Cauchy Problem for a Linear System of Ordinary Differential Equations of the Fractional Order

Мамчуев, М. О.

doi:10.3390/math8091475

Cited by 4 publications

(2 citation statements)

References 7 publications

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“…Problem (2) to the inhomogeneous linear Equation (1) with a sectorial unbounded linear operator A was studied in [30]. There are other works on various differential equations with Dzhrbashyan-Nersesyan fractional derivatives [31][32][33][34][35][36][37].…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Inverse Problems for Equations with Dzhrbashyan–Nersesyan Derivatives

Fedorov,

Plekhanova,

Melekhina

2023

Fractal Fract

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The unique solvability in the sense of classical solutions for nonlinear inverse problems to differential equations, solved for the oldest Dzhrbashyan–Nersesyan fractional derivative, is studied. The linear part of the equation contains a bounded operator, a continuous nonlinear operator that depends on lower-order Dzhrbashyan–Nersesyan derivatives, and an unknown element. The inverse problem is given by an equation, special initial value conditions for lower Dzhrbashyan–Nersesyan derivatives, and an overdetermination condition, which is defined by a linear continuous operator. Applying the fixed-point method for contraction mapping a theorem on the existence of a local unique solution is proved under the condition of local Lipschitz continuity of the nonlinear mapping. Analogous nonlocal results were obtained for the case of the nonlocally Lipschitz continuous nonlinear operator in the equation. The obtained results for the problem in arbitrary Banach spaces were used for the research of nonlinear inverse problems with time-dependent unknown coefficients at lower-order Dzhrbashyan–Nersesyan time-fractional derivatives for integro-differential equations and for a linearized system of dynamics of fractional Kelvin–Voigt viscoelastic media.

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Section: Introductionmentioning

confidence: 99%

Nonlinear Inverse Problems for Equations with Dzhrbashyan–Nersesyan Derivatives

Fedorov,

Plekhanova,

Melekhina

2023

Fractal Fract

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“…One of the rapidly developing areas of modern mathematics is the theory of fractional differential equations and their applications [1][2][3][4][5][6][7] (also see the references therein). Among the many different definitions of the fractional derivative, the Riemann-Liouville [8] and Gerasimov-Caputo [8][9][10] derivatives are most commonly used.…”

Section: Introductionmentioning

confidence: 99%

Initial Value Problems of Linear Equations with the Dzhrbashyan–Nersesyan Derivative in Banach Spaces

2021

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Among the many different definitions of the fractional derivative, the Riemann–Liouville and Gerasimov–Caputo derivatives are most commonly used. In this paper, we consider the equations with the Dzhrbashyan–Nersesyan fractional derivative, which generalizes the Riemann–Liouville and the Gerasimov–Caputo derivatives; it is transformed into such derivatives for two sets of parameters that are, in a certain sense, symmetric. The issues of the unique solvability of initial value problems for some classes of linear inhomogeneous equations of general form with the fractional Dzhrbashyan–Nersesyan derivative in Banach spaces are investigated. An inhomogeneous equation containing a bounded operator at the fractional derivative is considered, and the solution is presented using the Mittag–Leffler functions. The result obtained made it possible to study the initial value problems for a linear inhomogeneous equation with a degenerate operator at the fractional Dzhrbashyan–Nersesyan derivative in the case of relative p-boundedness of the operator pair from the equation. Abstract results were used to study a class of initial boundary value problems for equations with the time-fractional Dzhrbashyan–Nersesyan derivative and with polynomials in a self-adjoint elliptic differential operator with respect to spatial variables.

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Non-local boundary value problem for a system of ordinary differential equations with Riemann–Liouville derivatives

Мамчуев¹,

Жабелова²

2022

Вестник КРАУНЦ. Физико-Математические Науки

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В работе исследуется нелокальная краевая задача для линейной системы обыкновенных дифференциальных уравнений дробного порядка с постоянными коэффициентами на отрезке [0,l]. Дробная производная порядка α∈(0,1] понимается в смысле Римана–Лиувилля. Краевые условия связывают след дробного интеграла от искомой вектор-функции на левом конце отрезка – в точке x=0, со следом самой вектор функции на правом конце отрезка – в точке x=l. Цель настоящей работы – построение явного представления решения данной задачи в терминах функции Грина. Исследована структура решения краевой задачи, определена и построена соответствующая функция Грина, получено представление решения. Доказана теорема об однозначной разрешимости исследуемой краевой задачи. We study a nonlocal boundary value problem for a linear system of ordinary differential equations of fractional order with constant coefficients on the interval [0,l]. The fractional derivative of order α∈(0,1] is understood in the Riemann–Liouville sense. The boundary conditions connect the trace of the fractional integral of the desired vector function at the left end of the segment – at the x=0, with the trace of the vector function itself at the right end of the segment at the point x=l. The purpose of this work is to construct an explicit representation of the solution of this problem in terms of the Green’s function. The structure of the solution to the boundary value problem is investigated, the corresponding Green’s function is defined and constructed, and the representation of the solution is obtained. A theorem on the unique solvability of the boundary value problem under study is proved.

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Cauchy Problem for a Linear System of Ordinary Differential Equations of the Fractional Order

Cited by 4 publications

References 7 publications

Nonlinear Inverse Problems for Equations with Dzhrbashyan–Nersesyan Derivatives

Nonlinear Inverse Problems for Equations with Dzhrbashyan–Nersesyan Derivatives

Initial Value Problems of Linear Equations with the Dzhrbashyan–Nersesyan Derivative in Banach Spaces

Non-local boundary value problem for a system of ordinary differential equations with Riemann–Liouville derivatives

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