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

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58 CHAPTER 3. APPROXIMATE SOLUTIONS For a system, one defines a function f[t,y] that returns a list, and then call ForwardEuler[f, {t 0 , y 0 }, h, t max ], as before, where the argument y 0 is also a list of the same length as f and y. The return value becomes instead {{t 0 , { y 00 , y 01 , . . . }}, {t 1 , {y 10 , y 11 ,. . . }}, . . . } ✬ Algorithm 3.2. Forward Euler Method for Systems To solve the system of initial value problems y ′ 1 = f 1 (t, y 1 , y 2 , . . . ), y 1 (t 0 ) = y 0,1 y ′ 2 = f 2 (t, y 1 , y 2 , . . . ), y 2 (t 0 ) = y 0,2 y ′ 3 = f 3 (t, y 1 , y 2 , . . . ), y 3 (t 0 ) = y 0,3 . on an interval [t 0 , t max ] with a fixed step size h. ✩ 1. input: f 1 (t, y 1 , . . . ), f 2 (t, y 1 , . . . ), . . . , t 0 , y 0,1 , y 0,2 , . . . , h, t max 2. output: (t 0 , y 10 , y 20 , . . . ) 3. let t = t 0 , u 1 = y 0,1 , u 2 = y 0,2 , u 3 = y 0,3 . . . 4. while t < t max (a) let y n,1 = u 1 + hf 1 (t, u 1 , u 2 , u 3 , . . . ) y n,2 = u 2 + hf 2 (t, u 1 , u 2 , u 3 , . . . ) y n,3 = u 3 + hf 3 (t, u 1 , u 2 , u 3 , . . . ) . (b) let t = t + h (c) let t n = t (d) output: (t n , y n ) (e) let u 1 = y n,1 , u 2 = y n,2 , y 3 = y n,3 , . . . ✫ ✪ Example 3.3. Solve the initial value problem y 1(t) ′ = y 2 (t) (3.145) y 2(t) ′ = −2y 1 (t) (3.146) y 1 (0) = 1 (3.147) y 2 (0) = 0 (3.148) Math 582B, Spring 2007 California State University Northridge c○2007, B.E.Shapiro Last revised: May 23, 2007
CHAPTER 3. APPROXIMATE SOLUTIONS 59 using ForwardEuler on [0, 2π] using h = 0.1π, and plot the results. Solution. First we define the function and the initial conditions, then we solve the system. y0={1,0}; f[t , y ]:= {y[[2]], -2y[[1]]}; r=ForwardEuler[f,{0, y0}, 0.1π, 2π]; The function ListPlot can be used to plot either points or a joined line, but not both on the same function call. To avoid having to repeatedly call ListPlot , we define the following function JoinedListPlotwhich will plot a set of points as a a sequence of connected dots. JoinedListPlot[data , color , opt ?OptionQ] := Show[ ListPlot[data, PlotJoined -> True, PlotStyle -> color, DisplayFunction -> Identity], ListPlot[data, PlotStyle -> color, PointSize[0.02], DisplayFunction -> Identity], opt, DisplayFunction -> $DisplayFunction] The two calls to ListPlot generate the plot with the connected lines (the first call, with PlotJoined->True); and then, the plot with the dots (the second call). The options DisplayFunction->Identity tell Mathematica to actually suppress actually showing the individual plots, and to just calculate the graphical elements that make up the plots. The call to Show then renders the two plots generated by ListPlot by using the option DisplayFunction->$DisplayFunction. The argument opt is used to pass any desired options to the Showfunction. We actually want to generate two plots for our simulation: one for y 1 , and the other for y 2 , but we want them plotted on a single graph. Our new function JoinedListPlot will plot a single data set, but not two data sets. But we can repeat the same trick we used with the definition of JoinedListPlot: set DisplayFunction->Identityand then call the function twice. In outline form, p1 = JoinedListPlot[FirstDataSet, Red, DisplayFunction->Identity]; p2 = JoinedListPlot[SecondDataSet, Blue, DisplayFunction->Identity]; Show[p1, p2, DisplayFunction -> $DisplayFunction] The main problem now is to extract the data sets. The data is in the form { {{t 0 , {y 0,1 , y 0,2 , . . . }}, {{t 1 , {y 1,1 , y 1,2 , . . . }}, . . . } and what we need for each plot is { {t 0 , y 0,1 }, {t 1 , y 1,1 }, {t 2 , y 2,1 }, . . . } 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 1. CLASSIFYING THE PROBLEM
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Page 31 and 32: Chapter 2 Successive Approximations
Page 33 and 34: CHAPTER 2. SUCCESSIVE APPROXIMATION
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Page 51 and 52: CHAPTER 3. APPROXIMATE SOLUTIONS 45
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CHAPTER 5. RUNGE-KUTTA METHODS 109
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Chapter 6 Linear Multistep Methods
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CHAPTER 6. LINEAR MULTISTEP METHODS
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Chapter 7 Delay Differential Equati
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CHAPTER 7. DELAY DIFFERENTIAL EQUAT
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Chapter 8 Boundary Value Problems 8
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CHAPTER 8. BOUNDARY VALUE PROBLEMS
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Chapter 9 Differential Algebraic Eq
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CHAPTER 9. DIFFERENTIAL ALGEBRAIC E
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Chapter 10 Appendix on Analytic Met
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Bibliography [1] Ascher, Uri M., Ma
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