Network for Biological Invasions and Dispersal Research

MITACS Seminar at the University of New Brunswick, Fredericton

Wednesday, 15 November, 2006,

Alexei Cheviakov

of the University of British Columbia

will speak on

Framework for Nonlocally Related PDE Systems Arising from Conservation Laws

3:30 PM, Tilley Hall 404

Abstract: For a given system of partial differential equations (PDE), a conservation law is a divergence expression Dx(A) + Dy(B) + . = 0, where Dx, Dy, etc. are total derivatives with respect to x, y, etc. Conservation laws describe essential properties of phenomena modeled by the given PDE system, such as conservation of mass, charge, energy, etc. Conservation laws are also used for analysis and development of numerical methods. In this talk, I will outline the algorithmic procedure of finding conservation laws.

Using conservation laws, for a given PDE system, one can systematically construct other PDE systems that are nonlocally related to the given one, but have solution sets equivalent to that of the given system. Therefore any general method of analysis (qualitative, numerical, perturbation, conservation law, symmetry, etc.) considered for a given PDE system may be tried again on any nonlocally related system. In this way, new results may be obtained. I will describe the procedure of construction of sets (``trees'') of nonlocally related PDE systems, give examples of such trees (for PDE systems of gas dynamics and nonlinear elasticity), and show new results that follow from such analysis.

This work is joint with G. Bluman.