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A numerical-asymptotic multicomponent averaging method for equations with contrast coefficients. (English. Russian original) Zbl 0718.73008

U.S.S.R. Comput. Math. Math. Phys. 30, No. 1, 178-186 (1990); translation from Zh. Vychisl. Mat. Mat. Fiz. 30, No. 2, 243-253 (1990).
Nonlinear partial differential equations with contrast coefficients are considered. They describe a process in a nonhomogeneous periodic medium, which is characterized by the coefficient \(\epsilon \ll 1\) of periodicity and the coefficient \(\omega\geq 1\) of contrast of medium properties. In this connection the coefficients of those equations are periodic functions of spatial coordinates with rapid oscillations and large amplitudes.
The asymptotic solution of an initial value problem is considered for \(\epsilon\to 0\), \(\omega\to \infty\) and \(\epsilon^ 2\omega =const\). or \(\epsilon^ 2\omega \to \infty\) in the form of power series of \(\epsilon\) and 1/\(\omega\). The differential equations which are obtained after applying this procedure can be solved making use of a numerical method.
A particular case of a linear differential equation with rapidly oscillating coefficients is studied and the proximity of asymptotic and rigorous solutions is estimated.

MSC:

74E05 Inhomogeneity in solid mechanics
65Z05 Applications to the sciences
35C20 Asymptotic expansions of solutions to PDEs
65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
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