Fourth Order Diffusion Equations with Increasing Entropy

Fourth Order Diffusion Equations with Increasing Entropy

The general quasi-linear autonomous fourth order diffusion equation ut = −[G(u)uxxx + h(u, ux, uxx)]x with positive variable diffusivity G(u) and lower-order flux component h is considered on the real line. A direct algorithm produces a general class of equations for which the Shannon entropy density obeys a reaction-diffusion equation with a positive irreducible source term. Such equations may...

Numerical method for singularly perturbed fourth order ordinary differential equations of convection-diffusion type

In this paper, we have proposed a numerical method for singularly perturbed  fourth order ordinary differential equations of convection-diffusion type. The numerical method combines boundary value technique, asymptotic expansion approximation, shooting method and  finite difference method. In order to get a numerical solution for the derivative of the solution, the given interval is divided  in...

Fourth Order Equations In

Entropy-stable and entropy-dissipative approximations of a fourth-order quantum diffusion equation

Structure-preserving numerical schemes for a nonlinear parabolic fourthorder equation, modeling the electron transport in quantum semiconductors, with periodic boundary conditions are analyzed. First, a two-step backward differentiation formula (BDF) semi-discretization in time is investigated. The scheme preserves the nonnegativity of the solution, is entropy stable and dissipates a modified e...

New family of Two-Parameters Iterative Methods for Non-Linear Equations with Fourth-Order Convergence