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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 19 "Numerical Solutions" }}
{PARA 18 "" 0 "" {TEXT -1 9 "of ODE's." }}{PARA 19 "" 0 "" {TEXT -1 
22 "Math 374    Fall  1999" }}{PARA 19 "" 0 "" {TEXT -1 11 "Jay Treima
n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "One of the most important us
es of computers in" }}{PARA 0 "" 0 "" {TEXT -1 49 "differential equati
ons is graphically viewing the" }}{PARA 0 "" 0 "" {TEXT -1 52 "behavio
r of a differential equation.  With this tool" }}{PARA 0 "" 0 "" 
{TEXT -1 44 "one can often tell if a model is reasonable." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Load the " }
{HYPERLNK 17 "plots" 2 "plots" "" }{TEXT -1 22 " package for plotting.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 
"restart;\nwith(plots):" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 19 "Numer
ical solutions" }}{PARA 0 "" 0 "" {TEXT -1 40 "Maple has built in rout
ines to do all of" }}{PARA 0 "" 0 "" {TEXT -1 41 "the classic numerica
l methods of solving " }}{PARA 0 "" 0 "" {TEXT -1 39 "differential equ
ations.  The help file," }}{PARA 0 "" 0 "" {HYPERLNK 17 "dsolve[classi
cal]" 2 "dsolve[classical]" "" }{TEXT -1 21 " contains information" }}
{PARA 0 "" 0 "" {TEXT -1 41 "on how to use Maple for the computations.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "The o
utput contains more information" }}{PARA 0 "" 0 "" {TEXT -1 34 "than w
e want.  In what follows the" }}{PARA 0 "" 0 "" {TEXT -1 29 "extra inf
ormation is removed." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 44 "Consider the following differential equation" }}{PARA 
0 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "diff(P(t),t) = t*P(t);" "6#/
-%%diffG6$-%\"PG6#%\"tGF**&F*\"\"\"-F(6#F*F," }{TEXT -1 1 "." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "de
1 := diff(P(t),t) = t*P(t);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "The exact solut
ion is" }}{PARA 0 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "P(t) = C*exp
(t^2/2);" "6#/-%\"PG6#%\"tG*&%\"CG\"\"\"-%$expG6#*&F'\"\"#\"\"#!\"\"F*
" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 
"C;" "6#%\"CG" }{TEXT -1 32 " is the initial population.  Our" }}
{PARA 0 "" 0 "" {TEXT -1 29 "initial population will be 1." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "Here is a plot of
 the exact solution.  Since we want to plot" }}{PARA 0 "" 0 "" {TEXT 
-1 61 "this solution with the numerical solutions the polt structure" 
}}{PARA 0 "" 0 "" {TEXT -1 34 "is stored and displayed using the " }
{HYPERLNK 17 "display" 2 "plots,display" "" }{TEXT -1 9 " command." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
101 "P_exact := t -> exp(t^2/2);\nExact_plot := plot(P_exact(t),t=0..2
.5,color=black):\ndisplay(Exact_plot);" }}}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 14 "Euler's Method" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Here is Euler's me
thod applied to" }}{PARA 0 "" 0 "" {TEXT -1 35 "our first differential
 equation for" }}{PARA 0 "" 0 "" {TEXT -1 33 "time from 0 to 2.5 using
 5 steps." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 134 "nsol_1 := \n   dsolve(\{de1, P(0)=1\}, P(t), numeric
,\n     method=classical[foreuler],stepsize=.5,\n     value=array([seq
(.5*i,i=0..5)]));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 12 "One can use " }{HYPERLNK 17 "pointplot" 2 "plots,pointplo
t" "" }{TEXT -1 28 " to plot this solution.  One" }}{PARA 0 "" 0 "" 
{TEXT -1 47 "needs to get the points from the table and then" }}{PARA 
0 "" 0 "" {TEXT -1 56 "convert the structure to a list of lists.  Reme
mber that" }}{PARA 0 "" 0 "" {TEXT -1 53 "a numerical method gives app
roximations at a descrete" }}{PARA 0 "" 0 "" {TEXT -1 14 "set of point
s." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 144 "points := convert(nsol_1[2,1],listlist):\np_plot1 :=
 pointplot(points,style=point,\n          symbol=box,color=red):\ndisp
lay(Exact_plot,p_plot1);\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 
1 {PARA 4 "" 0 "" {TEXT -1 15 " Improved Euler" }}{PARA 0 "" 0 "" 
{TEXT -1 41 "This method is also called Heun's method." }}{PARA 0 "" 
0 "" {TEXT -1 22 "In Maple it is called " }{HYPERLNK 17 "heunform" 2 "
dsolve,classical" "" }{TEXT -1 17 ".  It is used in " }}{PARA 0 "" 0 "
" {TEXT -1 31 "the same way as Euler's method." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "Here is the same differen
tial equation as above." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 279 "nsol_2 := \n   dsolve(\{de1, P(0)=
1\}, P(t), numeric,\n     method=classical[heunform],stepsize=.5,\n   \+
  value=array([seq(.5*i,i=0..5)]));\npoints := convert(nsol_2[2,1],lis
tlist):\np_plot2 := pointplot(points,symbol=cross,\n        style=poin
t,color=blue):\ndisplay(Exact_plot,p_plot2);" }}}}{SECT 1 {PARA 4 "" 
0 "" {TEXT -1 11 "Runge-Kutta" }}{PARA 0 "" 0 "" {TEXT -1 35 "The Rung
e-Kutta algorithm is a more" }}{PARA 0 "" 0 "" {TEXT -1 52 "accurate t
echnique than the Euler and Modified Euler" }}{PARA 0 "" 0 "" {TEXT 
-1 47 "algorithms.  It requires 4 function evaluations" }}{PARA 0 "" 
0 "" {TEXT -1 9 "per step." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 43 "Here is the Runge-Kutta-4 algorithm applied" }}
{PARA 0 "" 0 "" {TEXT -1 34 "to the same differential equation." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
277 "nsol_3 := \n   dsolve(\{de1, P(0)=1\}, P(t), numeric,\n     metho
d=classical[rk4],stepsize=.5,\n     value=array([seq(.5*i,i=0..5)]));
\npoints := convert(nsol_3[2,1],listlist):\np_plot3 := pointplot(point
s,symbol=diamond,\n        style=point,color=brown):\ndisplay(Exact_pl
ot,p_plot3);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 43 "One can plot all of the numerical solutions" }}{PARA 0 "
" 0 "" {TEXT -1 24 "together for comparison." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "display(\{Exact_pl
ot,p_plot1,p_plot2,p_plot3\});" }}}{PARA 0 "" 0 "" {TEXT -1 43 "Notice
 how close together the solutions are" }}{PARA 0 "" 0 "" {TEXT -1 13 "
in this case." }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Exercises" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Use Euler's method to solve the di
fferential" }}{PARA 0 "" 0 "" {TEXT -1 8 "equation" }}{PARA 0 "" 0 "" 
{TEXT -1 2 "  " }{XPPEDIT 18 0 "Diff(y(x),x) = 1+y(x)^2;" "6#/-%%DiffG
6$-%\"yG6#%\"xGF*,&\"\"\"\"\"\"*$)-%\"yG6#%\"xG\"\"#F-F-" }{TEXT -1 
11 ",  y(0) = 0" }}{PARA 0 "" 0 "" {TEXT -1 45 "on the interval [0,2] \+
with 5, 10 , 15, and 20" }}{PARA 0 "" 0 "" {TEXT -1 47 "equal steps.  \+
Then plot each of these numerical" }}{PARA 0 "" 0 "" {TEXT -1 49 "solu
tions along with the exact solution.  What do" }}{PARA 0 "" 0 "" 
{TEXT -1 43 "you conclude about Euler's method?  What do" }}{PARA 0 "
" 0 "" {TEXT -1 48 "you conclude about numerical methods in general?" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Numerically solve the different
ial equation" }}{PARA 0 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "diff(y
(x),x) = x^2+y(x);" "6#/-%%diffG6$-%\"yG6#%\"xGF*,&*$)F*\"\"#\"\"\"F/-
F(6#F*F/" }{TEXT -1 12 ",           " }{XPPEDIT 18 0 "y(0) = 1;" "6#/-
%\"yG6#\"\"!\"\"\"" }}{PARA 0 "" 0 "" {TEXT -1 44 "on the interval [0,
1] with 4 steps using the" }}{PARA 0 "" 0 "" {TEXT -1 40 "Euler, Modif
ied Euler, and Runge-Kutta 4" }}{PARA 0 "" 0 "" {TEXT -1 50 "algorithm
s and plot your results.  Do your results" }}{PARA 0 "" 0 "" {TEXT -1 
34 "match the theoretical predictions?" }}}}}{MARK "0 2 0" 16 }
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