Solving Differential EquationsandNumerical Solutions of Differential EquationsMath 374 Summer 2008Jay TreimanThis worksheet is an introduction to asolving differential equationsand approximating solutions to initial value problems. The first part isdone using dsolve and DEplot. Using the numerical options for dsolve, it is easy to do to numerical approximations and to look at errors.
<Text-field style="_cstyle5" layout="_pstyle4">Load the <Hyperlink linktarget="Help:DEtools" hyperlink="true"><Font style="_cstyle6">DEtools</Font></Hyperlink> and <Hyperlink linktarget="Help:plots" hyperlink="true"><Font style="_cstyle6">plots</Font></Hyperlink> Packages</Text-field>restart;
with(DEtools):
with(plots):
<Text-field style="Heading 1" layout="Heading 1">Solving and plotting differential equations</Text-field>One can easily solve many differential equations with Maple. The onlything to watch is that you must make the dependent variable a functionof the independent variable.Here is the solution to 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.DE_A := diff(y(x),x)=x*y(x)-x;dsolve(DE_A);One can find out what type of differential equation one has using theodeadvisor.odeadvisor(DE_A);One can also try to see how Maple solves the differential equationusing infolevel.infolevel[dsolve]:=2;
dsolve(DE_A);In order to see solutions one can use the DEplot command.infolevel[dsolve]:=0:
DEplot(DE_A,y(x),x=-4..4,y=-100..100,[[0,0],[0,1],[0,-1],[0,3],[0,-3]],
linecolor=[blue,cyan,orange,black,green],thickness=2,
stepsize=0.04);
<Text-field style="_cstyle5" layout="_pstyle4">Euler's Method</Text-field>The simplest method is Euler's method. It is a simple linear approximationto the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ4RidGNEY3Rj5GPkY+RitGPg== where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is the solution to 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. This approximationis 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.Using dsolve with the classical option one can get either a function or a table of values. Here is an example with 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.DE_1 := diff(y(x),x)=x+y(x);
DE_1_E_soln := dsolve({DE_1,y(0)=1},numeric,method=classical[foreuler],
stepsize=1/4);
DE_1_E_vals := dsolve({DE_1,y(0)=1},numeric,method=classical[foreuler],
output=array([seq(i/4,i=0..4)]),
stepsize=1/4);
Since this differential equation has a simple closed form solution, it is possibleto calculate the errors and show plots that are illustrative. Here is a plot the Euler approximation with the exact solution.Note the use of the options in pointplot to get both a picewise line plot and a plot of the points give with the stepsize of 1/4.DE_1_soln := dsolve({DE_1,y(0)=1});
DE_1_plot := plot(rhs(DE_1_soln),x=-0.1..1.1):
DE_1_pts := convert(DE_1_E_vals[2,1],listlist):
DE_1_Ptplot := pointplot(DE_1_pts,symbol=box,symbolsize=15):
DE_1_Ptplota := pointplot(DE_1_pts,connect=true):
display(DE_1_plot,DE_1_Ptplot,DE_1_Ptplota);
<Text-field style="_cstyle5" layout="_pstyle4">RK2 (Sometimes called Modified Euler or Heun's method)</Text-field>This method can be interpreted as trapazoid integration for the 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 from the differential equation 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. Given a pointLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPg== and a LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= value LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPg==, Set 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. One can approximate 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 by Euler's method to get 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. The approximation to 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 is 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. Using thisin the trapazoid method one gets 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 as the approximation.One uses dsolve in Maple with the rk2 option for this. Here is the same plot as abovefor this solution method.DE_1_rk2_soln := dsolve({DE_1,y(0)=1},numeric,method=classical[rk2],
stepsize=1/4);
DE_1_rk2_vals := dsolve({DE_1,y(0)=1},numeric,method=classical[rk2],
output=array([seq(i/4,i=0..4)]),
stepsize=1/4);DE_1_rk2_soln_4 := rhs(dsolve({DE_1,y(0)=1},numeric,method=classical[rk2],
stepsize=1/4,output=listprocedure)[2]);
pts_rk2_4 := [seq([i/4,DE_1_rk2_soln_4(i/4)],i=0..4)];
DE_1_Ptplot := pointplot(pts_rk2_4,symbol=box,symbolsize=15):
DE_1_Ptplota := pointplot(pts_rk2_4,connect=true):
display(DE_1_plot,DE_1_Ptplot,DE_1_Ptplota);Notice how much better this numerical solution matches the actual solution.
<Text-field style="_cstyle5" layout="_pstyle4">RK4</Text-field>This method corresponds to Simpson's method for numerical integration. Here is the formula. One calculates, in order, k1, k2, k3, and k4.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 that one uses two Euler type approximations for 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and one for 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. There is one value at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPg==, four at 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, andone at 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, this is the same as Simpson's method.Here is same plot as above for RK4.DE_1_RK4_soln := dsolve({DE_1,y(0)=1},numeric,method=classical[rk4],
stepsize=1/4);
DE_1_RK4_vals := dsolve({DE_1,y(0)=1},numeric,method=classical[rk4],
output=array([seq(i/4,i=0..4)]),
stepsize=1/4);DE_1_rk4_soln_4 := rhs(dsolve({DE_1,y(0)=1},numeric,method=classical[rk4],
stepsize=1/4,output=listprocedure)[2]);
pts_rk4_4 := [seq([i/4,DE_1_rk4_soln_4(i/4)],i=0..4)];
DE_1_Ptplot := pointplot(pts_rk4_4,symbol=box,symbolsize=15):
DE_1_Ptplota := pointplot(pts_rk4_4,connect=true):
display(DE_1_plot,DE_1_Ptplot,DE_1_Ptplota);It is hard to see, but the values are better than for RK2.
<Text-field style="_cstyle9" layout="_pstyle6">Exercises</Text-field>
<Text-field style="Heading 2" layout="Heading 2">1</Text-field>Solve the differential equation 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 with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkjbW5HRiQ2JFEiMUYnL0YzUSdub3JtYWxGJ0Y+Rj4tSSNtb0dGJDYtUSI9RidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRi8lKXN0cmV0Y2h5R0ZGLyUqc3ltbWV0cmljR0ZGLyUobGFyZ2VvcEdGRi8lLm1vdmFibGVsaW1pdHNHRkYvJSdhY2NlbnRHRkYvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZVLUY7NiRRIzIuRidGPkY+
<Text-field style="Heading 2" layout="Heading 2">2</Text-field>Solve the differential 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.
<Text-field style="_cstyle10" layout="_pstyle7">3</Text-field>Repeat the plot for the Euler's solution to DE1 with step sizes of 1/8, 1/16, and 1/32.
<Text-field style="_cstyle10" layout="_pstyle7">4</Text-field>Use pointplot to plot the errors for the Euler's solution to DE1 with step sizes of 1/4, 1/8, 1/16, and 1/32. Does this data coincide with the error 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? (You do not need to find LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=.)
<Text-field style="_cstyle10" layout="_pstyle7">5</Text-field>Use pointplot to plot the errors for the RK2 solution to DE1 with step sizes of 1/4, 1/8, 1/16, and 1/32. Does this data coincide with the error estimate 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? (You do not need to find LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=.)
<Text-field style="_cstyle10" layout="_pstyle7">6</Text-field>Use pointplot to plot the errors for the RK4 solution to DE1 with step sizes of 1/4, 1/8, 1/16, and 1/32. Does this data coincide with the error 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? (You do not need to find LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=.)
<Text-field style="_cstyle10" layout="_pstyle7">7</Text-field>Compare your errors with the error for the standard RKF45 method thatMaple uses for DE1. What do you conclude?