Computer Algebra Recipes

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INTRODUCTION A. Computer Algebra Systems Man is still the most extraordinary computer of all. John F. Kennedy, former American president. Speech, 21 May 1963. Unlike traditional programming languages such as Fortran and C, a computer algebra language such as Maple allows one to compute not only with numbers, but also with symbols, formulas, equations, and so on. Using a computer algebra system (CAS), symbolic computation can be done on the computer, replacing the traditional pen-and-paper approach with the keyboard/mouse and computer display. By entering short, simple, transparent commands on the computer keyboard (which will be referred to as the \classic" approach), or by selecting mathematical symbols from a palette with the mouse, the CAS user can quickly and accurately generate symbolic input and output on the computer screen. Mathematical operations such as di®erentiation, integration, and series expansion of functions can be done analytically on the computer. Because it also has numerical capability, a CAS allows the student or the researcher to tackle all aspects of mathematical modeling, from analytic derivation and manipulation of the model equations to the analytic or numerical solution of those equations, to the plotting or animation of the results. One of the most powerful computer algebra systems currently available is Maple 10, which will be used in this text. Useful reference books to this CAS are the Maple user manual [Map05] and the introductory and advanced programming guides [MGH + 05]. In the two volumes of Computer Algebra Recipes, we present classic Maple worksheets, or \recipes," that demonstrate how a CAS can serve as a valuable adjunct tool in easily deriving, solving, plotting, and exploring interesting, modern scienti¯c models chosen from a wide variety of disciplines ranging from the physical and biological sciences to the social sciences and engineering. The present book is the second volume and concentrates on mathematically more advanced models involving linear and nonlinear ordinary and partial di®erential equations (ODEs and PDEs). The classic Maple worksheet interface, which requires less computer memory than the standard interface, is used to generate all the mathematical and graphical output shown in this text. 1
2 INTRODUCTION B. Computer Algebra Recipes The mathematics is not there till we put it there. Sir Arthur Eddington, The Philosophy of Physical Science, 1939 The ODE and PDE recipes in our computer algebra \menu" have been organized into three sections of increasing mathematical sophistication. The Appetizers illustrate phase-plane portraits and analysis, the Entrees deal with linear and nonlinear ODEs and linear PDEs, and the Desserts feature the hunt for solitons and some nonlinear diagnostic tools. The recipes are intended to ful¯ll not only a useful and serious pedagogical purpose but also to titillate and stimulate the reader's intellect and imagination. Associated with each recipe is an important scienti¯c model or method and usually some historical background or an interesting story featuring an engineering or science student who will guide you through the steps of the recipe. These storybook characters are ¯ctitious composites of some of the more likeable, industrious, and brighter students that the authors have had the privilege of teaching over the years. Every topic or story in the text contains the Maple code or recipe to explore that particular topic. To make life easier for you, all recipes have been placed on the CD-ROM enclosed within the back cover of this text. The recipes are ordered according to the chapter, section, and subsection (story) number. For example, the recipe 01-2-3, entitled RÄossler's Strange Attractor, is associated with Chapter 1, Section 2, Subsection 3. Although the recipes can be directly accessed on the CD by clicking on the appropriate worksheet number, it is recommended that you access them through the menu index ¯le, 00menu. All recipes may be conveniently accessed from this menu using the hyperlinks. Complete instructions on how to do this may be found in the menu ¯le. The computer code on the CD is unannotated, so you will have to read the text in order to understand what the code is trying to accomplish. The code has been imported into the text and here is accompanied by detailed explanations of the underlying modeling concepts and computational methods. The recommended procedure for using this text is ¯rst to read a given topic/story for overall comprehension and enjoyment. If you are having any di±culty in understanding a piece of the text code, then you should execute the corresponding Maple worksheet and try variations on the code. Keep in mind that the same objective may often be achieved by a di®erent combination of Maple commands from those used by the authors. After reading the topic, you should execute the worksheet (if you have not already done so) to make sure the code works as expected. At this point feel free to explore the topic. Try rotating any three-dimensional graphs or running any animations in the ¯le. See what happens when changes in the model or Maple code are made and then try to interpret any new results. This book is intended to be open-ended and merely serve as a guide to what is possible in mathematical modeling using a CAS, the possibilities being limited only by your own background and desires. Each topic or story is self-contained and generally done completely, from
Page 1 and 2: Richard H. Enns George C. McGuire C
Page 3 and 4: PREFACE A computer algebra system (
Page 5 and 6: viii CONTENTS 2.3.1 Finite Di®eren
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Page 13 and 14: 6 INTRODUCTION The negative answer
Page 15 and 16: 8 INTRODUCTION D. Maple Help We tea
Page 17 and 18: Part I THE APPETIZERS A man ceases
Page 19 and 20: 14 CHAPTER 1. PHASE-PLANE PORTRAITS
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54 CHAPTER 2. PHASE-PLANE ANALYSIS
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Chapter 3 Linear ODE Models Among a
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3.1. FIRST-ORDER MODELS 111 Therate
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3.1. FIRST-ORDER MODELS 113 Pressur
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3.1. FIRST-ORDER MODELS 115 3.1.2 G
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3.1. FIRST-ORDER MODELS 117 Evil Kn
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3.2. SECOND-ORDER MODELS 119 > tf:=
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3.2. SECOND-ORDER MODELS 121 3.2.2
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3.2. SECOND-ORDER MODELS 123 Loadin
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3.2. SECOND-ORDER MODELS 125 0.4 0.
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3.2. SECOND-ORDER MODELS 127 On ent
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3.2. SECOND-ORDER MODELS 129 Using
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3.3. SPECIAL FUNCTION MODELS 131 3.
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3.3. SPECIAL FUNCTION MODELS 133 1
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3.3. SPECIAL FUNCTION MODELS 135 Th
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3.3. SPECIAL FUNCTION MODELS 137 3.
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3.3. SPECIAL FUNCTION MODELS 139 th
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3.3. SPECIAL FUNCTION MODELS 141 PR
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3.3. SPECIAL FUNCTION MODELS 143 ¡
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3.3. SPECIAL FUNCTION MODELS 145 μ
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3.3. SPECIAL FUNCTION MODELS 147 >
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150 CHAPTER 4. NONLINEAR ODE MODELS
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208 CHAPTER 5. LINEAR PDE MODELS. P
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Chapter 6 Linear PDE Models. Part 2
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6.1. WAVE EQUATION MODELS 249 > sol
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6.1. WAVE EQUATION MODELS 251 6.1.2
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6.1. WAVE EQUATION MODELS 253 The p
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6.1. WAVE EQUATION MODELS 255 Recog
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6.1. WAVE EQUATION MODELS 257 PROBL
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6.1. WAVE EQUATION MODELS 259 Then
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6.2. SEMI-INFINITE AND INFINITE DOM
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6.3. NUMERICAL SIMULATION OF PDES 2
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Part III THE DESSERTS The way a chi
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288 CHAPTER 7. THE HUNT FOR SOLITON
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320 CHAPTER 8. NONLINEAR DIAGNOSTIC
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356 BIBLIOGRAPHY [Dav62] H. T. Davi
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358 BIBLIOGRAPHY [Nyq28] H. Nyquist
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Index acoustical waveguide, 261 ade
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INDEX 363 eigenvalue, 130 Eiram Eir
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INDEX 365 Help, Full Text Search, 8
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INDEX 367 laplace, 121,267,268 lhs,
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INDEX 369 ¯sh harvesting, 100 ¯xe
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INDEX 371 quartic map, 345 Queen Di
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