Abstract
The paper considers the issues of constructive analysis of the de la Vallee – Poussin boundary-value problem for the second-order linear matrix differential Lyapunov equation with a parameter and variable coefficients. The initial problem is reduced to an equivalent integral problem, and to study its solvability a modification of the generalized contraction mapping principle is used. A connection between the approach used and the Green’s function method is established. The coefficient sufficient conditions for the unique solvability of this problem are obtained. Using the Lyapunov – Poincaré small parameter method, an algorithm for constructing a solution has been developed. The convergence and the rate of convergence of this algorithm have been investigated, and a constructive estimation of the region of solution localization is given. To illustrate the application of the results obtained, the linear problem of steady heat conduction for a cylindrical wall, as well as a two-dimensional matrix model problem is considered. With the help of the developed general algorithm, analytical approximate solutions of these problems have been constructed and on the basis of their exact solutions a comparative numerical analysis has been carried out.
Highlights
The paper considers the issues of constructive analysis of the de la Vallee – Poussin boundary-value problem for the second-order linear matrix differential Lyapunov equation with a parameter and variable coefficients
Two point nonlinear Lyapunov systems associated with annth order nonlinear system of differential equations – existence and uniqueness / K
Information about the authorsЛаптинский Валерий Николаевич – доктор физико- математических наук, профессор, главный научный сотрудник, Институт технологии металлов Национальной академии наук Беларуси
Summary
Y= Y0 + lY1 + l 2Y2 + ... + l k Yk +. Подставляя (16), (17), (18) в (14), (15) и приравнивая коэффициенты при одинаковых степенях λ, получим последовательно. При этом = X 0(0) M, = X 0(ω) N, поскольку PUV (0) = 0, PUV (ω) = N - M. ∫ ∫ K U (τ, s)H m (s)K V (s, τ)ds d τ dj ≤ glU lV ∫ dj∫ ∫ K U (τ, s)H m (s)K V (s, τ)ds d τ ≤. ≤ glU lV ∫ dj∫ ∫ K U (τ, s)H m (s)K V (s, τ) ds d τ ≤ glU lV ∫ dj∫ ∫ K U (τ, s) H m (s) K V (s, τ) ds d τ ≤. ≤ glU2 lV2 ∫ dj∫ ∫ H m (s) ds d τ ≤glU2 lV2 ∫ dj∫ j - τ d τ (α1 + b1 ) X m C + (α2 + b2 ) Ym C =.
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