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We consider quasilinear and linear parabolic problems with rapidly oscillating coefficients in a domain Ω that is ε-periodically perforated by small holes of order [MediaObject not available: see fulltext.]. The holes are divided into three ε-periodical sets depending on boundary conditions. The homogeneous Dirichlet boundary conditions are imposed for holes of one set, whereas, for holes in the remaining sets, different inhomogeneous Neumann and nonlinear Robin boundary conditions involving additional perturbation parameters are imposed. For a solution to the quasilinear problem we find the leading terms of the asymptotic expansion and prove asymptotic estimates that show the influence of perturbation parameters. In the linear case, we construct and justify a complete asymptotic expansion of the solution by using the two-scale asymptotic expansion method. Bibliography: 25 titles. Illustrations: 1 figure. [ABSTRACT FROM AUTHOR]

This paper considers the structure of a set of systems of continuous solutions in a number of cases depending on the hypotheses for the matrices A, B, number q and their properties. Using the methods of the theory of differential and difference equations, we define new conditions for the existence of continuous solutions of these systems of equations. Specifically, we develop the method of their construction and examine their properties. In theorems 1 and 3 we obtain the results under the conditions ai1, i = 1,…, n, q > 1, (t≤0), 0 i < 1, i = 1,…,m, q > 1, (t ≥ 0), whilst in theorems 5, 6 we obtain the research results under Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. [ABSTRACT FROM AUTHOR]

We consider a parabolic semilinear problem with rapidly oscillating coefficients in a domain Ωε that is ε-periodically perforated by small holes of size $\mathcal {O}$(ε). The holes are divided into two ε-periodical sets depending on the boundary interaction at their surfaces, and two different nonlinear Robin boundary conditions σε( uε) + εκ m( uε) = ε g( m) ε, m = 1, 2, are imposed on the boundaries of holes. We study the asymptotics as ε → 0 and establish a convergence theorem without using extension operators. An asymptotic approximation of the solution and the corresponding error estimate are also obtained. Bibliography: 60 titles. Illustrations: 1 figure. [ABSTRACT FROM AUTHOR]

We consider a boundary-value problem for the second-order elliptic differential operator with rapidly oscillating coefficients in a domain Ω ε that is ε-periodically perforated by small holes. The holes are split into two ε-periodic sets depending on the boundary interaction via their boundary surfaces. Therefore, two different nonlinear boundary conditions σ ε( u ε) + εκ m( u ε) = εg ε ( m) , m = 1 , 2 , are given on the corresponding boundaries of the small holes. The asymptotic analysis of this problem is performed as ε → 0 , namely, the convergence theorem for both the solution and the energy integral is proved without using an extension operator, asymptotic approximations for the solution and the energy integral are constructed, and the corresponding approximation error estimates are obtained. [ABSTRACT FROM AUTHOR]

The structure of a set of continuous solutions of the systems of linear difference equations with rational deviations of continuous argument has been investigated and a method of construction of continuous solutions of such systems has been developed. [ABSTRACT FROM AUTHOR]