Construction of solutions of integro-differential equations with restrictions and control by projection-iterative method

Luchka, A. Yu.; Nesterenko, O. B.

doi:10.1007/s11072-009-0061-9

Cited by 7 publications

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References 2 publications

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“…The theory of control problems for a system of ordinary differential equations and for a system of integro-differential equations in partial derivatives, with parameters, is rapidly developing and used in various fields of applied mathematics, biophysics, biomedicine, chemistry, etc. Control problems, also called as boundary value problems with parameters and parameter identification problems for systems of ordinary differential and integro-differential equations with parameters, are intensively studied by many authors [3,4,8,9,17,18,19,20,24,25]. To find solutions to these problems, methods of the qualitative theory of differential equations, variational calculus and optimization theory, the method of upper and lower solutions, etc.…”

Section: Introductionmentioning

confidence: 99%

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A Problem With Parameter for the Integro-Differential Equations

Бакирова

Assanova

Kadirbayeva

2021

Mathematical Modelling and Analysis

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The article proposes a numerically approximate method for solving a boundary value problem for an integro-differential equation with a parameter and considers its convergence, stability, and accuracy. The integro-differential equation with a parameter is approximated by a loaded differential equation with a parameter. A new general solution to the loaded differential equation with a parameter is introduced and its properties are described. The solvability of the boundary value problem for the loaded differential equation with a parameter is reduced to the solvability of a system of linear algebraic equations with respect to arbitrary vectors of the introduced general solution. The coefficients and the right-hand sides of the system are compiled through solutions of the Cauchy problems for ordinary differential equations. Algorithms are proposed for solving the boundary value problem for the loaded differential equation with a parameter. The relationship between the qualitative properties of the initial and approximate problems is established, and estimates of the differences between their solutions are given.

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Section: Introductionmentioning

confidence: 99%

“…20) where P (t) is a square matrix or a vector of dimension n continuous on [0, T ].Denote by a r (P, t) the unique solution to the Cauchy problem(2.20). It is clear thata r (P, t) = X r (t) t θr−1 X −1 r (τ )P (τ )dτ, t ∈ [θ r−1 , θ r ], r = 1, m. (2.21)…”

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confidence: 99%

A Problem With Parameter for the Integro-Differential Equations

Бакирова

Assanova

Kadirbayeva

2021

Mathematical Modelling and Analysis

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“…max Параметрі бар интеграл-дифференциалдық теңдеулер үшін шеттік есептер, механика, физика, химия, техника, биология, экономика және басқада түрлі үдерістердің математикалық моделі бола отырып, қолданбалы математиканың көптеген бөлімдерінде кездеседі. Интеграл-дифференциалдық теңдеулер үшін және параметрі бар интеграл-дифференциалдық теңдеулер үшін шеттік есептердің шешілімділік, жәнеде олардың шешімін табу әдістерін құру мәселелері [1][2][3][4][5][6][7][8][9][10] жұмыстарда зерттелді.…”

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On a Method for Solving a Linear Boundary Value Problem for a System of Integro-Differential Equations Containing the Parameter

Iskakova¹,

Alihanova²,

Duisen³

2021

Bullet. Ser. of Phys. & Math. Sc.

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In the present work for a limited period, we consider the system of integro-differential equations of containing the parameter. The kernel of the integral term is assumed to be degenerate, and as additional conditions for finding the values of the parameter and the solution of the given integro-differential equation, the values of the solution at the initial and final points of the given segment are given. The boundary value problem under consideration is investigated by D.S. Dzhumabaev's parametrization method. Based on the parameterization method, additional parameters are introduced. For a fixed value of the desired parameter, the solvability of the special Cauchy problem for a system of integro-differential equations with a degenerate kernel is established. Using the fundamental matrix of the differential part of the integro-differential equation and assuming the solvability of the special Cauchy problem, the original boundary value problem is reduced to a system of linear algebraic equations with respect to the introduced additional parameters. The existence of a solution to this system ensures the solvability of the problem under study. An algorithm for finding the solution of the initial problem based on the construction and solutions of a system of linear algebraic equations is proposed.

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Construction of approximate solutions of systems of differential equations with parameters and restrictions

Luchka

Feruk

2011

Nonlinear Oscill

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We propose a new approach to the investigation of boundary-value problems for systems of differential equations with parameters and restrictions. This approach uses ideas of both iterative and projection methods. We develop and substantiate a new modification of the projection-iterative method for the construction of approximate solutions of boundary-value problems that has significant advantages over the existing versions of the method.The solution of numerous problems of contemporary natural sciences is impossible without the construction of various mathematical models, the investigation and analysis of which stimulate the development of many fields of mathematics. Among these mathematical models, an important place is occupied by problems with parameters. There are many papers devoted to the development of the theory of these problems. In particular, in [1-5], problems of analytic and qualitative theory and development of approximation methods for the construction of solutions of differential, integral, and integro-differential equations with restrictions were considered. In the present paper, we propose a new approach to the investigation of boundary-value problems for systems of differential equations with parameters and restrictions and develop an efficient method for the construction of their solutions. Statement of the ProblemConsider a system of ordinary differential equations of the form(1) whose solutions satisfy the conditionswhere P .t /; C.t/; and S.t / are m m; m l; and l m matrices, respectively, whose elements are square summable on the segment OE0; T : Furthermore, the columns of the matrix C.t/ are linearly independent, D is a constant m m matrix, f 2 L 2 .OE0; T ; R m /;˛2 R l ; and 2 R m : A solution of problem (1), (2) is understood as a vector 2 R l and vector function x 2 W 1 2 .OE0; T ; R m / that satisfy Eq. (1) and conditions (2). If this is not true, then the problem is inconsistent. In what follows, for convenience, we use the term "function" instead of "vector function."

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Construction of solutions of integro-differential equations with restrictions and control by projection-iterative method

Cited by 7 publications

References 2 publications

A Problem With Parameter for the Integro-Differential Equations

A Problem With Parameter for the Integro-Differential Equations

On a Method for Solving a Linear Boundary Value Problem for a System of Integro-Differential Equations Containing the Parameter

Construction of approximate solutions of systems of differential equations with parameters and restrictions

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