A Nonlocal Problem from Conservation Biology

Derivation of conservation laws from nonlocal symmetries of differential equations

An identity is derived which yields a correspondence between symmetries and conservation laws for self-adjoint differential equations. This identity does not rely on use of a Lagrangian as needed to obtain conservation laws by Noether’s theorem. Moreover, unlike Noether’s theorem, which can only generate conservation laws from local symmetries, the derived identity generates conservation laws f...

On the instability of a nonlocal conservation law

We are interested in a nonlocal conservation law which describes the morphodynamics of sand dunes sheared by a fluid flow, recently proposed by Andrew C. Fowler and studied by [1, 2]. We prove that constant solutions of Fowler’s equation are non-linearly unstable. We also illustrate this fact using a finite difference scheme.

A new total variation diminishing implicit nonstandard finite difference scheme for conservation laws

In this paper, a new implicit nonstandard finite difference scheme for conservation laws, which preserving the property of TVD (total variation diminishing) of the solution, is proposed. This scheme is derived by using nonlocal approximation for nonlinear terms of partial differential equation. Schemes preserving the essential physical property of TVD are of great importance in practice. Such s...

Symbolic Computation of Nonlocal Symmetries and Nonlocal Conservation Laws of Partial Differential Equations Using the GeM Package for Maple

The use of the symbolic software package GeM for Maple is illustrated with examples of computation of nonlocal symmetries and nonlocal conservation laws of nonlinear partial differential equations. In the considered examples, the nonlocal symmetries and conservation laws arise as local symmetries and conservation laws of potential systems. FullMaple codewith detailed comments is presented. Exam...

Nonlocal Boundary Conditions for the Navier – Stokes Equations

In this paper nonlocal boundary conditions for the Navier–Stokes equations are derived, starting from the Boltzmann equation in the hydrodynamic limit. Basing on phenomenological arguments, two scattering kernels which model non–local interactions between the gas molecules and the wall boundary are proposed. They satisfy the global mass conservation and a generalized reciprocity relation. The a...

Quenching Time for a Nonlocal Diffusion Problem with Large Initial Data