Averaging in Boundary-Value Problems for Systems of Differential and Integrodifferential Equations
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AVERAGING IN BOUNDARY-VALUE PROBLEMS FOR SYSTEMS OF DIFFERENTIAL AND INTEGRODIFFERENTIAL EQUATIONS A. N. Stanzhitskii,1,2 S. T. Mynbayeva,3 and N. A. Marchuk4
UDC 517.9
The averaging method is applied to the investigation of the problem of existence of solutions of boundaryvalue problems for systems of differential and integrodifferential equations. It is shown that if the averaged boundary-value problem has a solution, then the original problem also has a solution. Note that, in this case, the system obtained as a result of averaging of a system of integrodifferential equations has the form of a simpler system of ordinary differential equations.
Introduction We consider boundary-value problems for systems of differential and integrodifferential equations with small parameter of the form ✓ ✓ ◆◆ T =0 (1) x˙ = "X(t, x), F x(0), x " and 0
x˙ = "X @t, x,
Zt 0
1 � � ' t, s, x(s) dsA,
✓ ◆◆ T = 0, F x(0), x " ✓
(2)
where " > 0 is a small parameter, X and F are d-dimensional vector functions, and ' is an m-dimensional vector function. Under the condition of the existence of the integral mean 1 X0 (x) = lim T !1 T
ZT
X(t, x) dt,
(3)
0
problem (1) is associated with the following averaged boundary-value problem: y˙ = "X0 (y),
✓ ✓ ◆◆ T =0 F y(0), y "
(4)
1
T. Shevchenko Kiev National University, Kiev, Ukraine; e-mail: [email protected]. Corresponding author. 3 Zhubanov Aktyubinsk Regional State University, Aktobe, Kazakhstan; Institute of Mathematics and Mathematical Modelling, Almaty, Kazakhstan; e-mail: [email protected]. 4 Podol’skii Agrarian-Technical University, Kamenets-Podol’skii, Ukraine; e-mail: [email protected]. 2
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 2, pp. 245–266, February, 2020. Original article submitted September 18, 2019. 0041-5995/20/7202–0277
© 2020
Springer Science+Business Media, LLC
277
A. N. S TANZHITSKII , S. T. M YNBAYEVA ,
278
AND
N. A. M ARCHUK
or, in terms of “slow time” ⌧ = "t, with � � F y(0), y(T ) = 0.
dy = X0 (y), d⌧ Similarly, for problem (2), if '1 (t, x) =
Z
t 0
then the integral mean has the form 1 X0 (x) = lim T !1 T
ZT 0
(5)
� � ' t, s, x ds,
� � X t, x, '1 (t, x) dt,
(6)
and the averaged boundary-value problem has the form (4) or (5). The aim of the present paper is to prove that, in the case where averaged boundary-value problems are solvable, the original boundary-value problems (1) and (2) also have solutions for small values of the parameter " and to show that the corresponding solutions of the exact and averaged problems are close. The exact statement of the problem and main results are given in Sec. 2. Integrodifferential equations serve as mathematical models of numerous actual processes in various fields of natural sciences, e.g., in fluid dynamics and kinetic chemistry (see [1–3] and the references therein). These equations are also encountered in the investigation of multivolatile populations [4], space-time spread of epidemics [5], etc. In these models, it is, as a rule, necessary to solve boundary-value pr
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