## Solving ODEs with MATLABThis book is a text for a one-semester course for upper-level undergraduates and beginning graduate students in engineering, science, and mathematics. Prerequisites are a first course in the theory of ODEs and a survey course in numerical analysis, in addition to specific programming experience, preferably in MATLAB, and knowledge of elementary matrix theory. Professionals will also find that this useful concise reference contains reviews of technical issues and realistic and detailed examples. The programs for the examples are supplied on the accompanying web site and can serve as templates for solving other problems. Each chapter begins with a discussion of the "facts of life" for the problem, mainly by means of examples. Numerical methods for the problem are then developed, but only those methods most widely used. The treatment of each method is brief and technical issues are minimized, but all the issues important in practice and for understaning the codes are discussed. The last part of each chapter is a tutorial that shows how to solve problems by means of small, but realistic, examples. |

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### Contents

Getting Started | 1 |

12 Existence Uniqueness and WellPosedness | 6 |

13 Standard Form | 19 |

14 Control of the Error | 27 |

15 Qualitative Properties | 34 |

Initial Value Problems | 39 |

22 Numerical Methods for IVPs | 40 |

221 OneStep Methods | 41 |

32 Boundary Value Problems | 135 |

33 Boundary Conditions | 138 |

331 Boundary Conditions at Singular Points | 139 |

332 Boundary Conditions at Infinity | 146 |

34 Numerical Methods for BVPs | 156 |

35 Solving BVPs in MATLAB | 168 |

Delay Differential Equations | 213 |

42 Delay Differential Equations | 214 |

222 Methods with Memory | 57 |

23 Solving IVPs in MATLAB | 81 |

231 Event Location | 92 |

232 ODEs Involving a Mass Matrix | 105 |

233 Large Systems and the Method of Lines | 114 |

234 Singularities | 127 |

Boundary Value Problems | 133 |

43 Numerical Methods for DDEs | 217 |

44 Solving DDEs in MATLAB | 221 |

45 Other Kinds of DDEs and Software | 247 |

251 | |

257 | |

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### Common terms and phrases

absolute error tolerance accuracy accurate algebraic equations approximate solution argument array asymptotic backward Euler method BDFs behavior boundary conditions bvp4c coefficients compute condition at infinity continuous extension convergence DDEs default delay Delay Differential Equations differential equations discontinuities discussed eigenvalue error control estimate event function event location example EXERCISE explicit Runge-Kutta Figure first-order system formula function dydt global guess Heun's method initial point initial value interpolation interval of integration Jacobian linear linear multistep method mass matrix MATLAB IVP solvers mesh points nonlinear numerical methods numerical solution odel5s option partial derivatives PDEs phase plane plot the solution polynomial quadrature relative error result Runge-Kutta formula Runge-Kutta method Shampine smooth solution components solution structure solve the BVP solve the DDEs solving BVPs Solving ODEs sparse matrix specified stability region stiff problems system of ODEs Taylor series tion trapezoidal rule truncation error unknown parameter vector zero