic structures of general and multiperiodic solutions of homogeneous and inhomogeneous systems with perturbed input data are investigated. In this spirit, the zeros of the operator D and the matricant of the system are studied. The conditions for the absence and existence of multiperiodic solutions of both homogeneous and inhomogeneous systems are established.

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“…The research of multi-frequency oscillations led to the concept of multidimensional time. In this connection, of the theory solutions of partial differential equations that are periodic in multidimensional time is being developed, both in time and in space independent variables [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35]. It is known that the system of canonical Hamilton equations, under fairly general conditions, can be solved by the Jacobi method, the essence of which is the transition from its integration to the integration of a partial differential equation.…”

mentioning

confidence: 99%

“…A similar approach is implemented in [19], where quasiperiodic solutions of ordinary differential equations are studied with a transition to the research of multiperiodic solutions of partial differential equations. This method was developed in [20][21][22][23][24][25][26][27][28][29][30] with its extension to the solution of a number of oscillation problems in systems of integro-differential equations.…”

mentioning

confidence: 99%

MULTIPERIODIC SOLUTIONS OF LINEAR SYSTEMS INTEGRO-DIFFERENTIAL EQUATIONS WITH Dc -OPERATOR AND  -PERIOD OF HEREDITARY

Sartabanov

Aitenova

2019

OF THENATIONAL ACADEMY OF SCIENCES OF THE REPUBLIC OF KAZAKHSTA

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Pseudoperiodic solution of a system of integrodifferential equations

Cited by 5 publications

References 1 publication

Research of Multiperiodic Solutions of Perturbed Linear Autonomous Systems With Differentiation Operator on the Vector Field

Research of Multiperiodic Solutions of Perturbed Linear Autonomous Systems With Differentiation Operator on the Vector Field

Research of Multiperiodic Solutions of Perturbed Linear Autonomous Systems With Differentiation Operator on the Vector Field

MULTIPERIODIC SOLUTIONS OF LINEAR SYSTEMS INTEGRO-DIFFERENTIAL EQUATIONS WITH Dc -OPERATOR AND  -PERIOD OF HEREDITARY

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