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Numerical Analysis 2000 : Volume 6 - C.T.H. Baker

Numerical Analysis 2000

Volume 6


Published: 4th July 2001
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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 "n"th 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 Henon 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 "stochastic differential equations" has developed as a relatively new item of study in the area.
Keven Burrage, Pamela Burrage and Taketomo Mitsui review the way numerical methods for solving stochastic differential equations (SDE's) are constructed.
One of the more recent areas to attract scrutiny has been the area of "differential equations with after-effect" (retarded, delay, or neutral delay differential equations) and in this volume we include a number of papers on evolutionary problems in this area.
The paper of Genna Bocharov and Fathalla Rihan conveys the importance in mathematical biology of models using retarded differential equations.
The contribution by Christopher Baker is intended to convey much of the background necessary for the application of numerical methods and includes some original results on stability and on the solution of approximating equations.
Alfredo Bellen, Nicola Guglielmi and Marino Zennaro contribute to the analysis of stability of numerical solutions of nonlinear neutral differential equations.
Koen Engelborghs, Tatyana Luzyanina, Dirk Roose, Neville Ford and Volker Wulf consider the numerics of bifurcation in delay differential equations.
Evelyn Buckwar contributes a paper indicating the construction and analysis of a numerical strategy for stochastic delay differential equations (SDDEs).
This volume contains contributions on both "Volterra and Fredholm-type integral equations."
Christopher Baker responded to a late challenge to craft a review of the theory of the basic numerics of Volterra integral and integro-differential equations.
Simon Shaw and John Whiteman discuss Galerkin methods for a type of Volterra integral equation that arises in modelling viscoelasticity.
A subclass of "boundary-value problems" for ordinary differential equation comprises "eigenvalue problems" such as Sturm-Liouville problems (SLP) and Schrodinger equations.
Liviu Ixaru describes the advances made over the

Preface: Numerical Analysis 2000. Vol. VI: Ordinary Differential Equations and Integral Equationsp. xi
Numerical methods for ordinary differential equations in the 20th centuryp. 1
Initial value problems for ODEs in problem solving environmentsp. 31
Resolvent conditions and bounds on the powers of matrices, with relevance to numerical stability of initial value problemsp. 41
Numerical bifurcation analysis for ODEsp. 57
Preserving algebraic invariants with Runge-Kutta methodsp. 69
Performance of two methods for solving separable Hamiltonian systemsp. 83
Order stars and stiff integratorsp. 93
Exponentially fitted Runge-Kutta methodsp. 107
Modified extended backward differentiation formulae for the numerical solution of stiff initial value problems in ODEs and DAEsp. 117
Software and algorithms for sensitivity analysis of large-scale differential algebraic systemsp. 131
Compensating for order variation in mesh refinement for direct transcription methodsp. 147
Continuous numerical methods for ODEs with defect controlp. 159
Numerical solutions of stochastic differential equations--implementation and stability issuesp. 171
Numerical modelling in biosciences using delay differential equationsp. 183
Dynamics of constrained differential delay equationsp. 201
A perspective on the numerical treatment of Volterra equationsp. 217
Numerical stability of nonlinear delay differential equations of neutral typep. 251
Numerical bifurcation analysis of delay differential equationsp. 265
How do numerical methods perform for delay differential equations undergoing a Hopf bifurcation?p. 277
Designing efficient software for solving delay differential equationsp. 287
Introduction to the numerical analysis of stochastic delay differential equationsp. 297
Retarded differential equationsp. 309
Adaptive space-time finite element solution for Volterra equations arising in viscoelasticity problemsp. 337
CP methods for the Schrodinger equationp. 347
Asymptotic correction of Numerov's eigenvalue estimates with natural boundary conditionsp. 359
Numerical methods for higher order Sturm-Liouville problemsp. 367
On a computer assisted proof of the existence of eigenvalues below the essential spectrum of the Sturm-Liouville problemp. 385
Numerical analysis for one-dimensional Cauchy singular integral equationsp. 395
Numerical methods for integral equations of Mellin typep. 423
Quadrature methods for 2D and 3D problemsp. 439
Qualocationp. 461
A sparse ye-matrix arithmetic: general complexity estimatesp. 479
Multilevel methods for the h-, p-, and hp-versions of the boundary element methodp. 503
Domain decomposition methods via boundary integral equationsp. 521
Author Index Volume 125p. 539
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780444506009
ISBN-10: 0444506004
Series: Numerical Analysis 2000
Audience: General
Format: Paperback
Language: English
Number Of Pages: 558
Published: 4th July 2001
Publisher: Elsevier Science & Technology
Country of Publication: NL
Dimensions (cm): 27.94 x 20.96  x 2.9
Weight (kg): 1.24