Alessandro De Iaco Veris. Practical Methods for Ordinary Differential Equations

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2 Alessandro De Iaco Veris Practical Methods for Ordinary Differential Equations

3 Copyright MMVIII ARACNE editrice S.r.l. via Raffaele Garofalo, 133 A/B Roma (06) ISBN No part of this book may be reproduced in any form, by print, photoprint, microfilm, microfiche, or any other means, without written permission from the publishers. 1 st edition: February 2008

4 My father, who was a lawyer, wanted me to be an engineer. This book is dedicated to his memory.

5 Contents Preface 13 Chapter 1: Fundamental concepts on ordinary differential equations Statement of the problem Existence and uniqueness of the solution in the initial-value problem Fundamental concepts of functional analysis and their application Some applications of differential equations What numerical methods can be expected to do Accuracy and error analysis Consistency, convergence and stability Inherent and induced instability Stiffness How to choose a numerical method The historical origins of differential equations and related numerical methods 74 Chapter 2: Single-step methods Classification of finite-difference methods Order of a single-step method Euler's method and its modifications Taylor-series method Runge-Kutta methods Runge-Kutta-Gill methods Runge-Kutta fourth-order methods with local truncation error control Runge-Kutta methods with order higher than four Runge-Kutta-Nyström methods and their applications 140 7

6 10. Step-size control with Runge-Kutta-Nyström methods Runge-Kutta methods for total error control Bulirsch-Stoer methods Implicit Runge-Kutta methods Rosenbrock-Wanner methods 161 Chapter 3: Multi-step methods Fundamental concepts Difference equations and their coefficients Predictor-corrector methods Adams formulae Milne-Simpson formulae Explicit Nyström's formulae Hamming's formulae Gear's methods Störmer-Cowell formulae Gauss-Jackson formulae Calculation of the starting values Halving the step size 223 Chapter 4: Multi-step methods applied to orbit computation problems Introduction Starting procedures for multi-step methods Ordinate formulae Application of starting procedures to orbit computation problems Computation of the velocity vector by single integration 237 8

7 6. Computation of the position vector by double integration Summary of the procedure Application of f and g time-series Integration for highly-eccentric elliptical orbits Variable-step integration Interpolation based on divided differences Shampine-Gordon integration method Continuous extension to the Shampine-Gordon integration method Step-size control for the Shampine-Gordon integration method Variable order for the Shampine-Gordon integration method 268 Chapter 5: Single-step methods applied to orbit computation problems Introduction General definitions and examples on Runge-Kutta methods General definitions and examples on Runge-Kutta-Nyström methods Interpolants How to construct an interpolant starting from a discrete Runge-Kutta formula 6. Symplectic explicit special Nyström methods Performance comparison for Runge-Kutta(-Nyström) methods Chapter 6: Stability analysis of finite-difference methods Intuitive concepts of weak and strong stability The linear theory of stability Absolute stability of the forward Euler method Absolute stability of explicit Runge-Kutta formulae Absolute stability of the Adams method 326 9

8 6. Absolute stability of the Nyström predictor Absolute stability of the Milne-Simpson corrector Zero-stability of linear multi-step methods Zero-stability of linear multi-step predictor-corrector pairs Stability of linear multi-step methods for finite step sizes Absolute stability of backward differentiation formulae Absolute stability of implicit Runge-Kutta formulae 346 Chapter 7: Boundary-value problems Statement of the problem Existence and uniqueness of a solution Semi-homogeneous and fully-homogeneous boundary-value problems Resolvability of linear boundary-value problems The shooting method The method of central differences The method of collocation Galërkin's method Method of weighted residuals Green's function for the solution of boundary-value problems The Rayleigh-Ritz method 408 Chapter 8: Methods of integration based on Chebyshev polynomials Spectral and pseudo-spectral methods Chebyshev polynomials Properties of the Chebyshev polynomials Differentiation and integration of Chebyshev polynomials

9 5. Use of Chebyshev polynomials over intervals other than [-1, 1] Lanczos' first tau method Clenshaw's method Lanczos' second tau method The collocation method for linear differential equations The collocation method for non-linear differential equations Differentiation matrices 437 Chapter 9: Methods of integration based on Fourier series Introduction The Euler coefficients in a Fourier-series expansion Gibbs' phenomenon An example of computation of Euler coefficients Periodic functions with periods other than Half-range expansions in Fourier series Other expressions of Fourier series Differentiation and integration of Fourier series Application of Fourier series to differential equations The Fourier transform and its applications to differential equations The Laplace transform and its applications to differential equations 484 Chapter 10: Eigenvalue problems Statement of the problem Examples of eigenvalue problems Orthogonal matrices and their properties Eigenvalues of symmetric matrices

10 5. The QR algorithm Sturm-Liouville eigenvalue problems The expansion of functions into infinite series of the eigenfunctions n (x) 8. Lagrange's and Green's identities Solution of differential equations by means of an eigenfunction expansion 10. The Fredholm alternative theorem 540 Appendix Table XV A-1 2. Table XVI A-2 3. Table XVII A-3 4. Table XVIII A

11 Ingegnati, se puoi, d'esser palese (Dante, Vita nova, XIX). I could have done it in a much more complicated way, said the Red Queen, immensely proud (Lewis Carroll). Knowledge puffeth up, but charity edifieth (I Corinthians, VIII.1). What hast thou that thou didst not receive? now if thou didst receive it, why dost thou glory, as if thou hadst not received it? (I Corinthians, IV.7). Preface The principles that have guided me in writing this book are summarised above. It has been written for the purpose of making differential equations and numerical methods for solving them accessible to the largest possible number of readers, and in particular to those of them who apply mathematics to solve problems arising in engineering. Pythagoras divided his disciples into two categories: the novices and the Pythagoreans. The disciples of the former category were allowed to know part of his teaching, whereas those of the latter received the full extent of it, on condition that they bounded themselves by an oath not to reveal the doctrine and the secrets of the school. Other philosophers of past times, Plato for one, were reluctant to apply their knowledge of mathematics to any practical use. By contrast, Archimedes, whose fame rests on great advances in both pure and applied mathematics, wanted his discoveries to be made readily available to his contemporaries, especially to those teaching at the university of Alexandria, where he had been educated. After the example given by this type of mathematicians, the present book is aimed at: placing numerical methods for ordinary differential equations at disposal of everybody, thus removing a difference of class which is still existing at present, though under different semblances; and providing scientists and engineers with such mathematical methods as are necessary in the practice of their respective professions. To this end, the formal and sometimes esoteric language which is still used in most mathematical books has been abandoned, lest the reader should prematurely close this book and put it on a shelf, which fact would perpetuate the difference of class mentioned above. On the contrary, all possible means have been used to direct and keep the attention of the reader to the subject. Numerous illustrations, almost all of which are taken from the works cited, have been added to the text. The presentation of theoretical concepts has always been followed by their immediate application to practical cases. The principal methods have been presented together with numerical examples carried out from beginning to end, in order to guide those who make their first attempts to apply such methods to cases of their interest. Equations and formulae appearing in the text have been rewritten as many times as necessary, instead of identifying them by numerals, which would have compelled the reader to search the pages in which they had been presented the first time. No exercises have been proposed without a complete guide to solve them. Several numerical tables have been given to enable the reader to apply immediately the methods presented. Many items of the so-called grey literature (unpublished works, technical reports, Ph.D. theses, lecture notes, et c.) have been 13

12 consulted and cited. The most complex concepts have been introduced to the reader by citing and comparing the opinions expressed by various authors who have previously written on the matter. Care has been taken not to lose sight of the physical problem which has given rise to the equation to be solved. My goal has been rather the search for a clear and simple manner of expressing the concepts to be introduced than a strict observance of the classical form. Such are, if not the results obtained, at least the means by which I have striven to attain my object. It is to be hoped that these efforts will be favourably received by my readers. There cannot be mathematics without mathematicians. Each one of them starts from the results found by one's predecessors and adds possibly one's own contribution. In the present book, what is due to the authors who have taught me about the matter has been duly identified and cited as such. In addition to consulting the works cited, I have also got in touch with some of their authors, who have provided me with suggestions, permitted me to use freely their concepts and illustrations, or helped me in any other way. Among these authors, I am particularly indebted to Matthew Berry, Gary Brookfield, Richard Fitzpatrick, Joseph Flaherty, Ernst Hairer, Evans Harrell, Russell Herman, Robert Hunt, Taras Lakoba, Erin McNelis, Oliver Montenbruck, Michael Olinick, Peter Olver, Brynjulf Owren, Hermann Riecke, Jørgen Sand, Lawrence Shampine and Jim Verner. Alessandro de Iaco Veris 5 July