Search results for "Mathematical analysis"

Zero Viscosity Limit for Analytic Solutions, of the Navier-Stokes Equation on a Half-Space.¶I. Existence for Euler and Prandtl Equations

1998

This is the first of two papers on the zero-viscosity limit for the incompressible Navier-Stokes equations in a half-space. In this paper we prove short time existence theorems for the Euler and Prandtl equations with analytic initial data in either two or three spatial dimensions. The main technical tool in this analysis is the abstract Cauchy-Kowalewski theorem. For the Euler equations, the projection method is used in the primitive variables, to which the Cauchy-Kowalewski theorem is directly applicable. For the Prandtl equations, Cauchy-Kowalewski is applicable once the diffusion operator in the vertical direction is inverted.

Efficient and accurate computation of Green's function for the Poisson equation in rectangular waveguides

2009

[1] In this paper, a new algorithm for the fast and precise computation of Green's function for the 2-D Poisson equation in rectangular waveguides is presented. For this purpose, Green's function is written in terms of Jacobian elliptic functions involving complex arguments. A new algorithm for the fast and accurate evaluation of such Green's function is detailed. The main benefit of this algorithm is successfully shown within the frame of the Boundary Integral Resonant Mode Expansion method, where a substantial reduction of the computational effort related to the evaluation of the cited Green's function is obtained.

A theorem of Radò’s type for the solutions of a quasi-linear equation

2004

Zero Viscosity Limit for Analytic Solutions of the Navier-Stokes Equation on a Half-Space.¶ II. Construction of the Navier-Stokes Solution

1998

This is the second of two papers on the zero-viscosity limit for the incompressible Navier-Stokes equations in a half-space in either 2D or 3D. Under the assumption of analytic initial data, we construct solutions of Navier-Stokes for a short time which is independent of the viscosity. The Navier-Stokes solution is constructed through a composite asymptotic expansion involving the solutions of the Euler and Prandtl equations, which were constructed in the first paper, plus an error term. This shows that the Navier-Stokes solution goes to an Euler solution outside a boundary layer and to a solution of the Prandtl equations within the boundary layer. The error term is written as a sum of firs…

The Method of Fundamental Solutions in Solving Coupled Boundary Value Problems for M/EEG

2015

The estimation of neuronal activity in the human brain from electroencephalography (EEG) and magnetoencephalography (MEG) signals is a typical inverse problem whose solution pro- cess requires an accurate and fast forward solver. In this paper the method of fundamental solutions is, for the first time, proposed as a meshfree, boundary-type, and easy-to-implement alternative to the boundary element method (BEM) for solving the M/EEG forward problem. The solution of the forward problem is obtained by numerically solving a set of coupled boundary value problems for the three-dimensional Laplace equation. Numerical accuracy, convergence, and computational load are investigated. The proposed met…

A remark on infinite initial values for quasilinear parabolic equations

2020

Abstract We study the possibility of prescribing infinite initial values for solutions of the Evolutionary p -Laplace Equation in the fast diffusion case p > 2 . This expository note has been extracted from our previous work. When infinite values are prescribed on the whole initial surface, such solutions can exist only if the domain is a space–time cylinder.

Radial growth of solutions to the poisson equation

2001

We establish a radial growth estimate of the type of the iterated law of the logarithm for solutions to the Poisson equation in the unit ball.

A Numerical Method for the Analysis of Plane-Strain Forming Processes with Unilateral Constraints

1983

The Authors propose a numerical model for the solution of plane-strain forming processes, based on the linearization of the yield surface. This allows to employ the linear programming technique for the solution of the variational problem derived from the application of the upper-bound theorem. Such a model permits to take into account the unilateral constraints in a very simple way. Therefore it is well suited to solve a large class of problems, such as sheet forming, in which the unilateral constraints are often present.

Modelling of Systems with a Dispersed Phase: “Measuring” Small Sets in the Presence of Elliptic Operators

2016

When modelling systems with a dispersed phase involving elliptic operators, as is the case of the Stokes or Navier-Stokes problem or the heat equation in a bounded domain, the geometrical structure of the space occupied by the dispersed phase enters in the homogenization process through its capacity, a quantity which can be used to define the equivalence classes in \(H^1\). We shall review the relationship between capacity and homogenization terms in the limit when the number of inclusions becomes large, focusing in particular on the situation where the distribution of inclusions is not necessarily too regular (i.e. it is not periodic).

Discrete KP Equation and Momentum Mapping of Toda System

2003

Abstract A new approach to discrete KP equation is considered, starting from the Gelfand-Zakhharevich theory for the research of Casimir function for Toda Poisson pencil. The link between the usual approach through the use of discrete Lax operators, is emphasized. We show that these two different formulations of the discrete KP equation are equivalent and they are different representations of the same equations. The relation between the two approaches to the KP equation is obtained by a change of frame in the space of upper truncated Laurent series and translated into the space of shift operators.