In this paper we shall study the evolution of the zero set of the solution of the heat equation perturbed by a potential c and Neumann boundary conditions in one and two dimensions. 1. Introduction Let IR d be a bounded connected domain with a smooth boundary @ (we assume throughout that is of class C 3) and let c be a continuous function on , the closure of. Consider the heat equation with a p...

Three new fully implicit methods which are based on the (5,5) Crank–Nicolson method, the (5,5) N-H (Noye–Hayman) implicit method and the (9,9) N-H implicit method are developed for solving the heat equation in two dimensional space with non-local boundary conditions. The latter is fourth-order while the others are second-order. While the implicit methods developed here, like the scheme based on...

The flow and heat transfer characteristics of incompressible viscous flow over a nonlinearly stretching sheet with the presence of viscous dissipation is investigated numerically. The similarity transformation reduces the time-independent boundary layer equations for momentum and thermal energy into a set of coupled ordinary differential equations. The obtained equations, including nonlinear eq...

We study the asymptotic behavior of solutions of the initialboundary value problem, with periodic boundary conditions, for a fourth-order nonlinear degenerate diffusion equation with a logarithmic nonlinearity. For strictly positive and suitably small initial data we show that a positive solution exponentially approaches its mean as time tends to infinity. These results are derived by analyzing...

Fourth-order quantum master equations ~FQMEs! are derived in both time nonlocal and local forms for a general system Hamiltonian, with new detailed expressions for the fourth-order kernel, where the bath correlation functions are explicitly decoupled from the system superoperators. Further simplifications can be made for the model of linearly coupled harmonic oscillator bath. Consideration of t...

In this paper, we present fourth-order finite difference method for solving nonlinear one-dimensional Burgers equation. This method is unconditionally stable. The convergence analysis of the present method is studied and an upper bound for the error is derived. Numerical comparisons are made with most of the existing numerical methods for solving this equation. 2005 Elsevier Inc. All rights res...

Abstract. A nonlinear fourth-order parabolic equation with nonhomogeneous Dirichlet–Neumann boundary conditions in one space dimension is analyzed. This equation appears, for instance, in quantum semiconductor modeling. The existence and uniqueness of strictly positive classical solutions to the stationary problem are shown. Furthermore, the existence of global nonnegative weak solutions to the...

In this paper we discuss the numerical solution of initial-boundary value problem for the fourth order parabolic partial integro-differential equation. We use the forward Euler scheme for time discretization and the quasi-wavelet method for space discretization. Sometimes, we give a new method about the treatment of boundary condition. Numerical experiment is included to demonstrate the validit...