Asymptotic analysis of an elastic rod with rounded ends

We derive a one-dimensional model for an elastic shuttle, that is, a thin rod with rounded ends and small fixed terminals, by means of an asymptotic procedure of dimension reduction. In the model, deformation of the shuttle is described by a system of ordinary differential equations with variable degenerating coefficients, and the number of the required boundary conditions at the end points of the one-dimensional image of the rod depends on the roundness exponent m is an element of(0,1). Error estimates are obtained in the case m is an element of(0,1/4) by using an anisotropic weighted Korn inequality, which was derived in an earlier paper by the authors. We also briefly discuss boundary layer effects, which can be neglected in the case m is an element of(0,1/4) but play a crucial role in the formulation of the limit problem for m >= 1/4.Peer reviewe

A time-fractional Borel-Pompeiu formula and a related hypercomplex operator calculus

In this paper we develop a time-fractional operator calculus in fractional Clifford analysis. Initially we study the $L_p$-integrability of the fundamental solutions of the multi-dimensional time-fractional diffusion operator and the associated time-fractional parabolic Dirac operator. Then we introduce the time-fractional analogues of the Teodorescu and Cauchy-Bitsadze operators in a cylindrical domain, and we investigate their main mapping properties. As a main result, we prove a time-fractional version of the Borel-Pompeiu formula based on a time-fractional Stokes' formula. This tool in hand allows us to present a Hodge-type decomposition for the forward time-fractional parabolic Dirac operator with left Caputo fractional derivative in the time coordinate. The obtained results exhibit an interesting duality relation between forward and backward parabolic Dirac operators and Caputo and Riemann-Liouville time-fractional derivatives. We round off this paper by giving a direct application of the obtained results for solving time-fractional boundary value problems.The work of M. Ferreira, M.M. Rodrigues and N. Vieira was supported by Portuguese funds through CIDMA-Center for Research and Development in Mathematics and Applications, and FCT–Fundação para a Ciência e a Tecnologia, within project UID/MAT/04106/2019. The work of the authors was supported by the project New Function Theoretical Methods in Computational Electrodynamics / Neue funktionentheoretische Methoden für instationäre PDE, funded by Programme for Cooperation in Science between Portugal and Germany (“Programa de Ações Integradas Luso-Alemãs 2017” - DAAD-CRUP - Acção No. A-15/17 / DAAD-PPP Deutschland-Portugal, Ref: 57340281). N. Vieira was also supported by FCT via the FCT Researcher Program 2014 (Ref: IF/00271/2014).publishe

On multi-scale asymptotic structure of eigenfunctions in a boundary value problem with concentrated masses near the boundary

We construct two-term asymptotics ?? k = ?m?2(M + ??k + O(?3/2)) of eigenvalues of a mixed boundary-value problem in ? R2 with many heavy (m > 2) concentrated masses near a straight part of the boundary ? . ? is a small positive parameter related to size and periodicity of the masses; k ? N. The main term M > 0 is common for all eigenvalues but the correction terms ?k , which are eigenvalues of a limit problem with the spectral Steklov boundary conditions on , exhibit the effect of asymptotic splitting in the eigenvalue sequence enabling the detection of asymptotic forms of eigenfunctions. The justification scheme implies isolating and purifying singularities of eigenfunctions and leads to a new spectral problem in weighed spaces with a "strongly" singular weight.This research work has been partially supported by Spanish MINECO, MTM2013-44883-P. Also, the research work of the first author has been partially supported by Russian Foundation of Basic research (project 15–01–02175)

Multilayer Modelling of Lubricated Contacts: A New Approach Based on a Potential Field Description

A first integral approach, derived in an analogous fashion to Maxwell’s use of potential fields, is employed to investigate the flow characteristics, with a view to minimising friction, of shear-driven fluid motion between rigid surfaces in parallel alignment as a model for a lubricated joint, whether naturally occurring or engineered replacement. For a viscous bilayer arrangement comprised of immiscible liquids, it is shown how the flow and the shear stress along the separating interface is influenced by the mean thickness of the layers and the ratio of their respective viscosities. Considered in addition, is how the method can be extended for application to the more challenging problem of when one, or both, of the layers is a viscoelastic material