The intention of this special session on ”Advances in the numerical solution of nonlinear evolution equations” is to gather mathematicians and theoretical physicists, interconnected through their field of application, the analytical tools, or the numerical methods used. The scope of topics includes but is not limited to Schrödinger type equations, highly oscillatory equations, parabolic problem...

This report discusses a mathematical software package DKLAG5 which consists of a suite of FORTRAN subroutines for the numerical solution of systems of functional di erential equations with state dependent delays. The package implements continuously imbedded Runge{ Kutta methods which are based on C polynomial interpolants. These interpolants are exploited by the software to handle the necessary...

We present a method to solve initial and boundary value problems using arti ial neural networks. A trial solution of the di erential equation is written as a sum of two parts. The rst part satis es the initial/boundary onditions and ontains no adjustable parameters. The se ond part is onstru ted so as not to a e t the initial/boundary onditions. This part involves a feedforward neural network, ...

This work develops fast and adaptive algorithms for numerically solving nonlinear partial di erential equations of the form ut = Lu +N f(u) where L and N are linear di erential operators and f(u) is a nonlinear function. These equations are adaptively solved by projecting the solution u and the operators L and N into a wavelet basis. Vanishing moments of the basis functions permit a sparse repr...

The in ltration process is generally described by a nonlinear di erential equation, which can be solved by iteration methods such as a Newton-Raphson method. In this paper we propose a Discrete Event System Speci cation(DEVS) model for Green-Ampt in ltration. We show that this model can be approximated using Genetic Algorithm optimization of a fuzzy system. The fuzzy approximation is shown to b...

Observers are usually formulated as explicit systems of di erential equations and implemented using standard ODE solvers. In this paper, we show that there can be advantages in formulating the observer as a DAE (Di erential-Algebraic Equation). We review the general idea of DAE observer design of [10]. We give two special normal forms for which DAE observer design yields an observer with linear...

This paper is devoted to the study of periodic boundary value problems for nonlinear third order di¤erential equations subjected to impulsive e¤ects. We provide su¢ cient conditions on the nonlinearity and the impulse functions that guarantee the existence of at least one solution. Our approach is based on a priori estimates, the method of upper and lower solutions combined with an iterative te...

Stochastic di erential games are considered in a non-Markovian setting. Typically, in stochastic di erential games the modulating process of the di usion equation describing the state ow is taken to be Markovian. Then Nash equilibria or other types of solution such as Pareto equilibria are constructed using Hamilton-Jacobi-Bellman (HJB) equations. But in a non-Markovian setting the HJB method i...

This paper describes the design philosophy behind the recent replacement of the NAG Ordinary Di erential Equation (ODE) sti integrators. This replacement was is intended to update the ODE chapter algorithmically but, more importantly in the context of this paper, it provides a more exible interface than has been available in the past. This interface is designed to permit a wide variety of probl...

After a series of application papers have proven the approach to be numerically e ective, this paper gives the rst theoretical foundation for methods solving partial di erential equations by collocation with (possibly radial) basis functions.