This paper investigates the effect of different methods of carbon nanotubes distribution in a thin matrix on static and dynamic behavior of the nanocomposite. Five different symmetric patterns of distribution are considered, including four parabolic patterns and a linear one. For each pattern, the effective mechanical properties of the resultant nanocomposite are calculated using the rule of mi...

This paper investigates the effect of different methods of carbon nanotubes distribution in a thin matrix on static and dynamic behavior of the nanocomposite. Five different symmetric patterns of distribution are considered, including four parabolic patterns and a linear one. For each pattern, the effective mechanical properties of the resultant nanocomposite are calculated using the rule of mi...

In the present paper, stability and convergence properties of linear multistep methods are investigated. The attention is focused on parabolic problems and variable stepsizes. Under weak assumptions on the method and the stepsize sequence an asymptotic stability result is shown. Further, stability bounds for linear nonautonomous parabolic problems with Hölder continuous operator are given. With...

We establish a Lipschitz stability estimate for the inverse problem consisting in the determination of the coe cient σ(t), appearing in a Dirichlet initial-boundary value problem for the parabolic equation ∂tu − ∆xu + σ(t)f(x)u = 0, from Neumann boundary data. We extend this result to the same inverse problem when the previous linear parabolic equation is changed to the semi-linear parabolic eq...

These notes contain a short exposition of selected results about parabolic equations: Schauder estimates for linear parabolic equations with Hölder coefficients, some existence, uniqueness and regularity results for viscosity solutions of fully nonlinear parabolic equations (including degenerate ones), the Harnack inequality for fully nonlinear uniformly parabolic equations. MSC. 35K55, 35D40, ...