We describe a MIMD parallel code to solve a general class of ordinary di erential equations, with particular emphasis on the large, sparse systems arising from space discretization of systems of parabolic partial di erential equations. The main goals of this work are sharp bounds on the accuracy of the computed solution and exibility of the software. We discuss the sources of error in solving d...

We investigate the derivation and the mathematical properties of a Saint-Venant model with an energy equation and with temperature-dependent transport coe±cients. These equations model shallow water °ows as well as thin viscous sheets over °uid substrates like oil slicks, atlantic waters in the Strait of Gilbraltar or °oat glasses. We exhibit an entropy function for the system of partial di®ere...

In this paper, we study general linear systems of delay di erential-algebraic equations (DDAEs) of arbitrary order. We show that under some consistency conditions, every linear high-order DAE can be reformulated as an underlying high-order ordinary di erential equation (ODE) and that every linear DDAE with single delay can be reformulated as a high-order delay di erential equation (DDE). We der...

In this paper we discuss the formulation of the governing equations that describe ow of uids in porous media Various types of uid ow ranging from single phase ow to compositional ow are considered It is shown that all the di erential equations governing these types of ow can be e ectively rewritten in a fractional ow formulation i e in terms of a global pressure and saturation or saturations an...

This paper is concerned with linear uniformly elliptic and par abolic partial di erential equations in divergence form It is assumed that the coe cients of the equations are random variables constant in time The Green s functions for the equations are then random variables Regularity properties for expectation values of Green s functions are obtained In par ticular it is shown that the expectat...

The theory of optimal foraging predicts abrupt changes in consumer behavior which lead to discontinuities in the functional response. Therefore population dynamical models with optimal foraging behavior can be appropriately described by di erential equations with discontinuous right{hand sides. In this paper we analyze the behavior of three di erent Lotka{Volterra predator{prey systems with opt...

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...