Solving ODEs with MATLAB

Solving ODEs with MATLAB - Solving ODEs with Matlab...

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Solving ODEs with Matlab : 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 5 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Existence, Uniqueness, and Well-Posedness . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Standard Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 Control of the Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.5 Qualitative Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2 Initial Value Problems 13 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Numerical Methods for IVPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2.1 One–Step Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Local Error Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Runge–Kutta Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Explicit Runge–Kutta Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Continuous Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2.2 Methods with Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Adams Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 BDF methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Error Estimation and Change of Order . . . . . . . . . . . . . . . . . . . . . . 15 Continuous Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Solving IVPs in Matlab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3.1 Event Location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3.2 ODEs Involving a Mass Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3.3 Large Systems and the Method of Lines . . . . . . . . . . . . . . . . . . . . . 22 2.3.4 Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3 Boundary Value Problems 25 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2 Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.3.1 Boundary Conditions at Singular Points . . . . . . . . . . . . . . . . . . . . . 25 3.3.2 Boundary Conditions at Infinity . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.4 Numerical Methods for BVPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.5 Solving BVPs in Matlab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4 Delay Differential Equations 33 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.2 Delay Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.3 Numerical Methods for DDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.4 Solving DDEs in Matlab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.5 Other Kinds of DDEs and Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3
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4 CONTENTS
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Chapter 1 Getting Started 1.1 Introduction 1.2 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 0 = f ( t, y ) , y ( t 0 ) = y 0 In fact, when we write the given IVP in this form, we obtain two IVPs, y 0 = f 1 ( t, y ) = p 1 - y 2 , y (0) = 0 and y 0 = f 2 ( t, y ) = - p 1 - y 2 , y (0) = 0 It is easily verified that both functions, f 1 and f 2 , satisfy a Lipschitz condition on a region containing the initial condition, hence both IVPs have a unique solution. For example, ∂f 1 ∂y = - y p 1 - y 2 0 . 5 0 . 75 for - 0 . 5 y 0 . 5 and for all t . In this way we find that the given IVP has exactly two solutions. Solution for Exercise 1.2. By definition, f ( t, y ) satisfies a Lipschitz condition with constant L in a region if | f ( t, u ) - f ( t, v ) | ≤ L | u - v | for all ( t, u ) , ( t, v ) in the region. If this function f ( t, y ) satisfies a Lipschitz condition on | t | ≤ 1 , | y | ≤ 1, then | p | u | - p | 0 | | = p | u | ≤ L | u | = L | u - 0 | This implies that 1 p | u | L 5
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6 CHAPTER 1. GETTING STARTED However, if we let u 0, we see that this is not possible. Accordingly, f ( t, y ) does not satisfy a Lipschitz condition on this rectangle. Two solutions to the IVP are y ( t ) 0 and y ( t ) = t 2 4 . On the rectangle | t | ≤ 1 , 0 < α y 1, ∂f ∂y = 1 2 y 1 2 α This upper bound on the magnitude of the partial derivative serves as a Lipschitz constant for f ( t, y ) on this rectangle.
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