# Solving Ordinary Differential Equations I: Nonstiff Problems

Springer Science & Business Media, Apr 16, 2008 - Mathematics - 528 pages

This book deals with methods for solving nonstiff ordinary differential equations. The first chapter describes the historical development of the classical theory, and the second chapter includes a modern treatment of Runge-Kutta and extrapolation methods. Chapter three begins with the classical theory of multistep methods, and concludes with the theory of general linear methods. The reader will benefit from many illustrations, a historical and didactic approach, and computer programs which help him/her learn to solve all kinds of ordinary differential equations. This new edition has been rewritten and new material has been included.

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### Contents

 Classical Mathematical Theory 1 I1 Terminology 2 I2 The Oldest Differential Equations 4 Leibniz and the Bernoulli Brothers 6 Variational Calculus 7 Clairaut 9 Exercises 10 I3 Elementary Integration Methods 12
 Dense Output for the GBS Method 237 Control of the Interpolation Error 240 Exercises 241 II10 Numerical Comparisons 244 Performance of the Codes 249 A Stretched Error Estimator for DOP853 254 Effect of StepNumber Sequence in ODEX 256 II11 Parallel Methods 257

 Second Order Equations 13 Exercises 14 I4 Linear Differential Equations 16 Variation of Constants 18 Exercises 19 I5 Equations with Weak Singularities 20 Nonlinear Equations 23 Exercises 24 I6 Systems of Equations 26 Fourier 29 Lagrangian Mechanics 30 Hamiltonian Mechanics 32 Exercises 34 I7 A General Existence Theorem 35 Existence Theorem of Peano 41 Exercises 43 I8 Existence Theory using Iteration Methods and Taylor Series 44 PicardLindelöf Iteration 45 Taylor Series 46 Recursive Computation of Taylor Coefficients 47 Exercises 49 19 Existence Theory for Systems of Equations 51 Vector Notation 52 Subordinate Matrix Norms 53 Exercises 55 I10 Differential Inequalities 56 The Fundamental Theorems 57 Estimates Using OneSided Lipschitz Conditions 60 Exercises 62 111 Systems of Linear Differential Equations 64 Resolvent and Wronskian 65 Inhomogeneous Linear Equations 66 Exercises 67 I12 Systems with Constant Coefficients 69 The Schur Decomposition 70 Numerical Computations 72 The Jordan Canonical Form 73 Geometric Representation 77 Exercises 78 I13 Stability 80 The RouthHurwitz Criterion 81 Computational Considerations 85 Liapunov Functions 86 Stability of Nonlinear Systems 87 Stability of NonAutonomous Systems 88 Exercises 89 I14 Derivatives with Respect to Parameters and Initial Values 92 The Derivative with Respect to a Parameter 93 Derivatives with Respect to Initial Values 95 The Nonlinear VariationofConstants Formula 96 Flows and VolumePreserving Flows 97 Canonical Equations and Symplectic Mappings 100 Exercises 104 I15 Boundary Value and Eigenvalue Problems 105 SturmLiouville Eigenvalue Problems 107 Exercises 110 I16 Periodic Solutions Limit Cycles Strange Attractors 111 Chemical Reactions 115 Limit Cycles in Higher Dimensions Hopf Bifurcation 117 Strange Attractors 120 The Ups and Downs of the Lorenz Model 123 Feigenbaum Cascades 124 Exercises 126 RungeKutta and Extrapolation Methods 129 II1 The First RungeKutta Methods 132 General Formulation of RungeKutta Methods 134 Discussion of Methods of Order 4 135 Optimal Formulas 139 Numerical Example 140 Exercises 141 II2 Order Conditions for RungeKutta Methods 143 The Derivatives of the True Solution 145 The Taylor Expansion of the True Solution 148 Faá di Brunos Formula 149 The Derivatives of the Numerical Solution 151 The Order Conditions 153 Exercises 154 II3 Error Estimation and Convergence for RK Methods 156 The Principal Error Term 158 Estimation of the Global Error 159 Exercises 163 II4 Practical Error Estimation and Step Size Selection 164 Embedded RungeKutta Formulas 165 Automatic Step Size Control 167 Starting Step Size 169 Numerical Experiments 170 Exercises 172 II5 Explicit RungeKutta Methods of Higher Order 173 6Stage 5th Order Processes 175 Embedded Formulas of Order 5 176 Higher Order Processes 179 Embedded Formulas of High Order 180 An 8th Order Embedded Method 181 Exercises 185 II6 Dense Output Discontinuities Derivatives 188 Continuous Dormand Prince Pairs 191 Dense Output for DOP853 194 Event Location 195 Discontinuous Equations 196 Numerical Computation of Derivatives with Respect to Initial Values and Parameters 200 Exercises 202 II7 Implicit RungeKutta Methods 204 Existence of a Numerical Solution 206 The Methods of Kuntzmann and Butcher of Order 2s 208 IRK Methods Based on Lobatto Quadrature 210 Collocation Methods 211 Exercises 214 II8 Asymptotic Expansion of the Global Error 216 Variable h 218 Negative h 219 Properties of the Adjoint Method 220 Symmetric Methods 221 Exercises 223 II9 Extrapolation Methods 224 The Aitken Neville Algorithm 226 The Gragg or GBS Method 228 Asymptotic Expansion for Odd Indices 231 Existence of Explicit RK Methods of Arbitrary Order 232 Order and Step Size Control 233
 Parallel RungeKutta Methods 258 Parallel Iterated RungeKutta Methods 259 Extrapolation Methods 261 Exercises 263 II12 Composition of BSeries 264 BSeries 266 Order Conditions for RungeKutta Methods 269 Butchers Effective Order 270 Exercises 272 1113 Higher Derivative Methods 274 Collocation Methods 275 HermiteObreschkoff Methods 277 Fehlberg Methods 278 General Theory of Order Conditions 280 Exercises 281 1114 Numerical Methods for Second Order Differential Equations 283 Nyström Methods 284 The Derivatives of the Exact Solution 286 The Derivatives of the Numerical Solution 288 The Order Conditions 290 On the Construction of Nyström Methods 291 An Extrapolation Method for y¹¹fxy 294 Problems for Numerical Comparisons 296 Performance of the Codes 298 Exercises 300 II15 PSeries for Partitioned Differential Equations 302 Derivatives of the Exact Solution PTrees 303 PSeries 306 Order Conditions for Partitioned RungeKutta Methods 307 Further Applications of PSeries 308 Exercises 311 II16 Symplectic Integration Methods 312 Symplectic RungeKutta Methods 315 An Example from Galactic Dynamics 319 Partitioned RungeKutta Methods 326 Symplectic Nystrom Methods 330 Conservation of the Hamiltonian Backward Analysis 333 Exercises 337 II17 Delay Differential Equations 339 Constant Step Size Methods for Constant Delay 341 Variable Step Size Methods 342 Stability 343 An Example from Population Dynamics 345 Infectious Disease Modelling 347 An Example from Enzyme Kinetics 348 A Mathematical Model in Immunology 349 IntegroDifferential Equations 351 Exercises 352 Multistep Methods and General Linear Methods 354 III1 Classical Linear Multistep Formulas 356 Explicit Adams Methods 357 Implicit Adams Methods 359 Numerical Experiment 361 Explicit Nyström Methods 362 MilneSimpson Methods 363 Methods Based on Differentiation BDF 364 Exercises 366 III2 Local Error and Order Conditions 368 Order of a Multistep Method 370 Error Constant 372 Irreducible Methods 374 The Peano Kernel of a Multistep Method 375 Exercises 377 III3 Stability and the First Dahlquist Barrier 378 Stability of the BDFFormulas 380 Highest Attainable Order of Stable Multistep Methods 383 Exercises 387 III4 Convergence of Multistep Methods 391 Formulation as OneStep Method 393 Proof of Convergence 395 Exercises 396 III5 Variable Step Size Multistep Methods 397 Recurrence Relations for gⱼnϕⱼn and ϕⱼn 399 Variable Step Size BDF 400 General Variable Step Size Methods and Their Orders 401 Stability 402 Convergence 407 Exercises 409 III6 Nordsieck Methods 410 Equivalence with Multistep Methods 412 Implicit Adams Methods 417 BDFMethods 419 Exercises 420 III7 Implementation and Numerical Comparisons 421 Some Available Codes 423 Numerical Comparisons 427 III8 General Linear Methods 430 A General Integration Procedure 431 Stability and Order 436 Convergence 438 Order Conditions for General Linear Methods 441 Construction of General Linear Methods 443 Exercises 445 III9 Asymptotic Expansion of the Global Error 448 Asymptotic Expansion for Strictly Stable Methods 84 450 Weakly Stable Methods 454 The Adjoint Method 457 Symmetric Methods 459 Exercises 460 III10 Multistep Methods for Second Order Differential Equations 461 Explicit Störmer Methods 462 Implicit Störmer Methods 464 Numerical Example 465 General Formulation 467 Convergence 468 Asymptotic Formula for the Global Error 471 Rounding Errors 472 Exercises 473 Appendix Fortran Codes 475 Subroutine DOPRI5 477 Subroutine DOP853 481 Subroutine ODEX 482 Subroutine ODEX2 484 Driver for the Code RETARD 486 Subroutine RETARD 488 Bibliography 491 Symbol Index 521 Subject Index 523 Copyright

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