Numerical Solution of Ill Posed Problems in Partial Differential Equations

摘要

This project is concerned with several questions concerning the existence, uniqueness, continuous data dependence and numerical computation of solutions of various ill posed problems in partial differential equations. Several problems involving reaction diffusion equations with and without convection terms present were studied. In the latter case the ability of finite element approximate solutions to reproduce the continuous time dynamics was investigated. In the former case, a convective diffusion equation with a semilinear source in the boundary conditions was analyzed. A fairly complete picture of the dynamics was obtained. With the source term in the equation, computations revealed a rich structure which has been partially analyzed theoretically. Several problems for reaction diffusion equations in unbounded regimes were also investigated. It was shown that under certain conditions in the rate law all nonzero solutions blow up in finite time, while for other conditions in the rate law, solutions damp out. It was shown that a potential well theory is possible for certain hyperbolic problems in which a nonlinear boundary condition is prescribed and not possible in certain cases when forcing term in the differential equation is singular. Numerical experiments performed on the wave equation with a singular forcing term have down that when quenching occurs, the time and exact derivatives blow up in finite time. The nature of the blowup was studied computationally. (jhd)