Search results for "partial differential equation"

Regularity and polar sets for supersolutions of certain degenerate elliptic equations

1988

On considere l'equation ⊇•⊇ h F(x,⊇u(x))=0. Cette equation est non lineaire et degeneree avec des coefficients mesurables. On etudie la regularite des supersolutions

EFFECTIVE FINITE-DIFFERENCE METHODS FOR THE SOLUTIONS OF FILTRATION PROBLEMS IN MULTILAYER DOMAINS

1997

In papers [1,2] there were consider different assumptions for averaging methods along the vertical coordinate.These methods were applied for the mathematical simulation of the mass transfer process in multilayered underground systems. A specific feature of these problems is that it is necessity to solve the 3‐D initial‐boundary‐value problems for parabolic type partial differential equations of second order with piece‐wise parameters in multilayer domain.Therefore here an effective finite‐difference method for solving a problem of the above type is developed.This method may be considered as a generalization of the method of finite volumes [3] for the layered systems. In the case of constant…

An Automatic Differentiation Based Approach to the Level Set Method

2015

This paper discusses an implementation of the parametric level set method. Adjoint approach is used to perform the sensitivity analysis, but contrary to standard implementations, the state problem is differentiated in its discretized form. The required partial derivatives are computed using tools of automatic differentiation, which avoids the need to derive the adjoint problem from the governing partial differential equation. The augmented Lagrangian approach is used to enforce volume constraints, and a gradient based optimization method is used to solve the subproblems. Applicability of the method is demonstrated by repeating well known compliance minimization studies of a cantilever beam …

Shakedown Analysis Within the Framework of Strain Gradient Plasticity

2015

A class of rate-independent material models is addressed within the framework of isotropic strain gradient plasticity. These models exhibit a size dependence through the strengthening effects (Hall–Petch effects), whereby the yield stress is related to the effective plastic strain by a suitable second-order partial differential equation with related boundary conditions. For a perfectly plastic material with strengthening effects, the classical concepts of plastic and shakedown limit analysis do hold, which lead to size dependent plastic and shakedown limit loads according to the dictum: smaller is stronger. In the perspective of a development of direct methods for applications to small-scal…

M/M/1 queue in two alternating environments and its heavy traffic approximation

2018

We investigate an M/M/1 queue operating in two switching environments, where the switch is governed by a two-state time-homogeneous Markov chain. This model allows to describe a system that is subject to regular operating phases alternating with anomalous working phases or random repairing periods. We first obtain the steady-state distribution of the process in terms of a generalized mixture of two geometric distributions. In the special case when only one kind of switch is allowed, we analyze the transient distribution, and investigate the busy period problem. The analysis is also performed by means of a suitable heavy-traffic approximation which leads to a continuous random process. Its d…

Partial differential equations and quasiregular mappings

1992

Singular distributed parameter systems

1993

The paper deals with the distributed parameter systems described by coupled partial differential equations with singular matrix coefficients. Initial-boundary-value problems are considered in the light of both singular 1d systems theory and the Fourier approach to distributed parameter systems. The method presented in this paper gives the possibility of determining acceptable initial-boundary conditions. An illustrative example is given.

Boundary-layer effects in wedges of piezoelectric laminates

2005

An approach to investigate boundary-layer effects in wedges of piezoelectric laminated structures is presented with the aim of ascertaining the electromechanical response characteristics. The wedge layer behavior is described in terms of generalized stress functions, which lead to a model consisting of a set of three coupled partial differential equations. The strength of the solution singularity is determined by solving the eigenvalue problem associated with the resolving system. The solution of the model is obtained by an eigenfunction expansion method coupled with a boundary collocation technique. Correspondingly, the singularity amplitude is assessed by introducing and calculating the g…

SPECIAL SPLINES OF HYPERBOLIC TYPE FOR THE SOLUTIONS OF HEAT AND MASS TRANSFER 3-D PROBLEMS IN POROUS MULTI-LAYERED AXIAL SYMMETRY DOMAIN

2017

In this paper we study the problem of the diffusion of one substance through the pores of a porous multi layered material which may absorb and immobilize some of the diffusing substances with the evolution or absorption of heat. As an example we consider circular cross section wood-block with two layers in the radial direction. We consider the transfer of heat process. We derive the system of two partial differential equations (PDEs) - one expressing the rate of change of concentration of water vapour in the air spaces and the other - the rate of change of temperature in every layer. The approximation of corresponding initial boundary value problem of the system of PDEs is based on the cons…

Mauro Picone, Sandro Faedo, and the numerical solution of partial differential equations in Italy (1928-1953)

2013

In this paper we revisit the pioneering work on the numerical analysis of partial differential equations (PDEs) by two Italian mathematicians, Mauro Picone (1885-1977) and Sandro Faedo (1913-2001). We argue that while the development of constructive methods for the solution of PDEs was central to Picone's vision of applied mathematics, his own work in this area had relatively little direct influence on the emerging field of modern numerical analysis. We contrast this with Picone's influence through his students and collaborators, in particular on the work of Faedo which, while not the result of immediate applied concerns, turned out to be of lasting importance for the numerical analysis of …