Ordinary Differential Equations and Integral Equations

C.T.H. Baker, G. Monegato, G. vanden Berghe, J.D. Pryce

• Publisher: North Holland
• Year: 2001
• Print ISBN: 9780444506009
• eBook ISBN: 9780080929552
• Topic: Mathematics
• Subtopic: Differential Equations
• Index: Mathematics

/homepage/sac/cam/na2000/index.html7-Volume Set now available at special set price ! 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 ordinary differential equations appear in this volume. Numerical methods for initial-value problems in ordinary differential equations fall naturally into two classes: those which use one starting value at each step (one-step methods) and those which are based on several values of the solution (multistep methods).John Butcher has supplied an expert's perspective of the development of numerical methods for ordinary differential equations in the 20th century. 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 Problem Solving Environments?"Natalia Borovykh and Marc Spijker study the problem of establishing upper bounds for the norm of the nth power of square matrices.The dynamical system viewpoint has been of great benefit to ODE theory and numerical methods. Related is the study of chaotic behaviour.Willy Govaerts discusses the numerical methods for the computation and continuation of equilibria and bifurcation points of equilibria of dynamical systems.Arieh Iserles and Antonella Zanna survey the construction of Runge-Kutta methods which preserve algebraic invariant functions.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.Stiff differential equations first became recognized as special during the 1950s. In 1963 two seminal publications laid to the foundations for later development: Dahlquist's paper on A-stable multistep methods and Butcher's first paper on implicit Runge-Kutta methods.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.Guido Vanden Berghe, Hans De Meyer, Marnix Van Daele and Tanja Van Hecke construct exponentially fitted Runge-Kutta methods with s stages.Differential-algebraic equations arise in control, in modelling of mechanical systems and in many other fields.Jeff Cash describes a fairly recent class of formulae for the numerical solution of initial-value problems for stiff and differential-algebraic systems.Shengtai Li and Linda Petzold describe methods and software for sensitivity analysis of solutions of DAE initial-value problems.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.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 error (i.e., the difference between the true and the approximate solutions) and (ii) attempting to estimate the defect - the amount by which the approximation fails to satisfy the given equation and any side-conditions.The paper by Wayne Enright on defect control relates to carefully analyzed techniques that have been proposed both for ordinary differential equations and for delay differential equations in which an attempt is made to control an estimate of the size of the defect.Many phenomena incorporate noise, and the numerical solution of

Table of contents

• Front Cover
• Ordinary Differential Equations and Integral Equations
• Copyright Page
• Table of Contents
• Preface
• Chapter 1. Numerical methods for ordinary differential equations in the 20th century
• Chapter 2. Initial value problems for ODEs in problem solving environments
• Chapter 3. Resolvent conditions and bounds on the powers of matrices, with relevance to numerical stability of initial value problems
• Chapter 4. Numerical bifurcation analysis for ODEs
• Chapter 5. Preserving algebraic invariants with Runge–Kutta methods
• Chapter 6. Performance of two methods for solving separable Hamiltonian systems
• Chapter 7. Order stars and stiff integrators
• Chapter 8. Exponentially fitted Runge–Kutta methods
• Chapter 9. Modified extended backward differentiation formulae for the numerical solution of stiff initial value problems in ODEs and DAEs
• Chapter 10. Software and algorithms for sensitivity analysis of large-scale differential algebraic systems
• Chapter 11. Compensating for order variation in mesh refinement for direct transcription methods
• Chapter 12. Continuous numerical methods for ODEs with defect control
• Chapter 13. Numerical solutions of stochastic differential equations—implementation and stability issues
• Chapter 14. Numerical modelling in biosciences using delay differential equations
• Chapter 15. Dynamics of constrained differential delay equations
• Chapter 16. A perspective on the numerical treatment of Volterra equations
• Chapter 17. Numerical stability of nonlinear delay differential equations of neutral type
• Chapter 18. Numerical bifurcation analysis of delay differential equations
• Chapter 19. How do numerical methods perform for delay differential equations undergoing a Hopf bifurcation?
• Chapter 20. Designing efficient software for solving delay differential equations
• Chapter 21. Introduction to the numerical analysis of stochastic delay differential equations
• Chapter 22. Retarded differential equations
• Chapter 23. Adaptive space—time finite element solution for Volterra equations arising in viscoelasticity problems
• Chapter 24. CP methods for the Schrodinger equation
• Chapter 25. Asymptotic correction of Numerov's eigenvalue estimates with natural boundary conditions
• Chapter 26. Numerical methods for higher order Sturm—Liouville problems
• Chapter 27. On a computer assisted proof of the existence of eigenvalues below the essential spectrum of the Sturm—Liouville problem
• Chapter 28. Numerical analysis for one-dimensional Cauchy singular integral equations
• Chapter 29. Numerical methods for integral equations of Mellin type
• Chapter 30. Quadrature methods for 2D and 3D problems
• Chapter 31. Qualocation
• Chapter 32. A sparse H-matrix arithmetic: general complexity estimates
• Chapter 33. Multilevel methods for the h-, p-, and hp-versions of the boundary element method
• Chapter 34. Domain decomposition methods via boundary integral equations
• Author Index Volume 125 (2000)