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

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$NAME: Math 103L: Exercise on Functions ... - Bruce E. Shapiro$

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140 CHAPTER 7. DELAY DIFFERENTIAL EQUATIONS y = y0 + Integrate[g[s - tau], {s, 0, t} ]; Print["y(t)=", y, " on [0, ", tau, "]"]; 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]; Print["y(t)=", y, " on [", a, ", ", b, "]"]; ] ] Figure 7.1: Solution found in example for 7.1 with λ = 1. The dots show the endpoints of each step. The solution of the initial value problem y ′ = λy, y(1) = 1 is indicated by the dashed lines. To run the program one must first define the function g(t). For example, to solve the DDE the following must be typed into mathematica: y ′ = g(t − τ) (7.60) y(t) = λe t , −τ < t < 0 (7.61) g[t ]:=λExp[t] methodOfSteps[g, t, 1, 5] The corresponding output that will be printed is: Math 582B, Spring 2007 California State University Northridge c○2007, B.E.Shapiro Last revised: May 23, 2007
CHAPTER 7. DELAY DIFFERENTIAL EQUATIONS 141 While the solution in the first example behaves in a manner that is similar (albeit numerically different) from the IVP with y(0) = g(0) the following slightly different example behaves quite differently. Example 7.2. Use Mathematica to solve the DDE on the interval [0, 10] and plot the results. y ′ (t) = −y(1 − t) (7.62) y(t) = 1, −τ < t < 0 (7.63) Solution. We can (almost) trivially modify our previous code to solve the new differential equation, which has a minus sign on the right hand side. methodOfSteps2[g , t , tau , n ] := Module[{y0, y, s, i}, y0 = g[0]; y = y0 - Integrate[g[s - tau], {s, 0, t} ]; Print["y(t)=", y, " on [0, ", tau, "]"]; 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]; Print["y(t)=", y, " on [", a, ", ", b, "]"]; ] ] The calling sequence is g[t ]:= 1; methodOfSteps2[g, t, 1, 10]; 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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CHAPTER 3. APPROXIMATE SOLUTIONS 45
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Bibliography [1] Ascher, Uri M., Ma
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The Computable Differential Equation Lecture ... - Bruce E. Shapiro

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