MATLAB Differential Equations. César Pérez López

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1 MATLAB Differential Equations César Pérez López

2 MATLAB Differential Equations Copyright 2014 by César Pérez López 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. ISBN-13 (pbk): ISBN-13 (electronic): Trademarked names, logos, and images may appear in this book. Rather than use a trademark symbol with every occurrence of a trademarked name, logo, or image we use the names, logos, and images only in an editorial fashion and to the benefit of the trademark owner, with no intention of infringement of the trademark. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. 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. Publisher: Heinz Weinheimer Lead Editor: Dominic Shakeshaft Editorial Board: Steve Anglin, Mark Beckner, Ewan Buckingham, Gary Cornell, Louise Corrigan, Jim DeWolf, Jonathan Gennick, Jonathan Hassell, Robert Hutchinson, Michelle Lowman, James Markham, Matthew Moodie, Jeff Olson, Jeffrey Pepper, Douglas Pundick, Ben Renow-Clarke, Dominic Shakeshaft, Gwenan Spearing, Matt Wade, Steve Weiss Coordinating Editor: Melissa Maldonado Copy Editor: Barnaby Sheppard Compositor: SPi Global Indexer: SPi Global Artist: SPi Global Cover Designer: Anna Ishchenko Distributed to the book trade worldwide by Springer Science+Business Media New York, 233 Spring Street, 6th Floor, New York, NY Phone SPRINGER, fax (201) , orders-ny@springer-sbm.com, or visit Apress Media, LLC is a California LLC and the sole member (owner) is Springer Science + Business Media Finance Inc (SSBM Finance Inc). SSBM Finance Inc is a Delaware corporation. For information on translations, please rights@apress.com, or visit Apress and friends of ED books may be purchased in bulk for academic, corporate, or promotional use. ebook versions and licenses are also available for most titles. For more information, reference our Special Bulk Sales ebook Licensing web page at Any source code or other supplementary material referenced by the author in this text is available to readers at For detailed information about how to locate your book s source code, go to

3 Contents at a Glance About the Author...ix Chapter 1: Introducing MATLAB and the MATLAB Working Environment...1 Chapter 2: First Order Differential Equations. Exact Equations, Separation of Variables, Homogeneous and Linear Equations Chapter 3: Higher Order Differential Equations. The Laplace Transform and Special Types of Equations...45 Chapter 4: Differential Equations Via Approximation Methods...61 Chapter 5: Systems of Differential Equations and Finite Difference Equations...67 Chapter 6: Numerical Calclus with MATLAB. Applications to Differential Equations Chapter 7: Ordinary and Partial Differential Equations with Initial and Boundary Values Chapter 8: Symbolic Differential and Integral Calculus iii

4

5 Contents About the Author...ix Chapter 1: Introducing MATLAB and the MATLAB Working Environment...1 Introduction...1 Developing Algorithms and Applications... 2 Data Access and Analysis... 5 Data Visualization... 6 Numerical Calculation... 9 Publication of Results and Distribution of Applications The MATLAB working environment...11 Help in MATLAB...15 Numerical Computation with MATLAB...18 Symbolic Calculations with MATLAB...19 Graphics with MATLAB...21 General Notation...25 Help with Commands...26 MATLAB and Programming...28 Commands to Escape and Exit to the MS-DOS Environment...29 Chapter 2: First Order Differential Equations. Exact Equations, Separation of Variables, Homogeneous and Linear Equations...33 First Order Differential Equations...33 Separation of Variables...36 Homogeneous Differential Equations...38 v

6 Contents Exact Differential Equations...41 Linear Differential Equations...43 Chapter 3: Higher Order Differential Equations. The Laplace Transform and Special Types of Equations...45 Ordinary High-Order Equations...45 Linear Higher-Order Equations. Homogeneous Equations with Constant Coefficients...47 Non-Homogeneous Equations with Constant Coefficients. Variation of Parameters...48 Non-Homogeneous Equations with Variable Coefficients. Cauchy Euler Equations...52 The Laplace Transform...52 Orthogonal Polynomials...55 Chebychev Polynomials of the First and Second Kind Legendre Polynomials Associated Legendre Polynomials Hermite Polynomials Generalized Laguerre Polynomials Laguerre Polynomials Jacobi Polynomials Gegenbauer Polynomials Bessel and Airy Functions...58 Chapter 4: Differential Equations Via Approximation Methods...61 Higher Order Equations and Approximation Methods...61 The Taylor Series Method...61 The Runge Kutta Method...64 Chapter 5: Systems of Differential Equations and Finite Difference Equations...67 Systems of Linear Homogeneous Equations with Constant Coefficients...67 Systems of Linear Non-Homogeneous Equations with Constant Coefficients...68 Finite Difference Equations...69 Partial Differential Equations...71 vi

7 Contents Chapter 6: Numerical Calclus with MATLAB. Applications to Differential Equations...73 MATLAB and Programming...73 Text Editor...73 Scripts...76 Functions and M-Files. Function, Eval and Feval...78 Local and Global Variables...82 Data Types...84 Flow Control: FOR Loops, WHILE and IF ELSEIF...85 The FOR Loop The WHILE Loop IF ELSEIF ELSE END Loops Switch and Case Continue Break Try... Catch Return Subfunctions...92 Ordinary Differential Equations Using Numerical Analysis...93 Euler s Method Heun s Method The Taylor Series Method Chapter 7: Ordinary and Partial Differential Equations with Initial and Boundary Values Numerical Solutions of Differential Equations Ordinary Differential Equations with Initial Values Ordinary Differential Equations with Boundary Values Partial Differential Equations vii

8 Contents Chapter 8: Symbolic Differential and Integral Calculus Symbolic Computation with MATLAB. Symbolic Variables Symbolic Functions. Substitution and Functional Operations Mathematical Analysis Functions. Limits, Continuity, and Series Derivatives, Integrals and Differential Equations Linear Algebra: Simplifying and Solving Equations viii

9 About the Author César Pérez López is a Professor at the Department of Statistics and Operations Research at the University of Madrid. César is also a Mathematician and Economist at the National Statistics Institute (INE) in Madrid, a body which belongs to the Superior Systems and Information Technology Department of the Spanish Government. César also currently works at the Institute for Fiscal Studies in Madrid. ix

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11 Coming Soon MATLAB Programming for Numerical Analysis, MATLAB Control Systems Engineering, MATLAB Linear Algebra, MATLAB Differential and Integral Calculus, MATLAB Matrix Algebra, xi

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