{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "" -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 268 21 "Improved Euler Method" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "define function " }{TEXT 262 6 "f(x,y)" }{TEXT -1 1 ":" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f:=(x,y)->(x+y-1)^2;" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "define ends of the interval [" } {TEXT 259 1 "a" }{TEXT -1 1 "," }{TEXT 260 1 "b" }{TEXT -1 28 "] and t he initial condition " }{TEXT 261 8 "y(a)=y_0" }{TEXT -1 2 " :" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "a:=0; b:=0.5; y_0:=2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "define the step size:" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 7 "h:=0.1;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "N - \+ number of calculations ; reserve space to store calculated values of \+ x and y :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "N:=floor((b-a)/h); \nx :=Array(0..N,datatype=float):\ny:=Array(0..N,datatype=float): " } {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "initial values of " }{TEXT 257 1 "x" }{TEXT -1 5 " and " }{TEXT 258 1 "y" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 19 "x[0]:=a; y[0]:=y_0;" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 3 "do " }{TEXT 263 1 "N" }{TEXT -1 35 " iterations using sc heme (5), p.362" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 167 "for k from 1 to N do\n x[k]:=a+k*h:\n y_star:=y[k-1]+h*f(x[k-1],y[k-1]) :\n y[k]:=y[k-1]+h*(f(x[k-1],y[k-1])+f(x[k],y_star))/2:\n \+ end do:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "store the val ues of previous calculations in a list " }{TEXT 256 1 "q" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "q:=[seq([ x[n], y[n] ], n = 0..N)]:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "plot the results stored in " } {TEXT 264 1 "q" }{TEXT -1 64 " using a blue line; indicate the receive d values with black dots" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "plot([q ,q],axes=boxed,color=[blue,black], style=[line,point]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 265 39 "find the exact solution of the equation " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(DEtools):" }}{PARA 0 "" 0 "" {TEXT -1 19 "define the equation" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "ODE:=diff(Y(X),X)=f(X,Y(X));" }}{PARA 0 "" 0 "" {TEXT -1 22 "use t he Matlab solver " }{TEXT 266 6 "dsolve" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "dsolve(ODE);" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 54 "plot th e solution using the standard Matlab function " }{TEXT 267 6 "DEplot " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "DEplot(ODE,Y(X),X=0..0.5,[[Y(0) =2]],\ncolour=coral,linecolour=blue,stepsize=.01);" }}}}{MARK "8 0 2" 9 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }