A First Course in Ordinary Differential Equations

A First Course in Ordinary Differential Equations

Martin Hermann Masoud Saravi A First Course in Ordinary Differential Equations Analytical and Numerical Methods 123

Martin Hermann Institute of Applied Mathematics Friedrich Schiller University Jena, Germany Masoud Saravi Department of Mathematics Islamic Azad University Nour Branch Nour, Iran ISBN 978-81-322-1834-0 ISBN 978-81-322-1835-7 (ebook) DOI 10.1007/978-81-322-1835-7 Springer New Delhi Heidelberg New York Dordrecht London Library of Congress Control Number: 2014937667 Springer India 2014 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

To our wives, Gudrun and Mahnaz, and our children, Alexander, Sina, and Sayeh

Preface This book presents a modern introduction to mathematical techniques for solving initial and boundary value problems in linear ordinary differential equations (ODEs). The focus on analytical and numerical methods makes this book particularly attractive. ODEs play an important role in modeling complex scientific, technological, and economic processes and phenomena. Therefore, to address students and scientists of various disciplines, we have circumvented the traditional definition-theorem-proof format. Instead, we describe the mathematical background by means of a variety of problems, examples, and exercises ranging from the elementary to the challenging problems. The book is intended as a primary text for courses in theory of ODEs and numerical treatment of ODEs for advanced undergraduate and early graduate students. It is assumed that the reader has a basic knowledge of elementary calculus, in particular methods of integration, and numerical analysis. Physicists, chemists, biologists, computer scientists, and engineers whose work involves the solution of ODEs will also find the book useful both as a reference and as a tool for self-studying. The book has been prepared within the scope of a German-Iranian research project on mathematical methods for ODEs, which was started in early 2012. We now outline the contents of the book. In Chap. 1, an introduction to ODEs and some basic concepts are presented. Chapter 2 deals with scalar first-order ODEs. The existence and uniqueness of solutions is studied on the basis of the well-known theorems of Peano and Picard-Lindelöf. The analytical standard techniques for the determination of the exact solution are given. Moreover, the solution of high-degree first-order ODEs (Clairaut and Lagrange equation), which are often encountered in the applications, is discussed. This chapter ends with the discussion about the family of curves and orthogonal trajectories. Chapter 3 is devoted to analytical methods for the solution of second-order ODEs. Subsequent to the introduction to higher-order ODEs, solution methods for homogeneous and inhomogeneous equations are presented. Chapter 4 focuses on the Laplace transform for scalar ODEs that is also used in Chap. 5 for the solution of systems of first-order ODEs. vii

viii Preface The Laplace transform is a widely applied integral transform in mathematics used to switch a function from the time domain to the s-domain. The Laplace transform can be used in some cases to solve linear ODEs with given initial conditions. At the end of Chap. 5, it is shown how the solution for the homogeneous system can be determined on the basis of the eigenvalues of the characteristic equation. Then, the corresponding solution for the inhomogeneous system is developed by the method of variation of parameters, and the solution for the ODE is presented by the superposition principle. In Chap. 6, it is shown how power series techniques can be used to represent the solution for scalar first- and second-order ODEs. Special attention is paid to Legendre s equation, Bessel s equation, and the hypergeometric equation since these equations often occur in the applications. The numerical part of this book begins with Chap. 7. Here, numerical methods for initial value problems of systems of first-order ODEs are studied. Starting with the concept of discretizing ODEs, the class of Runge Kutta methods is introduced. The Butcher schemes of a variety of Runge Kutta methods are given. Further topics discussed are consistency, convergence, estimation of the local discretization error, step-size control, A-stability, and stiffness. Finally, Chap. 8 deals with shooting methods for the solution for linear two-point boundary value problems. The following numerical techniques are studied in detail: simple shooting method, multiple shooting method, and the method of complementary functions and the stabilized march method for the solution to ODEs with partially separated boundary conditions. We would like to express our thanks to those colleagues who have supported our German-Iranian joint work by their criticism and encouragement, particularly Dr. Omid Jalili (IAU, Nour Branch) for discussing most of the physical problems and Dr. Dieter Kaiser (FSU Jena) for his expert guidance on our computer programming. It has been a pleasure working with the Springer India publication staff, in particular Mr. Shamim Ahmad and Ms. Nupoor Singh. Finally, the authors wish to emphasize that any helpful suggestion or comment for improving this book will be warmly appreciated and can be sent by e-mail to masoud@saravi.info and martin.hermann@uni-jena.de. Nour, Iran Jena, Germany Masoud Saravi Martin Hermann

Contents 1 Basic Concepts of Differential Equations... 1 1.1 Introduction... 1 1.2 Some ElementaryTerminology... 2 1.3 Solutionsof a DifferentialEquation... 3 1.4 Exercises... 6 2 First-Order Differential Equations... 11 2.1 Discussion of the Main Problem... 11 2.2 AnalyticalSolutionof First-Order ODEs... 14 2.3 High-DegreeFirst-OrderODEs... 31 2.4 Family of Curves, Orthogonal Trajectories... 35 2.5 Exercises... 38 Reference... 44 3 Second-Order Differential Equations... 45 3.1 LinearDifferentialEquations... 45 3.2 Solution Methods for Homogeneous Equations... 54 3.2.1 Methodof OrderReduction... 54 3.2.2 Exact Equations... 61 3.2.3 Pseudo-exact Equations... 63 3.2.4 Equationswith Constant Coefficients... 66 3.2.5 NormalForm... 72 3.3 Solution Methods for Inhomogeneous Equations... 74 3.3.1 Methodof UndeterminedCoefficients... 74 3.3.2 Methodof Variationof Parameters... 76 3.3.3 Methodof Operators... 84 3.4 Exercises... 88 References... 92 4 Laplace Transforms... 93 4.1 Introduction... 93 4.2 LaplaceTransform... 94 ix

x Contents 4.3 The LaplaceTransformof Periodic Functions... 100 4.4 The LaplaceTransformsof DerivativesandIntegrals... 103 4.5 DerivativesandIntegralsof the Laplace Transforms... 108 4.6 The ConvolutionTheoremand IntegralEquations... 111 4.7 Exercises... 114 5 Systems of Linear Differential Equations... 119 5.1 Introduction... 119 5.2 Elimination Methods... 121 5.2.1 Elimination Using the D Operator... 121 5.2.2 EliminationUsing LaplaceTransforms... 123 5.3 Methodof Vector-MatrixNotation... 125 5.3.1 Simple Real Eigenvalues... 126 5.3.2 Multiple Real Eigenvalues... 129 5.3.3 Simple ComplexEigenvalues... 131 5.4 Solution of an Inhomogeneous System of ODEs... 132 5.5 Methodof UndeterminedCoefficients... 136 5.6 Exercises... 142 Reference... 144 6 Power Series Solutions... 145 6.1 Introduction... 145 6.2 Review of Power Series... 146 6.3 Series Solutions About an Ordinary Point... 150 6.4 Series Solutions About a Regular Singular Point... 156 6.5 Special Functions... 174 6.5.1 Legendre sequation... 174 6.5.2 Bessel s Equation... 178 6.5.3 HypergeometricEquation... 181 6.6 Exercises... 184 Reference... 188 7 Numerical Methods for Initial Value Problems... 189 7.1 Initial Value Problems... 189 7.2 Discretizationof an ODE... 190 7.3 Runge-Kutta Methods... 192 7.4 LocalTruncationErrorandConsistency... 199 7.5 Construction of Runge-Kutta Methods... 206 7.6 Collocation and Implicit Runge-Kutta Methods... 212 7.7 GlobalErrorand Convergence... 219 7.8 Estimation of the Local Discretization Error andstep-size Control... 220 7.9 Absolute Stability and Stiffness... 224 7.10 Exercises... 231 References... 239

Contents xi 8 Shooting Methods for Linear Boundary Value Problems... 241 8.1 Two-Point Boundary Value Problems... 241 8.2 Simple ShootingMethod... 244 8.3 Methodof ComplementaryFunctions... 248 8.4 Stability of Simple Shooting... 256 8.5 Multiple Shooting Method... 258 8.6 Stability of Multiple Shooting... 261 8.7 NumericalTreatmentof the LinearAlgebraic System... 264 8.8 Stabilized MarchMethod... 265 8.9 Exercises... 272 References... 276 A Power Series... 279 B Some Elementary Integration Formulas... 281 C Table of Laplace Transforms... 283 Index... 285

About the Authors MARTIN HERMANN is professor of numerical mathematics at the Friedrich Schiller University (FSU) Jena (Germany). His activities and research interests are in the field of scientific computing and numerical analysis of nonlinear parameterdependent ordinary differential equations (ODEs). He is also the founder of the Interdisciplinary Centre for Scientific Computing (1999), where scientists of different faculties at the FSU Jena work together in the fields of applied mathematics, computer sciences and applications. Since 2003, he has headed an international collaborative project with the Institute of Mathematics at the National Academy of Sciences, Kiev, Ukraine, studying, e.g., the sloshing of liquids in tanks. Since 2003, Dr. Hermann has been a curator at the Collegium Europaeum Jenense of the FSU Jena (CEJ) and the first chairman of the Friends of the CEJ. In addition to his professional activities, he volunteers in various organizations and associations. In German-speaking countries, his books Numerical Mathematics and Numerical Treatment of ODEs: Initial and Boundary Value Problems count among the standard works on numerical analysis. He has also contributed over 70 articles for refereed journals. MASOUD SARAVI is professor of mathematics at the Islamic Azad University (IAU), Nour Branch, Iran. His research interests include the numerical solution of ODEs, partial differential equations (PDEs) and integral equations, as well as differential algebraic equations (DAEs) and spectral methods. In addition to publishing several papers with German colleagues, Dr. Saravi has published more than 15 successful titles on mathematics. The immense popularity of his books is deemed as a reflection of more than 20 years of educational experience and a result of his accessible style of writing, as well as a broad coverage of well-laid-out and easy-to-follow subjects. He is currently a board member at IAU and is working together with the Numerical Analysis Group and the Faculty of Mathematics and Computer Sciences of FSU Jena (Germany). He started off his academic studies at the UK s Dudley Technical College in the UK before receiving his first degree in mathematics and statistics from the Polytechnic of North London and his advanced xiii

xiv About the Authors degree in numerical analysis from Brunel University. After obtaining his M.Phil. in applied mathematics from the Amir Kabir University in Iran, he completed his Ph.D. in numerical analysis on solutions of ODEs and DAEs using spectral methods at the Open University in the UK.