Solvability and construction of solution to the de la Vallee – Poussin problem for the second-order matrix Lyapunov equation with a parameter
https://doi.org/10.29235/1561-2430-2019-55-1-50-61
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.
About the Authors
A. I. KashparBelarus
Assistant Rector.
43, Mira Ave., 212000, Mogilev.
V. N. Laptinskiy
Belarus
Dr. Sc. (Physics and Mathematics), Chief Researcher.
11, Byalynitskii-Birulya Str., 212030, Mogilev.
References
1. Sansone G. Equazioni differeziali nel campo reale [Ordinary Differential Equations]. Bologna, N. Zanichelli, 1948. 875 p. (in Italiano).
2. Hartman F. Ordinary Differential Equations. NY, John Wiley & Sons, 1964. 624 p. https://doi.org/10.1137/1.9780898719222
3. Avduevskii V. S., Galitseiskii B. M., Glebov G. A., Danilov Iu. I., Dreitser G. A., Kalinin E. K., Koshkin V. K., Mikhailov T. V., Molchanov A. M., Ryzhov Iu. A., Solntsev V. P. Fundamentals of Heat Transfer in Aviation and Rocket and Space Technology. Moscоw, Mashinostroenie Publ., 1970. 624 p. (in Russian).
4. Isaev S. I., Kozhinov I. A., Kofanov V. I., Leont'ev A. I., Mironov B. M., Nikitin V. M., Petrazhitskii G. B., Khvostov V. I., Chukaev A. G., Shishov E. V., Shkola V. V. Theory of Heat and Mass Transfer. Moscоw, Vysshaia Shkola Publ., 1979. 495 p. (in Russian).
5. Murty K. N., Howell G. W., Sarma G. V. R. L. Two (multi) point nonlinear Lyapunov systems associated with an nth order nonlinear system of differential equations – existence and uniqueness. Mathematical Problems in Engineering, 2000, vol. 6, no. 4, pp. 395–410. https://doi.org/10.1155/s1024123x00001393
6. Murty K. N, Howell G. W., Sivasundaram S. Two (multi) point nonlinear Lyapunov systems – Existence and uniqueness. Journal of Mathematical Analysis and Applications, 1992, vol. 167, no. 2, pp. 505–515. https://doi.org/10.1016/0022-247x(92)90221-x
7. Derevenskii V. P. Matrix two-sided linear differential equations. Mathematical Notes, 1994, vol. 55, no. 1. pp. 24-29. https://doi.org/10.1007/bf02110760
8. Derevenskii V. P. Matrix linear differential equations of higher orders. Differentsial'nye uravneniia = Differential equations, 1993, vol. 29, no. 4, pp. 711–714 (in Russian).
9. Derevenskii V. P. Matrix linear differential equations of the second order. Differentsial'nye uravneniia = Differential equations, 1995, vol. 31, no. 11, pp. 1926–1927 (in Russian).
10. Laptinskii V. N., Kashpar A. I. Constructive analysis of boundary value problem de la Vallee Poussin for linear matrix equation Lyapunov second order. Preprint, Part I. Mogilev, Belarusian-Russian University Publ., 2015. 48 р. (in Russian).
11. Laptinskii V. N. Constructive Analysis of Controlled Oscillatory Systems. Minsk, Institute of Mathematics of the National Academy of Sciences of Belarus Publ., 1998. 300 p. (in Russian).
12. Kantorovich L. V., Akilov G. P. Functional Analysis. Moscоw, Nauka Publ., 1977. 744 р. (in Russian).
13. Riess F., Sz.-Nad B. Leçons Dʹnalyse Fonctionnelle. Budapest, Akadémia Kiadó, 1972.
14. Krasnosel'skii M. A., Vainikko G. M., Zabreiko P. P., Rutitskii Ia. B., Stetsenko V. Ia. Approximate Solution of Operator Equations. Moscоw, Nauka Publ., 1969. 456 р. (in Russian).