A Sixth-Order Nonlinear Parabolic Equation for Quantum Systems

A Sixth-Order Nonlinear Parabolic Equation for Quantum Systems

The global-in-time existence of weak nonnegative solutions to a sixth-order nonlinear parabolic equation in one space dimension with periodic boundary conditions is proved. The equation arises from an approximation of the quantum drift-diffusion model for semiconductors and describes the evolution of the electron density in the semiconductor crystal. The existence result is based on two techniq...

Global Nonnegative Solutions of a Nonlinear Fourth-Order Parabolic Equation for Quantum Systems

A SIXTH ORDER METHOD FOR SOLVING NONLINEAR EQUATIONS

In this paper, we present a new iterative method with order of convergence eighth for solving nonlinear equations. Periteration this method requires three evaluations of the function and one evaluation of its first derivative. A general error analysis providing the eighth order of convergence is given. Several numerical examples are given to illustrate the efficiency and performance of the new ...

Finite element approximation of a sixth order nonlinear degenerate parabolic equation

We consider a finite element approximation of the sixth order nonlinear degenerate parabolic equation ut = ∇.( b(u)∇∆u), where generically b(u) := |u| for any given γ ∈ (0,∞). In addition to showing well-posedness of our approximation, we prove convergence in space dimensions d ≤ 3. Furthermore an iterative scheme for solving the resulting nonlinear discrete system is analysed. Finally some num...

Long-time Behavior for a Nonlinear Fourth-order Parabolic Equation

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...