# Solving Ode in Matlab

**Topics:**Numerical analysis, Boundary value problem, Initial value problem

**Pages:**43 (12466 words)

**Published:**February 8, 2013

Instructor’s Manual

L.F. Shampine and I. Gladwell Mathematics Department Southern Methodist University Dallas, TX 75275 S. Thompson Department of Mathematics & Statistics Radford University Radford, VA 24142

c 2002, L.F. Shampine, I. Gladwell & S. Thompson

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Contents

1 Getting Started 1.1 Introduction . . . . . . 1.2 Existence, Uniqueness, 1.3 Standard Form . . . . 1.4 Control of the Error . 1.5 Qualitative Properties . . . . . . . . . . . . and Well-Posedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 5 5 7 10 11 13 13 13 13 13 13 13 14 15 15 15 15 15 20 21 22 22 23 25 25 25 25 25 26 28 29 33 33 33 34 34 36

2 Initial Value Problems 2.1 Introduction . . . . . . . . . . . . . . . . . . . . 2.2 Numerical Methods for IVPs . . . . . . . . . . 2.2.1 One–Step Methods . . . . . . . . . . . . Local Error Estimation . . . . . . . . . Runge–Kutta Methods . . . . . . . . . . Explicit Runge–Kutta Formulas . . . . . Continuous Extensions . . . . . . . . . . 2.2.2 Methods with Memory . . . . . . . . . . Adams Methods . . . . . . . . . . . . . BDF methods . . . . . . . . . . . . . . . Error Estimation and Change of Order . Continuous Extensions . . . . . . . . . . 2.3 Solving IVPs in Matlab . . . . . . . . . . . . 2.3.1 Event Location . . . . . . . . . . . . . . 2.3.2 ODEs Involving a Mass Matrix . . . . . 2.3.3 Large Systems and the Method of Lines 2.3.4 Singularities . . . . . . . . . . . . . . . . 3 Boundary Value Problems 3.1 Introduction . . . . . . . . . . . . . . . . 3.2 Boundary Value Problems . . . . . . . . 3.3 Boundary Conditions . . . . . . . . . . . 3.3.1 Boundary Conditions at Singular 3.3.2 Boundary Conditions at Inﬁnity 3.4 Numerical Methods for BVPs . . . . . . 3.5 Solving BVPs in Matlab . . . . . . . . 4 Delay Diﬀerential Equations 4.1 Introduction . . . . . . . . . . . . . 4.2 Delay Diﬀerential Equations . . . . 4.3 Numerical Methods for DDEs . . . 4.4 Solving DDEs in Matlab . . . . . 4.5 Other Kinds of DDEs and Software . . . . . . . . . . . . . . . . . . . . . . . . . . . Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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CONTENTS

Chapter 1

Getting Started

1.1 1.2 Introduction Existence, Uniqueness, and Well-Posedness

Solution for Exercise 1.1. It is easily veriﬁed that both solutions returned by dsolve are solutions of the IVP. This fact does not conﬂict with the basic existence and uniqueness result because that result is for IVPs written in the standard (explicit) form y = f...

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