Verfügbarkeit: An introduction to ordinary differential equations

An introduction to ordinary differential equations:

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1. Verfasser:	Robinson, James C. (VerfasserIn)
Format:	Buch
Sprache:	English
Veröffentlicht:	Cambridge [u.a.] Cambridge Univ. Press 2007
Ausgabe:	1. publ., repr. with corr.
Schlagworte:	Equações diferenciais ordinárias Differential equations Gewöhnliche Differentialgleichung
Online-Zugang:	Inhaltsverzeichnis
Beschreibung:	XIV, 399 S. Ill., graph. Darst.
ISBN:	9780521533911

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MARC


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Datensatz im Suchindex

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adam_text	Contents Preface page xiii Introduction 1 Part I First order differential equations 3 1 Radioactive decay and carbon dating 5 1.1 Radioactive decay 5 1.2 Radiocarbon dating 6 Exercises 8 2 Integration variables 9 3 Classification of differential equations 11 3.1 Ordinary and partial differential equations 11 3.2 The order of a differential equation 13 3.3 Linear and nonlinear 13 3.4 Different types of solution 14 Exercises 16 4 Graphical representation of solutions using Matlab 18 Exercises 21 5 Trivial differential equations 22 5.1 The Fundamental Theorem of Calculus 22 5.2 General solutions and initial conditions 25 5.3 Velocity, acceleration and Newton s second law of motion 29 5.4 An equation that we cannot solve explicitly 32 Exercises 33 VUl Contents Part II 6 Existence and uniqueness of solutions 38 6.1 The case for an abstract result 38 6.2 The existence and uniqueness theorem 40 6.3 Maximal interval of existence 41 6.4 The Clay Mathematics Institute s $1 000 000 question 42 Exercises 44 7 Scalar autonomous ODEs 46 „ 7.1 The qualitative approach 46 7.2 Stability, instability and bifurcation 48 7.3 Analytic conditions for stability and instability 49 7.4 Structural stability and bifurcations 50 7.5 Some examples 50 7.6 The pitchfork bifurcation 54 7.7 Dynamical systems 56 Exercises 56 8 Separable equations 59 8.1 The solution recipe 59 8.2 The linear equation і = λχ 61 8.3 Malthus population model 62 8.4 Justifying the method 64 8.5 A more realistic population model 66 8.6 Further examples 68 Exercises 72 9 First order linear equations and the integrating factor 75 9.1 Constant coefficients 75 9.2 Integrating factors 76 9.3 Examples 78 9.4 Newton s law of cooling 79 Exercises 86 10 Two tricks for nonlinear equations 89 10.1 Exact equations 89 10.2 Substitution methods 94 Exercises 97 Second order linear equations with constant coefficients 99 11 Second order linear equations: general theory 101 11.1 Existence and uniqueness 101 1 1 .2 Linearity 102 11.3 Linearly independent solutions 104 11.4 The Wronskian 106 Contents їх 11.5 Linear algebra 1 07 Exercises 109 12 Homogeneous second order linear equations 111 12.1 Two distinct real roots 112 12.2 A repeated real root 113 12.3 No real roots 115 Exercises 118 13 Oscillations 120 13.1 The spring 120 13.2 The simple pendulum 122 13.3 Damped oscillations 123 Exercises 126 14 Inhomogeneous second order linear equations 131 14.1 Complementary function and particular integral 131 14.2 When ƒ (/) is a polynomial 133 14.3 When ƒ (0 is an exponential 135 14.4 When ƒ (?) is a sine or cosine 137 14.5 Rule of thumb 139 14.6 More complicated functions ƒ (ŕ) 139 Exercises 140 15 Resonance 141 15.1 Periodic forcing 141 15.2 Pseudo resonance in physical systems 145 Exercises 148 16 Higher order linear equations 150 16.1 Complementary function and particular integral 150 16.2 The general theory for nth order equations 152 Exercises 153 Part III Linear second order equations with variable coefficients 157 17 Reduction of order 159 Exercises 162 18 The variation of constants formula 164 Exercises 168 19 Cauchy-Euler equations 170 19.1 Two real roots 171 19.2 A repeated root 171 19.3 Complex roots 173 Exercises 174 χ Contents 20 Series solutions of second order linear equations 176 20.1 Power series 176 20.2 Ordinary points 178 20.3 Regular singular points 183 20.4 Besseľs equation 187 Exercises 195 Part IV Numerical methods and difference equations 199 21 Euler s method 201 21.1 Euler s method 201 21.2 An example 203 21.3 Matlab implementation of Euler s method 204 21.4 Convergence of Euler s method 206 Exercises 209 22 Difference equations 213 22.1 First order difference equations 213 22.2 Second order difference equations 215 22.3 The homogeneous equation 215 22.4 Particular solutions 219 Exercises 222 23 Nonlinear first order difference equations 224 23.1 Fixed points and stability 224 23.2 Cobweb diagrams 225 23.3 Periodic orbits 226 23.4 Euler s method for autonomous equations 227 Exercises 230 24 The logistic map 233 24.1 Fixed points and their stability 234 24.2 Periodic orbits 234 24.3 The period-doubling cascade 237 24.4 The bifurcation diagram and more periodic orbits 238 24.5 Chaos 240 24.6 * Analysis of „+] =4јси(1 - xn) 242 Exercises 245 Part V Coupled linear equations 247 25 Vector first order equations and higher order equations 249 25.1 Existence and uniqueness for second order equations 251 Exercises 252 26 Explicit solutions of coupled linear systems 253 Exercises 257 Contents xi 27 Eigenvalues and eigenvectors 259 27.1 Rewriting the equation in matrix form 259 27.2 Eigenvalues and eigenvectors 260 27.3 Eigenvaiues and eigenvectors with Matlab 266 Exercises 267 28 Distinct real eigenvalues 269 28.1 The explicit solution 270 28.2 Changing coordinates 271 28.3 Phase diagrams for uncoupled equations 276 28.4 Phase diagrams for coupled equations 279 28.5 Stable and unstable manifolds 281 Exercises 282 29 Complex eigenvalues 285 29.1 The explicit solution 285 29.2 Changing coordinates and the phase portrait 287 29.3 The phase portrait for the original equation 291 Exercises 292 30 A repeated real eigenvalue 295 30.1 A is a multiple of the identity: stars 295 30.2 A is not a multiple of the identity: improper nodes 295 Exercises 299 31 Summary of phase portraits for linear equations 301 31.1 Jordan canonical form 301 Exercises 305 Part VI Coupled nonlinear equations 307 32 Coupled nonlinear equations 309 32.1 Some comments on phase portraits 309 32.2 Competition of species 310 32.3 Direction fields 311 32.4 Analytical method for phase portraits 314 Exercises 322 33 Ecological models 323 33.1 Competing species 323 33.2 Predator-prey models I 331 33.3 Predator-prey models II 334 Exercises 338 34 Newtonian dynamics 341 34.1 One-dimensional conservative systems 341 34.2 A bead on a wire 344 xii Contents 34.3 Dissipative systems 347 Exercises 350 35 The real pendulum 352 35.1 The undamped pendulum 352 35.2 The damped pendulum 356 35.3 Alternative phase space 358 Exercises 358 36 Periodic orbits 360 36.1 Dulac s criterion 360 36.2 The Poinacré-Bendixson Theorem 361 Exercises 362 -37 *The Lorenz equations 364 38 What next? 373 38.1 Partial differential equations and boundary value problems 373 38.2 Dynamical systems and chaos 374 Exercises 375 Appendix A Real and complex numbers 379 Appendix В Matrices, eigenvalues, and eigenvectors 382 Appendix С Derivatives and partial derivatives 387 Index 395
adam_txt	Contents Preface page xiii Introduction 1 Part I First order differential equations 3 1 Radioactive decay and carbon dating 5 1.1 Radioactive decay 5 1.2 Radiocarbon dating 6 Exercises 8 2 Integration variables 9 3 Classification of differential equations 11 3.1 Ordinary and partial differential equations 11 3.2 The order of a differential equation 13 3.3 Linear and nonlinear 13 3.4 Different types of solution 14 Exercises 16 4 Graphical representation of solutions using Matlab 18 Exercises 21 5 'Trivial' differential equations 22 5.1 The Fundamental Theorem of Calculus 22 5.2 General solutions and initial conditions 25 5.3 Velocity, acceleration and Newton's second law of motion 29 5.4 An equation that we cannot solve explicitly 32 Exercises 33 VUl Contents Part II 6 Existence and uniqueness of solutions 38 6.1 The case for an abstract result 38 6.2 The existence and uniqueness theorem 40 6.3 Maximal interval of existence 41 6.4 The Clay Mathematics Institute's $1 000 000 question 42 Exercises 44 7 Scalar autonomous ODEs 46 „ 7.1 The qualitative approach 46 7.2 Stability, instability and bifurcation 48 7.3 Analytic conditions for stability and instability 49 7.4 Structural stability and bifurcations 50 7.5 Some examples 50 7.6 The pitchfork bifurcation 54 7.7 Dynamical systems 56 Exercises 56 8 Separable equations 59 8.1 The solution'recipe' 59 8.2 The linear equation і = λχ 61 8.3 Malthus' population model 62 8.4 Justifying the method 64 8.5 A more realistic population model 66 8.6 Further examples 68 Exercises 72 9 First order linear equations and the integrating factor 75 9.1 Constant coefficients 75 9.2 Integrating factors 76 9.3 Examples 78 9.4 Newton's law of cooling 79 Exercises 86 10 Two 'tricks' for nonlinear equations 89 10.1 Exact equations 89 10.2 Substitution methods 94 Exercises 97 Second order linear equations with constant coefficients 99 11 Second order linear equations: general theory 101 11.1 Existence and uniqueness 101 1 1 .2 Linearity 102 11.3 Linearly independent solutions 104 11.4 The Wronskian 106 Contents їх 11.5 Linear algebra 1 07 Exercises 109 12 Homogeneous second order linear equations 111 12.1 Two distinct real roots 112 12.2 A repeated real root 113 12.3 No real roots 115 Exercises 118 13 Oscillations 120 13.1 The spring 120 13.2 The simple pendulum 122 13.3 Damped oscillations 123 Exercises 126 14 Inhomogeneous second order linear equations 131 14.1 Complementary function and particular integral 131 14.2 When ƒ (/) is a polynomial 133 14.3 When ƒ (0 is an exponential 135 14.4 When ƒ (?) is a sine or cosine 137 14.5 Rule of thumb 139 14.6 More complicated functions ƒ (ŕ) 139 Exercises 140 15 Resonance 141 15.1 Periodic forcing 141 15.2 Pseudo resonance in physical systems 145 Exercises 148 16 Higher order linear equations 150 16.1 Complementary function and particular integral 150 16.2 The general theory for nth order equations 152 Exercises 153 Part III Linear second order equations with variable coefficients 157 17 Reduction of order 159 Exercises 162 18 The variation of constants formula 164 Exercises 168 19 Cauchy-Euler equations 170 19.1 Two real roots 171 19.2 A repeated root 171 19.3 Complex roots 173 Exercises 174 χ Contents 20 Series solutions of second order linear equations 176 20.1 Power series 176 20.2 Ordinary points 178 20.3 Regular singular points 183 20.4 Besseľs equation 187 Exercises 195 Part IV Numerical methods and difference equations 199 21 Euler's method 201 21.1 Euler's method 201 21.2 An example 203 21.3 Matlab implementation of Euler's method 204 21.4 Convergence of Euler's method 206 Exercises 209 22 Difference equations 213 22.1 First order difference equations 213 22.2 Second order difference equations 215 22.3 The homogeneous equation 215 22.4 Particular solutions 219 Exercises 222 23 Nonlinear first order difference equations 224 23.1 Fixed points and stability 224 23.2 Cobweb diagrams 225 23.3 Periodic orbits 226 23.4 Euler's method for autonomous equations 227 Exercises 230 24 The logistic map 233 24.1 Fixed points and their stability 234 24.2 Periodic orbits 234 24.3 The period-doubling cascade 237 24.4 The bifurcation diagram and more periodic orbits 238 24.5 Chaos 240 24.6 * Analysis of „+] =4јси(1 - xn) 242 Exercises 245 Part V Coupled linear equations 247 25 Vector first order equations and higher order equations 249 25.1 Existence and uniqueness for second order equations 251 Exercises 252 26 Explicit solutions of coupled linear systems 253 Exercises 257 Contents xi 27 Eigenvalues and eigenvectors 259 27.1 Rewriting the equation in matrix form 259 27.2 Eigenvalues and eigenvectors 260 27.3 Eigenvaiues and eigenvectors with Matlab 266 Exercises 267 28 Distinct real eigenvalues 269 28.1 The explicit solution 270 28.2 Changing coordinates 271 28.3 Phase diagrams for uncoupled equations 276 28.4 Phase diagrams for coupled equations 279 28.5 Stable and unstable manifolds 281 Exercises 282 29 Complex eigenvalues 285 29.1 The explicit solution 285 29.2 Changing coordinates and the phase portrait 287 29.3 The phase portrait for the original equation 291 Exercises 292 30 A repeated real eigenvalue 295 30.1 A is a multiple of the identity: stars 295 30.2 A is not a multiple of the identity: improper nodes 295 Exercises 299 31 Summary of phase portraits for linear equations 301 31.1 Jordan canonical form 301 Exercises 305 Part VI Coupled nonlinear equations 307 32 Coupled nonlinear equations 309 32.1 Some comments on phase portraits 309 32.2 Competition of species 310 32.3 Direction fields 311 32.4 Analytical method for phase portraits 314 Exercises 322 33 Ecological models 323 33.1 Competing species 323 33.2 Predator-prey models I 331 33.3 Predator-prey models II 334 Exercises 338 34 Newtonian dynamics 341 34.1 One-dimensional conservative systems 341 34.2 A bead on a wire 344 xii Contents 34.3 Dissipative systems 347 Exercises 350 35 The 'real' pendulum 352 35.1 The undamped pendulum 352 35.2 The damped pendulum 356 35.3 Alternative phase space 358 Exercises 358 36 Periodic orbits 360 36.1 Dulac's criterion 360 36.2 The Poinacré-Bendixson Theorem 361 Exercises 362 -37 *The Lorenz equations 364 38 What next? 373 38.1 Partial differential equations and boundary value problems 373 38.2 Dynamical systems and chaos 374 Exercises 375 Appendix A Real and complex numbers 379 Appendix В Matrices, eigenvalues, and eigenvectors 382 Appendix С Derivatives and partial derivatives 387 Index 395
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spelling	Robinson, James C. Verfasser aut An introduction to ordinary differential equations James C. Robinson 1. publ., repr. with corr. Cambridge [u.a.] Cambridge Univ. Press 2007 XIV, 399 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Equações diferenciais ordinárias larpcal Differential equations Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd rswk-swf Gewöhnliche Differentialgleichung (DE-588)4020929-5 s DE-604 Digitalisierung UB Augsburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016072260&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Robinson, James C. An introduction to ordinary differential equations Equações diferenciais ordinárias larpcal Differential equations Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd
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title	An introduction to ordinary differential equations
title_auth	An introduction to ordinary differential equations
title_exact_search	An introduction to ordinary differential equations
title_exact_search_txtP	An introduction to ordinary differential equations
title_full	An introduction to ordinary differential equations James C. Robinson
title_fullStr	An introduction to ordinary differential equations James C. Robinson
title_full_unstemmed	An introduction to ordinary differential equations James C. Robinson
title_short	An introduction to ordinary differential equations
title_sort	an introduction to ordinary differential equations
topic	Equações diferenciais ordinárias larpcal Differential equations Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd
topic_facet	Equações diferenciais ordinárias Differential equations Gewöhnliche Differentialgleichung
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