<p><br><URL><HREF>/homepage/sac/cam/na2000/index.html</HREF><HTXT>7-Volume Set now available at special set price !</HTXT></URL><p><p>This volume contains contributions in the area of differential equations and integral equations. Many numerical methods have arisen in response to the need to solve "real-life" problems in applied mathematics, in particular problems that do not have a closed-form solution. Contributions on both initial-value problems and boundary-value problems in <i>ordinary differential equations</i> appear in this volume. Numerical methods for <i>initial-value problems</i> in ordinary differential equations fall naturally into two classes: those which use <i>one</i> starting value at each step (one-step methods) and those which are based on <i>several</i> values of the solution (multistep methods). <br>John Butcher has supplied an expert's perspective of the development of numerical methods for ordinary differential equations in the 20th century. <br>Rob Corless and Lawrence Shampine talk about established technology, namely software for initial-value problems using Runge-Kutta and Rosenbrock methods, with interpolants to fill in the solution between mesh-points, but the 'slant' is new - based on the question, "How should such software integrate into the current generation of <i>Problem Solving Environments?"</i> <br>Natalia Borovykh and Marc Spijker study the problem of establishing upper bounds for the norm of the <i>n</i>th power of square matrices. <br>The dynamical system viewpoint has been of great benefit to ODE theory and numerical methods. Related is the study of <i>chaotic behaviour.</i> <br>Willy Govaerts discusses the numerical methods for the computation and continuation of equilibria and bifurcation points of equilibria of dynamical systems. <br>Arieh Iserles and Antonella Zanna survey the construction of Runge-Kutta methods which preserve algebraic invariant functions. <br>Valeria Antohe and Ian Gladwell present numerical experiments on solving a Hamiltonian system of Hénon and Heiles with a symplectic and a nonsymplectic method with a variety of precisions and initial conditions. <br><i>Stiff differential equations</i> first became recognized as special during the 1950s. In 1963 two seminal publications laid to the foundations for later development: Dahlquist's paper on <i>A</i>-stable multistep methods and Butcher's first paper on implicit Runge-Kutta methods. <br>Ernst Hairer and Gerhard Wanner deliver a survey which retraces the discovery of the order stars as well as the principal achievements obtained by that theory. <br>Guido Vanden Berghe, Hans De Meyer, Marnix Van Daele and Tanja Van Hecke construct exponentially fitted Runge-Kutta methods with <i>s</i> stages. <br><i>Differential-algebraic equations</i> arise in control, in modelling of mechanical systems and in many other fields. <br>Jeff Cash describes a fairly recent class of formulae for the numerical solution of initial-value problems for <i>stiff and differential-algebraic systems.</i> <br>Shengtai Li and Linda Petzold describe methods and software for <i>sensitivity analysis</i> of solutions of DAE initial-value problems. <br>Again in the area of differential-algebraic systems, Neil Biehn, John Betts, Stephen Campbell and William Huffman present current work on mesh adaptation for DAE two-point boundary-value problems. <br>Contrasting approaches to the question of how good an approximation is as a solution of a given equation involve (i) attempting to estimate the actual <i>error</i> (i.e., the difference between the true and the approximate solutions) and (ii) attempting to estimate the <i>defect</i>
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