The Computable Differential Equation Lecture ... - Bruce E. Shapiro

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142 CHAPTER 7. DELAY DIFFERENTIAL EQUATIONS which prints the following output: To automatically plot the results instead of printing them out, we modify our code block as follows. methodOfSteps2p[g , t , tau , n ] := Module[{y0, y, s, i, p = {}}, y0 = g[0]; y = y0 - Integrate[g[s - tau], {s, 0, t} ]; AppendTo[p, Plot[y, t, 0, tau, DisplayFunction -> Identity]]; For[i = 1, i n - 1, i++, a = i*tau; b = a + tau; y0 = y /. {t -> a}; y = y0 - Integrate[y /. {t -> s - 1}, {s, a, t} ]; y = Expand[y]; AppendTo[p, Plot[y, {t, a, b}, DisplayFunction -> Identity]]; ]; Return[Show[p, DisplayFunction -> $DisplayFunction]]; ] Math 582B, Spring 2007 California State University Northridge c○2007, B.E.Shapiro Last revised: May 23, 2007
CHAPTER 7. DELAY DIFFERENTIAL EQUATIONS 143 The plot is then generated with methodOfSteps2p[g, t, 1, 10] We observe that the solution is a decaying oscillation, whereas the corresponding initial value problem without deal y would give a pure exponential decay. Thus the two solutions are qualitatively diffferent. 7.3 Numerical Implementation of Method of Steps In this section we illustrate the numerical solution of a delay differential equation using the method of steps via several examples. These examples illustrate that the solution of the DDE can vary radically by minor changes in the initial data or the parameters of the DDE, such as the size of the delay. We will begin with the example from the previous section: y ′ (t) = −y(1 − t) (7.64) y(t) = 1, −τ < t < 0 (7.65) One key drawback of the method of steps is that the step size must be an integral divisor of the delay. Thus for some integer n, we must have nh = τ. So long as there is only a single delay and the delay is not a function of time, we can get around this by automatically setting the mesh size based on the number of points we want to have in the mesh. For a fixed step size, this would give us h = n/τ with n an input parameter; for a variable step size, we would be limited to multiples of this step. c○2007, B.E.Shapiro Last revised: May 23, 2007 Math 582B, Spring 2007 California State University Northridge
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The Computable Differential Equatio
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Contents 1 Classifying The Problem
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CONTENTS v Timeline on Computable D
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Chapter 1 Classifying The Problem 1
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CHAPTER 1. CLASSIFYING THE PROBLEM
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Chapter 2 Successive Approximations
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CHAPTER 2. SUCCESSIVE APPROXIMATION
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Chapter 3 Approximate Solutions 3.1
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Bibliography [1] Ascher, Uri M., Ma
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