Numerical Spectral Analysis and Approximation of Functional Traveling Waves

University Of Kansas Center For Research Inc, Lawrence KS

Investigators

Abstract

Differential equations are used as models of physical and biological phenomena in many areas of science and engineering. The focus of the investigator and his colleagues in this proposal is on the approximation of solutions of differential equations. The investigator is interested in the analysis and computation of stability spectra (point spectrum of differential and difference operators, Sacker-Sell spectrum, and Lyapunov exponents) and techniques for analysis and approximation of discrete models similar to time dependent partial differential equations but with a difference operator instead of a spatial differential operator. Discrete models play a prominent role in the modeling of physical and biological systems. Of particular interest are traveling wave solutions of lattice differential equations. The approach taken is to combine dynamical systems and numerical analysis ideas with the modeling and analysis of ordinary, partial, and lattice differential equations. This project is concerned with the development and analysis of efficient, accurate numerical techniques that are useful for the computation and analysis of dynamical systems. Sacker-Sell and Lyapunov spectral intervals are natural analogues of the real parts of the eigenvalues that provide stability information for time varying differential equations. The investigator develops, analyzes, and justifies the use of numerical techniques for the approximation of these spectral intervals. A suite of computational modules for the computation of stability information and for functional traveling waves is being developed. It is backed by analysis of the numerical techniques in a form that should prove useful to working scientists and engineers. The investigator and his colleagues consider issues in the approximation and computation of solutions of differential equations. Differential equations are commonly used to model physical and biological phenomena in many areas of science and engineering. A differential equation is a rule, a relationship between the solution and the rate of change of the solution, that determines how an initial configuration evolves into future configurations. The focus of this project is on the approximation of Lyapunov exponents and related quantities that provide information on stability, the tendency for nearby configurations to evolve and stay nearby, and instability, the tendency for nearby configurations to move apart. This type of analysis is useful in understanding complex biological phenomena that occur in the environment and in identifying instabilities in, for example, models of weather prediction. The analysis and computation of lattice differential equations, i.e., differential equations that are discrete in space and continuous in time, are important in the modeling of physical and biological systems in which the spatial component is naturally discrete, in particular for microscopic models in materials and physiology.