Resistance scaling on 4N-carpets

The 4 ⁢ N {4N} -carpets are a class of infinitely ramified self-similar fractals with a large group of symmetries. For a 4 ⁢ N {4N} -carpet F, let { F n } n ≥ 0 {\{F_{n}\}_{n\geq 0}} be the natural decreasing sequence of compact pre-fractal approximations with ⋂ n F n = F {\bigcap_{n}F_{n}=F} . On each F n {F_{n}} , let ℰ ⁢ ( u , v ) = ∫ F N ∇ ⁡ u ⋅ ∇ ⁡ v ⁢ d ⁢ x {\mathcal{E}(u,v)=\int_{F_{N}}\nabla u\cdot\nabla v\,dx} be the classical Dirichlet form and u n {u_{n}} be the unique harmonic function on F n {F_{n}} satisfying a mixed boundary value problem corresponding to assigning a constant potential between two specific subsets of the boundary. Using a method introduced by [M. T. Barlow and R. F. Bass, On the resistance of the Sierpiński carpet, Proc. Roy. Soc. Lond. Ser. A 431 (1990), no. 1882, 345–360], we prove a resistance estimate of the following form: there is ρ = ρ ⁢ ( N ) > 1 {\rho=\rho(N)>1} such that ℰ ⁢ ( u n , u n ) ⁢ ρ n {\mathcal{E}(u_{n},u_{n})\rho^{n}} is bounded above and below by constants independent of n. Such estimates have implications for the existence and scaling properties of Brownian motion on F.

Mixed boundary value problem for an ordinary differential equation with fractional derivatives with different origins

Решается смешанная краевая задача для обыкновенного дифференциального уравнения, содержащего композицию лево- и правосторонних операторов дробного дифференцирования Римана-Лиувилля и Капуто. Задача эквивалентно редуцирована к интегральному уравнению Фредгольма второго рода, для которого найдено достаточное условие однозначной разрешимости. В качестве следствия,для исследуемой задачи доказано неравенство Ляпунова A mixed boundary value problem is solved for an ordinary differential equation containing a composition of left- and right-sided Riemann-Liouville and Caputo fractional differentiation operators. The problem is equivalently reduced to a Fredholm integral equation of the second kind, for which a sufficient condition for unique solvability is found. As a consequence, the Lyapunov inequality is proved for the problem under study.

Modelling and asymptotic analysis of the concentration difference in a nanoregion between an influx andoutflux diffusion acrossnarrow windows

When a flux of Brownian particles is injected in a narrow window located on the surface of a bounded domain, these particles diffuse and can eventually escape through a cluster of narrow windows. At steady state, we compute asymptotically the distribution of concentration between the different windows. The solution is obtained by solving Laplace’s equation using Green’s function techniques and second-order asymptotic analysis, and depends on the influx amplitude, the diffusion properties, as well as the geometrical organization of all the windows, such as their distances and the mean curvature. We explore the range of validity of the present asymptotic expansions using numerical simulations of the mixed boundary value problem. Finally, we introduce a length scale to estimate how deep inside a domain a local diffusion current can spread.

The stiff Neumann problem: Asymptotic specialty and “kissing” domains

We study the stiff spectral Neumann problem for the Laplace operator in a smooth bounded domain Ω ⊂ R d which is divided into two subdomains: an annulus Ω 1 and a core Ω 0 . The density and the stiffness constants are of order ε − 2 m and ε − 1 in Ω 0 , while they are of order 1 in Ω 1 . Here m ∈ R is fixed and ε > 0 is small. We provide asymptotics for the eigenvalues and the corresponding eigenfunctions as ε → 0 for any m. In dimension 2 the case when Ω 0 touches the exterior boundary ∂ Ω and Ω 1 gets two cusps at a point O is included into consideration. The possibility to apply the same asymptotic procedure as in the “smooth” case is based on the structure of eigenfunctions in the vicinity of the irregular part. The full asymptotic series as x → O for solutions of the mixed boundary value problem for the Laplace operator in the cuspidal domain is given.

Solvability for a New Class of Moore-Gibson-Thompson Equation with Viscoelastic Memory, Source Terms, and Integral Condition

This paper deals with the existence and uniqueness of solutions of a new class of Moore-Gibson-Thompson equation with respect to the nonlocal mixed boundary value problem, source term, and nonnegative memory kernel. Galerkin’s method was the main used tool for proving our result. This work is a generalization of recent homogenous work.

On a method for approximate solution of a mixed boundary value problem for an elliptic equation

A mixed boundary value problem for an elliptic equation of divergent type with variable coefficients is considered. It is assumed that the integration region is a rectangle, and the boundary of the integration region is the union of two disjoint pieces. The Dirichlet boundary condition is set on the first piece, and the Neumann boundary condition is set on the other one. The given problem is a problem with a discontinuous boundary condition. Such problems with mixed conditions at the boundary are most often encountered in practice in process modeling, and the methods for solving them are of considerable interest. This work is related to the paper [1] and complements it. It is focused on the approbation of the results established in [1] on the convergence of approximations of the original mixed boundary value problem with the main boundary condition of the third boundary value problem already with the natural boundary condition. On the basis of the results obtained in this paper and in [1], computational experiments on the approximate solution of model mixed boundary value problems are carried out.