Time Of Parabolic Equations Research Articles

On non-denseness for a method of fundamental solutions with source points fixed in time for parabolic equations

Linear combinations of fundamental solutions to the parabolic heat equation with source points fixed in time is investigated. The open problem whether these linear combinations generate a dense set in the space of square integrable functions on the lateral boundary of a space-time cylinder, is settled in the negative. Linear independence of the set of fundamental solutions is shown to hold. It is outlined at the end, for a particular example, that such linear combinations constitute a linearly independent and dense set in the space of square integrable functions on the upper top part (where time is fixed) of the boundary of this space-time cylinder.

Observation estimate for kinetic transport equations by diffusion approximation

We study the unique continuation property for the neutron transport equation and for a simplified model of the Fokker–Planck equation in a bounded domain with absorbing boundary condition. An observation estimate is derived. It depends on the smallness of the mean free path and the frequency of the velocity average of the initial data. The proof relies on the well-known diffusion approximation under convenience scaling and on the basic properties of this diffusion. Eventually, we propose a direct proof for the observation at one time of parabolic equations. It is based on the analysis of the heat kernel.

Single and multi‐interval Legendre spectral methods in time for parabolic equations

AbstractIn this article, we take the parabolic equation with Dirichlet boundary conditions as a model to present the Legendre spectral methods both in spatial and in time. Error analysis for the single/multi‐interval schemes in time is given. For the single interval spectral method in time, we obtain a better error estimate in L2‐norm. For the multi‐interval spectral method in time, we obtain the L2‐optimal error estimate in spatial. By choosing approximate trial and test functions, the methods result in algebraic systems with sparse forms. A parallel algorithm is constructed for the multi‐interval scheme in time. Numerical results show the efficiency of the methods. The methods are also applied to parabolic equations with Neumann boundary conditions, Robin boundary conditions and some nonlinear PDEs. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006

Single and Multi-Interval Legendre tau-Methods in Time for Parabolic Equations.

In this paper, we take the parabolic equation with periodic boundary conditions as a model to present a spectral method with the Fourier approximation in spatial and single/multi-interval Legendre Petrov–Galerkin method in time. For the single interval spectral method in time, we obtain the optimal error estimate in L 2-norm. For the multi-interval spectral method in time, the L 2-optimal error estimate is valid in spatial. Numerical results show the efficiency of the methods.

Strong-norm error estimates for the projective-difference method for approximately solving abstract parabolic equations

Solutions continuously differentiable with respect to time of parabolic equations in Hilbert space are obtained by the projective-difference method approximately. The discretization of the problem is carried out in the spatial variables using Galerkin's method, and in the time variable using Euler's implicit method. Strong-norm error estimates for approximate solutions are obtained. These estimates not only allow one to establish the convergence of the approximate solutions to the exact ones but also yield numerical characteristics of the rates of convergence. In particular, order-sharp error estimates for finite element subspaces are obtained.

Galerkin-Type Approximations which are Discontinuous in Time for Parabolic Equations in a Variable Domain

We consider general linear parabolic equations in a given time dependent domain and we describe a general class of Galerkin-type approximations which are continuous with respect to the space variables, but which admit discontinuities with respect to time at each step. Unconditional stability is proved and a general error estimate is established. These results are applied to certain finite element methods based on space-time finite elements.