A homogeneous boundary condition is constructed for the parabolic equation (∂t + I − Δ)u = f in an arbitrary cylindrical domain Ω×R (Ω ⊂ Rn being a bounded domain, I and Δ being the identity operator and the Laplacian) which generates an initial-boundary value problem with an explicit formula of the solution u. In the paper, the result is obtained not just for the operator ∂t + I −Δ, but also f...

In this paper, we study a boundary-value problem with discontinuous conjugation conditions on the line of type changing for model equation mixed hyperbolic-parabolic degeneration order in hyperbolicity domain. parabolic domain, is fractional diffusion equation, whereas hyperbolic domain it loaded one-speed transfer equation. We prove uniqueness and existence theorem propose an explicit solution...

In this paper we consider Galerkin-finite element methods that approximate the solutions of initial-boundary-value problems in one space dimension for parabolic and Schrödinger evolution equations with dynamical boundary conditions. Error estimates of optimal rates of convergence in L and H are proved for the accociated semidiscrete and fully discrete Crank-Nicolson-Galerkin approximations. The...

We study the asymptotic behavior of Lipschitz continuous solutions of nonlinear degenerate parabolic equations in the periodic setting. Our results apply to a large class of Hamilton-Jacobi-Bellman equations. Defining Σ as the set where the diffusion vanishes, i.e., where the equation is totally degenerate, we obtain the convergence when the equation is uniformly parabolic outside Σ and, on Σ, ...

A problem of sound propagation in a shallow-water waveguide with a weakly sloping penetrable bottom is considered. The adiabatic mode parabolic equations are used to approximate the solution of the three-dimensional (3D) Helmholtz equation by modal decomposition of the acoustic pressure field. The mode amplitudes satisfy parabolic equations that admit analytical solutions in the special case of...

We consider parabolic partial differential equations of Lotka-Volterra type, with a non-local nonlinear term. This models, at the population level, the darwinian evolution of a population; the Laplace term represents mutations and the nonlinear birth/death term represents competition leading to selection. Once rescaled with a small diffusion, we prove that the solutions converge to a moving Dir...

We propose a new technique for analyzing the error of finite element approximations of parabolic problems with non-smooth initial data. For homogeneous equation we prove optimal L-error estimate of order O ( h/t ) for t > 0 when the given initial data is in L. Further, for non-homogeneous parabolic equation with zero initial data we establish an optimal error estimate of order O(h) in L. Thus, ...

A classical result of Aleksandrov allows one to estimate the size of a convex function u at a point x in a bounded domain Ω in terms of the distance from x to the boundary of Ω if R Ω det Du dx < ∞. This estimate plays a prominent role in the existence and regularity theory of the Monge-Ampère equation. Jerison proved an extension of Aleksandrov’s result that provides a similar estimate, in som...

This paper rst reviews some basic facts about electron transport in semiconductor materials. Then, it develops several macroscopic models from the diiusion approximation of the semiconductor Boltz-mann equation. The rst model to be derived is the 'Spherical Harmonics Expansion model', which consists of a parabolic equation for the energy distribution function. A second model is then deduced, kn...

Periodic and solitary travelling-wave solutions of an extended reduced Ostrovsky equation are investigated. Attention is restricted to solutions that, for the appropriate choice of certain constant parameters, reduce to solutions of the reduced Ostrovsky equation. It is shown how the nature of the waves may be categorized in a simple way by considering the value of a certain single combination ...