{VERSION 5 0 "SUN SPARC SOLARIS" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 273 "" 1 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 1 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 1 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 1 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 1 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 278 "" 1 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 1 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 1 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 1 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 1 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 283 "" 1 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 1 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 285 "" 1 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 286 "" 1 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 287 "" 1 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 288 "" 1 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 289 "" 1 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 290 "" 1 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 291 "" 1 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 292 "" 1 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 293 "" 1 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 294 "" 1 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 295 "" 1 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 296 "" 1 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 297 "" 1 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 298 "" 1 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 299 "" 1 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier " 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 2 6 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 2 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple P lot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "MATH 2280 - 2 \+ SECOND MAPLE ASSIGNMENT" }}{PARA 0 "" 0 "" {TEXT -1 70 "Tr eibergs Due Tuesday March 5, 2002" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "This do cument may be downloaded directly to MAPLE. Go to the " }}{PARA 0 "" 0 "" {TEXT -1 63 "class home page http://www.math.utah.edu/~treiberg/ M2280.html/" }}{PARA 0 "" 0 "" {TEXT -1 59 "in your web browser, such \+ as NETSCAPE and download the file" }}{PARA 0 "" 0 "" {TEXT -1 58 "M228 0L2.mws. The \".mws\" suffix means MAPLE worksheet. Then" }}{PARA 0 " " 0 "" {TEXT -1 47 "you should be able to open the file from MAPLE." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "In this \+ lab we explore some of the ways computers approximate the" }}{PARA 0 " " 0 "" {TEXT -1 62 "solution of first order ODE's. In the basic Euler method, the" }}{PARA 0 "" 0 "" {TEXT -1 67 "computer calculates the s lope at a point and draws a short straight" }}{PARA 0 "" 0 "" {TEXT -1 63 "line with that slope from that point to get the next point. The " }}{PARA 0 "" 0 "" {TEXT -1 62 "modified Euler Method and the Runge-K utta methods are improved" }}{PARA 0 "" 0 "" {TEXT -1 69 "versions, th at take into account the derivative at more than just the" }}{PARA 0 " " 0 "" {TEXT -1 68 "left endpoint. This material is discussed in secti ons 2.4-2.6 of the" }}{PARA 0 "" 0 "" {TEXT -1 28 "text by Edwards and Penney. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "The initial value problem we wish to address is the general first " }}{PARA 0 "" 0 "" {TEXT -1 34 "order equation with initial point." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {MPLTEXT 1 0 36 "diff(y(x),x) = f(x,y(x)),y(x0) = y0;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "Let (x0,y 0) denote the initial point of the solution curve and we try" }}{PARA 0 "" 0 "" {TEXT -1 70 "to find an approximation the solution on the in terval [x0,x1]. Suppose" }}{PARA 0 "" 0 "" {TEXT -1 70 "that we wish t o take n equally spaced points in the interval [x0,x1]," }}{PARA 0 " " 0 "" {TEXT -1 65 "x[i]:=x0+i*h, where i=0,...,n and h:=(x1-x0)/n. S uppose that the" }}{PARA 0 "" 0 "" {TEXT -1 64 "point [x[i],y[i]] has \+ been determined. Then we compute the slope" }}{PARA 0 "" 0 "" {TEXT -1 64 "k:=f(x[i],y[i]) and approximate the solution of the differentia l" }}{PARA 0 "" 0 "" {TEXT -1 57 "equation by a line segment of slope \+ k to the new point " }}{PARA 0 "" 0 "" {TEXT -1 17 "y[i+1]:=y[i]+k*h ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "restart:with(DEtools):with(plots): #clear the memory" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefin ed\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "f:= (x,y)-> -3*x*(x -1.25)+y/(3+x); #enter slope function" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),&*&9$\"\"\",&F/F0 $\"$D\"!\"#!\"\"F0!\"$*&9%F0,&\"\"$F0F/F0F5F0F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "x0:=0.0; y0:= 1.0; x1:= 2.0; n:=16; #IC ,endpt,num.steps" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G$\"\"!F&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y0G$\"#5!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"#?!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"nG\"#;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 " h:= (x1-x0)/n; y[0]:=y0; #step size, copy IC" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG$\"++++]7!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> &%\"yG6#\"\"!$\"#5!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 " eqn:= diff(w(t),t) = f(t,w(t)); #enter DE for solver " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/-%%diffG6$-%\"wG6#%\"tGF,,&*&F,\"\" \",&F,F/$\"$D\"!\"#!\"\"F/!\"$*&F)F/,&\"\"$F/F,F/F4F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "soln:=dsolve(\{eqn,w( x0 )=y0\}); \+ #symbolic soln DE" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%solnG/-% \"wG6#%\"tG*&,,*$)F)\"\"#\"\"\"#!\"$F.*&#\"#^\"\"%F/F)F/F/*&#\"$`\"F5F /-%#lnG6#,&\"\"$F/F)F/F/!\"\"*&#F8F5F/-F:6#F=F/F/#F/F=F/F/F " 0 "" {MPLTEXT 1 0 23 "printf(\"EULER METHOD\");" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "printf(\"x[i] y[i] \+ Actual y error\"); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "for i \+ from 0 to n do; # loop. " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 " x[i]:=x0+h*i: # GUTS of the program" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 58 " k:= f(x[i],y[i]): #substitute \+ t=x[i] into " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 " y[i+1]:=y[i]+k *h; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 " valu:=subs(t= x[i],rhs(soln)): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 " erro:=y[i ]-valu:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 " printf(\"%6.2f%15.9f \+ %15.9f %15.9f\\n\"," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 " x[i],y[i] ,valu, erro);#right side of soln" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 " od:# " }}{PARA 6 "" 1 "" {TEXT -1 12 "EULER METHOD" }}{PARA 6 "" 1 "" {TEXT -1 44 "x[i] y[i] Actual y error" }}{PARA 6 " " 1 "" {TEXT -1 53 " 0.00 1.000000000 1.000000000 0.000000 000" }}{PARA 6 "" 1 "" {TEXT -1 53 " .13 1.041666667 1.069389 300 -.027722600" }}{PARA 6 "" 1 "" {TEXT -1 53 " .25 1.136067 709 1.187711800 -.051644100" }}{PARA 6 "" 1 "" {TEXT -1 53 " \+ .38 1.273512621 1.344751000 -.071238400" }}{PARA 6 "" 1 " " {TEXT -1 53 " .50 1.443726630 1.529744600 -.086018000" }}{PARA 6 "" 1 "" {TEXT -1 53 " .63 1.635913295 1.731442700 \+ -.095529400" }}{PARA 6 "" 1 "" {TEXT -1 53 " .75 1.838808473 \+ 1.938159400 -.099350900" }}{PARA 6 "" 1 "" {TEXT -1 53 " .88 \+ 2.040727089 2.137815300 -.097088200" }}{PARA 6 "" 1 "" {TEXT -1 53 " 1.00 2.229603870 2.317976300 -.088372400" }} {PARA 6 "" 1 "" {TEXT -1 53 " 1.13 2.393028991 2.465886200 \+ -.072857200" }}{PARA 6 "" 1 "" {TEXT -1 53 " 1.25 2.518279396 \+ 2.568497200 -.050217800" }}{PARA 6 "" 1 "" {TEXT -1 53 " 1.38 \+ 2.592346437 2.612494000 -.020147600" }}{PARA 6 "" 1 "" {TEXT -1 53 " 1.50 2.601960353 2.584318200 .017642200" }}{PARA 6 "" 1 "" {TEXT -1 53 " 1.63 2.533612029 2.470188000 .063 424000" }}{PARA 6 "" 1 "" {TEXT -1 53 " 1.75 2.373572405 2.256 115800 .117456600" }}{PARA 6 "" 1 "" {TEXT -1 53 " 1.88 2.107 909837 1.927925600 .179984200" }}{PARA 6 "" 1 "" {TEXT -1 53 " 2.00 1.722505682 1.471266300 .251239400" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "We superi mpose the actual solution of the IVP that MAPLE found by" }}{PARA 0 " " 0 "" {TEXT -1 70 "symbolic manipulation on top of the numerical solu tion. To do this, we" }}{PARA 0 "" 0 "" {TEXT -1 69 "store the plot st ructures from the plot of the curve, the plot of the" }}{PARA 0 "" 0 " " {TEXT -1 68 "polygon, and the plot of the points and display them to gether. The x" }}{PARA 0 "" 0 "" {TEXT -1 64 "and y scales are the sam e if we include scaling=constrained. The" }}{PARA 0 "" 0 "" {TEXT -1 63 "equation giving the solution is stored in the variable soln and" } }{PARA 0 "" 0 "" {TEXT -1 67 "rhs(soln) picks off the expression of th e function. Note that it is" }}{PARA 0 "" 0 "" {TEXT -1 69 "not a MAPL E function. Thus to evaluate it at the variable x[i] we had" }}{PARA 0 "" 0 "" {TEXT -1 66 "to substitute t=x[i] using the subs command. Be wary of semicolons" }}{PARA 0 "" 0 "" {TEXT -1 57 "which prevent the \+ display of the tedious plot structures!" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "P1:=plot([seq([x[i],y[i]],i=0..n)],color=magenta):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "P2:=plot([seq([x[i],y[i]],i=0..n)],style=po int," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 " symbol=circle,sym bolsize=5,color=red):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "P3:=plot(r hs(soln),t=x0..x1,color=blue):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "d isplay(\{P1,P2,P3\},scaling=constrained,title=\"Euler Polygon Method\" );" }}{PARA 13 "" 1 "" {GLPLOT2D 431 355 355 {PLOTDATA 2 "6)-%'CURVESG 6$737$$\"\"!F)$\"\"\"F)7$$\"3+++++++]7!#=$\"35+++nmmT5!#<7$$\"3+++++++ +DF/$\"3)*******3x1O6F27$$\"3+++++++]PF/$\"31+++@E^t7F27$$\"3++++++++] F/$\"3++++ImsV9F27$$\"3+++++++]iF/$\"3%******\\H8fj\"F27$$\"3++++++++v F/$\"30+++t%3)Q=F27$$\"3+++++++]()F/$\"3A+++*3F2/#F27$F*$\"3/+++qQgHAF 27$$\"3+++++++D6F2$\"3)******4**GIR#F27$$F.F2$\"31+++'Rz#=DF27$$\"3*** **********\\P\"F2$\"3y*****pVYBf#F27$$\"3++++++++:F2$\"3&******HNg>g#F 27$$\"3+++++++D;F2$\"3$*******G?hLDF27$$\"3+++++++] o0Ds\"F2-%'COLOURG6&%$RGBG$\"*++++\"!\")F(Fip-F$6&F&-Ffp6&FhpFipF(F(-% 'SYMBOLG6$%'CIRCLEG\"\"&-%&STYLEG6#%&POINTG-F$6$7UF'7$$\"39LLLL3VfV!#> $\"3M?$oPzG!=5F27$$\"3&pmm;H[D:)F_r$\"3PIo(>2.#R5F27$$\"3LLLLe0$=C\"F/ $\"3J?)\\H.m(o5F27$$\"3HLLL3RBr;F/$\"3!>R>-^#H/6F27$$\"3Ymm;zjf)4#F/$ \"3'>g%H#[$*\\9\"F27$$\"3=LL$e4;[\\#F/$\"3q:-lIQ8(=\"F27$$\"3p****\\i' y]!HF/$\"3oASpA<([B\"F27$$\"3,LL$ezs$HLF/$\"3ih[Sg/C)G\"F27$$\"3^**** \\7iI_PF/$\"3Y'R![$))p]M\"F27$$\"3#pmmm@Xt=%F/$\"3$\\x-VRioS\"F27$$\"3 QLLL3y_qXF/$\"3i*=J\")HBPY\"F27$$\"3i******\\1!>+&F/$\"3@30%*=8/I:F27$ $\"3')******\\Z/NaF/$\"3!*4vMCWj)f\"F27$$\"3&*******\\$fC&eF/$\"3[\"*) yax'=m;F27$$\"3ELL$ez6:B'F/$\"3'*zUcONRGF27$$\"3slmmm(y8!zF/$\"38(\\si5HO+#F27$$\"3V++]i.tK$)F/$\"3!H3h R18F2#F27$$\"39++](3zMu)F/$\"3KIvK%Q8o8#F27$$\"3\"pmm;H_?<*F/$\"3&yS)e GZX,AF27$$\"3dmm;zihl&*F/$\"3->$pBWp$eAF27$$\"39LLL3#G,***F/$\"3)3t`az imJ#F27$$\"3;LLezw5V5F2$\"3:\\[y'*GGtBF27$$\"3*)***\\PQ#\\\"3\"F2$\"33 @$R>-t)=CF27$$\"3ALL$e\"*[H7\"F2$\"3&zMnQz)yjCF27$$\"3#*******pvxl6F2$ \"3%*3H6r,)\\]#F27$$\"3z****\\_qn27F2$\"3wsq$\\ZZ'RDF27$$\"3%)***\\i&p @[7F2$\"3h'4c&QOTnDF27$$\"3#)****\\2'HKH\"F2$\"3!*[bH\"GE5f#F27$$\"3_m mmwanL8F2$\"3[\"fgJkk_g#F27$$\"3&******\\2goP\"F2$\"3\\[&=*ewj7EF27$$ \"3CLLeR<*fT\"F2$\"3k[(*3,`(=h#F27$$\"3'******\\)Hxe9F2$\"37N+2Q6V-EF2 7$$\"3Ymm\"H!o-*\\\"F2$\"3aI;\"*4;&[e#F27$$\"3))***\\7k.6a\"F2$\"3\"[d zmk`pb#F27$$\"3emmmT9C#e\"F2$\"3fR!f\"Hkw>DF27$$\"3!****\\i!*3`i\"F2$ \"3b)=D7)yypCF27$$\"3PLLL$*zym;F2$\"3Xirwb8W5CF27$$\"3GLL$3N1#4F27$$\"3/++DOl5;>F2$\"3#*=!*Ga_j#z\"F27$$\"3/+++q`KO>F2$ \"3kE-,Qt$3s\"F27$$\"3/++v.Uac>F2$\"36Rh\"R\"G\\X;F27$$\"3-+](=5s#y>F2 $\"3xT8^f5]g:F27$Fap$\"3zw4U@hEr9F2-Ffp6&FhpF(F(Fip-%+AXESLABELSG6%Q!6 \"Fjal-%%FONTG6#%(DEFAULTG-%(SCALINGG6#%,CONSTRAINEDG-%&TITLEG6#Q5Eule r~Polygon~MethodF[bl-%%VIEWG6$;F($\"#?!\"\"F_bl" 1 2 0 1 10 0 2 9 1 4 1 1.000000 -34.000000 -183.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }} }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "The Improved Euler Method is a p redictor - corrector method. The" }}{PARA 0 "" 0 "" {TEXT -1 70 "probl em is that the solution depends on the values of f(x,y) along the" }} {PARA 0 "" 0 "" {TEXT -1 67 "whole interval but the Euler polygon just uses the left endpoint to" }}{PARA 0 "" 0 "" {TEXT -1 68 "construct t he approximation. In numerical analysis classes, you will" }}{PARA 0 " " 0 "" {TEXT -1 67 "derive thae fact that the Improved Euler Method ha s higher order of" }}{PARA 0 "" 0 "" {TEXT -1 69 "accuracy. The basic \+ idea is that one uses an Euler step with slope k1" }}{PARA 0 "" 0 "" {TEXT -1 69 "to find a first guess (the predictor) and then use the fi rst guess to" }}{PARA 0 "" 0 "" {TEXT -1 65 "evaluate the function nea r the actual right hand endpoint to find" }}{PARA 0 "" 0 "" {TEXT -1 64 "another slope k2 (the corrector.) The better approximation of te" }}{PARA 0 "" 0 "" {TEXT -1 68 "average slope along the interval [x[i], x[i+1]] is the average of the" }}{PARA 0 "" 0 "" {TEXT -1 68 "two slop es k:=(k1+k2)/2. The resulting improvement is similar to the" }}{PARA 0 "" 0 "" {TEXT -1 70 "improvement from the left-hand rule to the trap ezoid rule in numerical" }}{PARA 0 "" 0 "" {TEXT -1 67 "integration. T o implement this improvement merely requires that the" }}{PARA 0 "" 0 "" {TEXT -1 49 "guts of the preceding code be slightly uprgraded." }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "z[0]:=y0:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "printf(\"IMPROVED EULER METHOD\");" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "printf(\"x[i] Impr. Euler y Actual y er ror\"); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "for i from 0 to n do; \+ " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 " k1:= f(x[i],z[i]) : # GUTS " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 " \+ k2:= f(x[i+1],z[i]+k1*h): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 " z[i+1]:=z[i]+(k1+k2)*h/2;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 " valu:=subs(t=x[i],rhs(soln)): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 " erro:=z[i]-valu:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 " p rintf(\"%6.2f%15.9f %15.9f %15.9f\\n\"," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 " x[i],z[i],valu, erro);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}{PARA 6 "" 1 "" {TEXT -1 21 "IMPROVED EULER METHOD" }}{PARA 6 "" 1 "" {TEXT -1 44 "x[i] Impr. Euler y Actual y error" }} {PARA 6 "" 1 "" {TEXT -1 53 " 0.00 1.000000000 1.000000000 \+ 0.000000000" }}{PARA 6 "" 1 "" {TEXT -1 53 " .13 1.068033854 \+ 1.069389300 -.001355400" }}{PARA 6 "" 1 "" {TEXT -1 53 " .25 \+ 1.185011518 1.187711800 -.002700300" }}{PARA 6 "" 1 "" {TEXT -1 53 " .38 1.340723433 1.344751000 -.004027600" }}{PARA 6 "" 1 "" {TEXT -1 53 " .50 1.524413060 1.529744600 -.005 331500" }}{PARA 6 "" 1 "" {TEXT -1 53 " .63 1.724835640 1.731 442700 -.006607100" }}{PARA 6 "" 1 "" {TEXT -1 53 " .75 1.930 308825 1.938159400 -.007850600" }}{PARA 6 "" 1 "" {TEXT -1 53 " .88 2.128756535 2.137815300 -.009058800" }}{PARA 6 "" 1 "" {TEXT -1 53 " 1.00 2.307747146 2.317976300 -.01022920 0" }}{PARA 6 "" 1 "" {TEXT -1 53 " 1.13 2.454526886 2.46588620 0 -.011359300" }}{PARA 6 "" 1 "" {TEXT -1 53 " 1.25 2.55604918 2 2.568497200 -.012448000" }}{PARA 6 "" 1 "" {TEXT -1 53 " 1. 38 2.599000537 2.612494000 -.013493500" }}{PARA 6 "" 1 "" {TEXT -1 53 " 1.50 2.569823450 2.584318200 -.014494800" }} {PARA 6 "" 1 "" {TEXT -1 53 " 1.63 2.454736784 2.470188000 \+ -.015451200" }}{PARA 6 "" 1 "" {TEXT -1 53 " 1.75 2.239753924 \+ 2.256115800 -.016361900" }}{PARA 6 "" 1 "" {TEXT -1 53 " 1.88 \+ 1.910699024 1.927925600 -.017226600" }}{PARA 6 "" 1 "" {TEXT -1 53 " 2.00 1.453221580 1.471266300 -.018044700" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "P1:=plot([seq([x[i],z[i]],i= 0..n)],color=coral):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "P2:=plot([s eq([x[i],z[i]],i=0..n)],style=point," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 " symbol=circle,symbolsize=5,color=red):" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 40 "P3:=plot(rhs(soln),t=x0..x1,color=blue):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "display(\{P1,P2,P3\},scaling=constr ained,title=\"Improved Euler Method\");" }}{PARA 13 "" 1 "" {GLPLOT2D 573 379 379 {PLOTDATA 2 "6)-%'CURVESG6&737$$\"\"!F)$\"\"\"F)7$$\"3++++ +++]7!#=$\"31+++aQ.o5!#<7$$\"3++++++++DF/$\"3.+++=:,&=\"F27$$\"3++++++ +]PF/$\"3'******HVB2M\"F27$$\"3++++++++]F/$\"35+++gITC:F27$$\"3+++++++ ]iF/$\"3++++Sc$[s\"F27$$\"3++++++++vF/$\"3++++D)3.$>F27$$\"3+++++++]() F/$\"3&******\\`c(G@F27$F*$\"33+++Yru2BF27$$\"3+++++++D6F2$\"3,+++')o_ aCF27$$F.F2$\"3>+++#=\\gb#F27$$\"3*************\\P\"F2$\"3A+++P0+*f#F2 7$$\"3++++++++:F2$\"3*)******\\M#)pDF27$$\"3+++++++D;F2$\"31+++%yOZX#F 27$$\"3+++++++]F27$$\"\"#F)$\"3!*******z:A`9F2-%'COLOURG6&%$RGBG$\"*++++\"! \")F(F(-%'SYMBOLG6$%'CIRCLEG\"\"&-%&STYLEG6#%&POINTG-F$6$7UF'7$$\"39LL LL3VfV!#>$\"3M?$oPzG!=5F27$$\"3&pmm;H[D:)F[r$\"3PIo(>2.#R5F27$$\"3LLLL e0$=C\"F/$\"3J?)\\H.m(o5F27$$\"3HLLL3RBr;F/$\"3!>R>-^#H/6F27$$\"3Ymm;z jf)4#F/$\"3'>g%H#[$*\\9\"F27$$\"3=LL$e4;[\\#F/$\"3q:-lIQ8(=\"F27$$\"3p ****\\i'y]!HF/$\"3oASpA<([B\"F27$$\"3,LL$ezs$HLF/$\"3ih[Sg/C)G\"F27$$ \"3^****\\7iI_PF/$\"3Y'R![$))p]M\"F27$$\"3#pmmm@Xt=%F/$\"3$\\x-VRioS\" F27$$\"3QLLL3y_qXF/$\"3i*=J\")HBPY\"F27$$\"3i******\\1!>+&F/$\"3@30%*= 8/I:F27$$\"3')******\\Z/NaF/$\"3!*4vMCWj)f\"F27$$\"3&*******\\$fC&eF/$ \"3[\"*)yax'=m;F27$$\"3ELL$ez6:B'F/$\"3'*zUcONRGF27$$\"3slmmm(y8!zF/$\"38(\\si5HO+#F27$$\"3V++]i.tK$)F/$ \"3!H3hR18F2#F27$$\"39++](3zMu)F/$\"3KIvK%Q8o8#F27$$\"3\"pmm;H_?<*F/$ \"3&yS)eGZX,AF27$$\"3dmm;zihl&*F/$\"3->$pBWp$eAF27$$\"39LLL3#G,***F/$ \"3)3t`azimJ#F27$$\"3;LLezw5V5F2$\"3:\\[y'*GGtBF27$$\"3*)***\\PQ#\\\"3 \"F2$\"33@$R>-t)=CF27$$\"3ALL$e\"*[H7\"F2$\"3&zMnQz)yjCF27$$\"3#****** *pvxl6F2$\"3%*3H6r,)\\]#F27$$\"3z****\\_qn27F2$\"3wsq$\\ZZ'RDF27$$\"3% )***\\i&p@[7F2$\"3h'4c&QOTnDF27$$\"3#)****\\2'HKH\"F2$\"3!*[bH\"GE5f#F 27$$\"3_mmmwanL8F2$\"3[\"fgJkk_g#F27$$\"3&******\\2goP\"F2$\"3\\[&=*ew j7EF27$$\"3CLLeR<*fT\"F2$\"3k[(*3,`(=h#F27$$\"3'******\\)Hxe9F2$\"37N+ 2Q6V-EF27$$\"3Ymm\"H!o-*\\\"F2$\"3aI;\"*4;&[e#F27$$\"3))***\\7k.6a\"F2 $\"3\"[dzmk`pb#F27$$\"3emmmT9C#e\"F2$\"3fR!f\"Hkw>DF27$$\"3!****\\i!*3 `i\"F2$\"3b)=D7)yypCF27$$\"3PLLL$*zym;F2$\"3Xirwb8W5CF27$$\"3GLL$3N1#4 F27$$\"3/++DOl5;>F2$\"3#*=!*Ga_j#z\"F27$$\"3/+++q `KO>F2$\"3kE-,Qt$3s\"F27$$\"3/++v.Uac>F2$\"36Rh\"R\"G\\X;F27$$\"3-+](= 5s#y>F2$\"3xT8^f5]g:F27$Fap$\"3zw4U@hEr9F2-Ffp6&FhpF(F(Fip-F$6$F&-Ffp6 &FhpFip$\")AR!)\\F[qF(-%+AXESLABELSG6%Q!6\"F\\bl-%%FONTG6#%(DEFAULTG-% (SCALINGG6#%,CONSTRAINEDG-%&TITLEG6#Q6Improved~Euler~MethodF]bl-%%VIEW G6$;F($\"#?!\"\"Fabl" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 69 "The Runge-Kutta Method is a fourth order method. This t ime, there are" }}{PARA 0 "" 0 "" {TEXT -1 66 "four evaluations of the function along the interval. The left hand" }}{PARA 0 "" 0 "" {TEXT -1 66 "slope k1 is used to find an prediction of the midpoint. The sec ond" }}{PARA 0 "" 0 "" {TEXT -1 69 "evaluation finds the slope near th e midpoint k2 and is used to find a" }}{PARA 0 "" 0 "" {TEXT -1 70 "be tter approximation k3 near the midpoint. This is used to predict the" }}{PARA 0 "" 0 "" {TEXT -1 63 "point near the right end where the four th slope k4 is computed." }}{PARA 0 "" 0 "" {TEXT -1 67 "Finally, the \+ new point is computed by taking an average of all four" }}{PARA 0 "" 0 "" {TEXT -1 52 "slopes. This resembles Simpson's rule in quadrature. " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "printf(\"RUNGE - KUTTA METHOD\");" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "printf(\"x[i] Runge-Kutta y Actual y er ror\");" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "rk[0]:=y0:h2:=h/2:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "for i from 0 to n do; #RU NGE - KUTTA METHOD " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 " k1:= f(x[ i],rk[i]): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 " k2:= f( x[i]+h2,rk[i]+k1*h2):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 " k3:= f( x[i]+h2,rk[i]+k2*h2):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 " k4:= f( x[i+1],rk[i]+k3*h): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 " rk[i+1]:=rk[i]+(k1+2*k2+2*k3+k4)*h/6;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 " valu:=subs(t=x[i],rhs(soln)): " }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 20 " erro:=rk[i]-valu:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 " printf(\"%6.2f%15.9f %15.9f %15.9f\\n\"," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 " x[i],rk[i],valu, erro);#right side of soln" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}{PARA 6 "" 1 "" {TEXT -1 21 " RUNGE - KUTTA METHOD" }}{PARA 6 "" 1 "" {TEXT -1 44 "x[i] Runge-K utta y Actual y error" }}{PARA 6 "" 1 "" {TEXT -1 53 " 0.00 \+ 1.000000000 1.000000000 0.000000000" }}{PARA 6 "" 1 "" {TEXT -1 53 " .13 1.069389084 1.069389300 -.000000200" }}{PARA 6 "" 1 "" {TEXT -1 53 " .25 1.187711517 1.187711800 -.000 000300" }}{PARA 6 "" 1 "" {TEXT -1 53 " .38 1.344750611 1.344 751000 -.000000400" }}{PARA 6 "" 1 "" {TEXT -1 53 " .50 1.529 743996 1.529744600 -.000000600" }}{PARA 6 "" 1 "" {TEXT -1 53 " .63 1.731442140 1.731442700 -.000000600" }}{PARA 6 "" 1 "" {TEXT -1 53 " .75 1.938158786 1.938159400 -.00000060 0" }}{PARA 6 "" 1 "" {TEXT -1 53 " .88 2.137814661 2.13781530 0 -.000000600" }}{PARA 6 "" 1 "" {TEXT -1 53 " 1.00 2.31797553 6 2.317976300 -.000000800" }}{PARA 6 "" 1 "" {TEXT -1 53 " 1. 13 2.465885525 2.465886200 -.000000700" }}{PARA 6 "" 1 "" {TEXT -1 53 " 1.25 2.568496348 2.568497200 -.000000900" }} {PARA 6 "" 1 "" {TEXT -1 53 " 1.38 2.612493144 2.612494000 \+ -.000000900" }}{PARA 6 "" 1 "" {TEXT -1 53 " 1.50 2.584317334 \+ 2.584318200 -.000000900" }}{PARA 6 "" 1 "" {TEXT -1 53 " 1.63 \+ 2.470186942 2.470188000 -.000001100" }}{PARA 6 "" 1 "" {TEXT -1 53 " 1.75 2.256114719 2.256115800 -.000001100" }}{PARA 6 "" 1 "" {TEXT -1 53 " 1.88 1.927924354 1.927925600 -.000 001200" }}{PARA 6 "" 1 "" {TEXT -1 53 " 2.00 1.471265026 1.471 266300 -.000001300" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "P 1:=plot([seq([x[i],rh[i]],i=0..n)],color=maroon):" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 48 "P2:=plot([seq([x[i],rh[i]],i=0..n)],style=point," } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 " symbol=circle,symbolsize =5,color=red):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "P3:=plot(rhs(soln ),t=x0..x1,color=blue):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "display( \{P1,P2,P3\},title=\"Runge-Kutta Method\");" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7U7$$\"\"!F)$\"\"\"F) 7$$\"39LLLL3VfV!#>$\"3M?$oPzG!=5!#<7$$\"3&pmm;H[D:)F/$\"3PIo(>2.#R5F27 $$\"3LLLLe0$=C\"!#=$\"3J?)\\H.m(o5F27$$\"3HLLL3RBr;F;$\"3!>R>-^#H/6F27 $$\"3Ymm;zjf)4#F;$\"3'>g%H#[$*\\9\"F27$$\"3=LL$e4;[\\#F;$\"3q:-lIQ8(= \"F27$$\"3p****\\i'y]!HF;$\"3oASpA<([B\"F27$$\"3,LL$ezs$HLF;$\"3ih[Sg/ C)G\"F27$$\"3^****\\7iI_PF;$\"3Y'R![$))p]M\"F27$$\"3#pmmm@Xt=%F;$\"3$ \\x-VRioS\"F27$$\"3QLLL3y_qXF;$\"3i*=J\")HBPY\"F27$$\"3i******\\1!>+&F ;$\"3@30%*=8/I:F27$$\"3')******\\Z/NaF;$\"3!*4vMCWj)f\"F27$$\"3&****** *\\$fC&eF;$\"3[\"*)yax'=m;F27$$\"3ELL$ez6:B'F;$\"3'*zUcONRGF27$$\"3slmmm(y8!zF;$\"38(\\si5HO+#F27$$\"3V+ +]i.tK$)F;$\"3!H3hR18F2#F27$$\"39++](3zMu)F;$\"3KIvK%Q8o8#F27$$\"3\"pm m;H_?<*F;$\"3&yS)eGZX,AF27$$\"3dmm;zihl&*F;$\"3->$pBWp$eAF27$$\"39LLL3 #G,***F;$\"3)3t`azimJ#F27$$\"3;LLezw5V5F2$\"3:\\[y'*GGtBF27$$\"3*)*** \\PQ#\\\"3\"F2$\"33@$R>-t)=CF27$$\"3ALL$e\"*[H7\"F2$\"3&zMnQz)yjCF27$$ \"3#*******pvxl6F2$\"3%*3H6r,)\\]#F27$$\"3z****\\_qn27F2$\"3wsq$\\ZZ'R DF27$$\"3%)***\\i&p@[7F2$\"3h'4c&QOTnDF27$$\"3#)****\\2'HKH\"F2$\"3!*[ bH\"GE5f#F27$$\"3_mmmwanL8F2$\"3[\"fgJkk_g#F27$$\"3&******\\2goP\"F2$ \"3\\[&=*ewj7EF27$$\"3CLLeR<*fT\"F2$\"3k[(*3,`(=h#F27$$\"3'******\\)Hx e9F2$\"37N+2Q6V-EF27$$\"3Ymm\"H!o-*\\\"F2$\"3aI;\"*4;&[e#F27$$\"3))*** \\7k.6a\"F2$\"3\"[dzmk`pb#F27$$\"3emmmT9C#e\"F2$\"3fR!f\"Hkw>DF27$$\"3 !****\\i!*3`i\"F2$\"3b)=D7)yypCF27$$\"3PLLL$*zym;F2$\"3Xirwb8W5CF27$$ \"3GLL$3N1#4F27$$\"3/++DOl5;>F2$\"3#*=!*Ga_j#z\"F 27$$\"3/+++q`KO>F2$\"3kE-,Qt$3s\"F27$$\"3/++v.Uac>F2$\"36Rh\"R\"G\\X;F 27$$\"3-+](=5s#y>F2$\"3xT8^f5]g:F27$$\"\"#F)$\"3zw4U@hEr9F2-%'COLOURG6 &%$RGBGF(F($\"*++++\"!\")-%+AXESLABELSG6%Q!6\"F]\\l-%%FONTG6#%(DEFAULT G-%&TITLEG6#Q3Runge-Kutta~MethodF^\\l-%%VIEWG6$;F($\"#?!\"\"Fb\\l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "We tabulate the three methods at some of the x[i]'s to compare the" }} {PARA 0 "" 0 "" {TEXT -1 19 "numerical methods. " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "printf (\"x[i] Actual y Euler E-error Impr.Euler IE-erro r Runge-Kutta RK-error\");" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 " for j from 2 to n by 2 do;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "valu: =subs(t=x[j],rhs(soln)): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "er1:=v alu-y[j]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "er2:=valu-z[j]:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "er3:=valu-rk[j]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "printf(\"%6.3f %12.8f %12.8f %12.8f %12.8f %12.8f \+ %12.8f %12.8f\\n\", " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 " x[j],valu ,y[j],er1,z[j],er2,rk[j],er3);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od :" }}{PARA 6 "" 1 "" {TEXT 273 94 "x[i] Actual y Euler E -error Impr.Euler IE-error Runge-Kutta RK-error" }}{PARA 6 "" 1 "" {TEXT 274 97 " .250 1.18771180 1.13606771 .05164410 \+ 1.18501152 .00270030 1.18771152 .00000030" }}{PARA 6 "" 1 " " {TEXT 275 97 " .500 1.52974460 1.44372663 .08601800 1.5244 1306 .00533150 1.52974400 .00000060" }}{PARA 6 "" 1 "" {TEXT 276 97 " .750 1.93815940 1.83880847 .09935090 1.93030883 \+ .00785060 1.93815879 .00000060" }}{PARA 6 "" 1 "" {TEXT 277 97 " \+ 1.000 2.31797630 2.22960387 .08837240 2.30774715 .01022920 2.31797554 .00000080" }}{PARA 6 "" 1 "" {TEXT 278 97 " 1.250 2 .56849720 2.51827940 .05021780 2.55604918 .01244800 2.5684 9635 .00000090" }}{PARA 6 "" 1 "" {TEXT 279 97 " 1.500 2.58431820 2.60196035 -.01764220 2.56982345 .01449480 2.58431733 . 00000090" }}{PARA 6 "" 1 "" {TEXT 280 97 " 1.750 2.25611580 2.3735 7241 -.11745660 2.23975392 .01636190 2.25611472 .00000110 " }}{PARA 6 "" 1 "" {TEXT 281 97 " 2.000 1.47126630 1.72250568 - .25123940 1.45322158 .01804470 1.47126503 .00000130" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "Fi nally, in order to study the effect of decreasing h, we solve a new" } }{PARA 0 "" 0 "" {TEXT -1 61 "equation by the Euler method several tim es for different h's." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "restart:wi th(DEtools):with(plots): #clear the memory" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "f:= (x,y)-> cos(3*x)-2*y; #enter slope fu nction" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "x0:=0.0; y0:= -2/13; x1:= 2.0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "eqn:= diff(w(t),t) = f(t,w (t)); #enter DE for solver " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "soln:=dsolve(\{eqn, w(x0)=y0\}); #symbolic soln DE" }} {PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been r edefined\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%\"xG%\"yG6\"6 $%)operatorG%&arrowGF),&-%$cosG6#,$9$\"\"$\"\"\"*&\"\"#F49%F4!\"\"F)F) F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G$\"\"!F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y0G#!\"#\"#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#x1G$\"#?!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/-%%diffG6$- %\"wG6#%\"tGF,,&-%$cosG6#,$F,\"\"$\"\"\"*&\"\"#F3F)F3!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%solnG/-%\"wG6#%\"tG,(-%$cosG6#,$F)\"\"$# \"\"#\"#8*&#F/F2\"\"\"-%$sinGF-F5F5*&#\"\"%F2F5-%$expG6#,$F)!\"#F5!\" \"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "printf(\"Comparison for different step sizes in EULER METHOD\\n\");" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "printf(\" n \+ h x=%5.3f %7.3f %7.3f %7.3f %7.3f %7.3f %7.3f %7.3f\\n\", " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "seq(x0+jj*(x1-x0)/8.0,jj=1..8)) ;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "for ll from 3 to 10 do:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "nn:=2^ll:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "hh:=(x1-x0)/nn:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 " y[0]:=y0:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 " for m from 0 to nn do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 " x[m]:=x0+m*hh:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " k:=f(x[m],y[m]):" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 22 " y[m+1]:=y[m]+k*hh;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 " od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "lis[ll]: =[seq([x[j],y[j]],j=0..nn)]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "jjs kip:=2^(ll-3):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "printf(\"%6d %9.5 f%9.5f%9.5f%9.5f%9.5f%9.5f%9.5f%9.5f%9.5f\\n\", nn,hh,seq(y[" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "jj*2^(ll-3) ],jj=1..8));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "printf(\" D/h = %9.5f%9.5f%9.5f%9.5f%9 .5f%9.5f%9.5f%9.5f\\n\"," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "seq((y[ jj*jjskip]-evalf(subs(t=x[jj*jjskip],rhs(soln))))/hh,jj=1..8));" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}{PARA 6 "" 1 "" {TEXT 282 51 "Comparison for different step sizes in EULER METHOD" }}{PARA 6 "" 1 " " {TEXT 283 85 " n h x= .250 .500 .750 1.000 1 .250 1.500 1.750 2.000" }}{PARA 6 "" 1 "" {TEXT 284 88 " \+ 8 .25000 .17308 .26946 .15241 -.08084 -.28792 -.34910 -.2 2725 .01440" }}{PARA 6 "" 1 "" {TEXT 285 89 " D/h = .35 933 .56632 .55263 .32218 -.01808 -.30306 -.39409 -.25282" }} {PARA 6 "" 1 "" {TEXT 286 88 " 16 .12500 .12352 .19198 .07 718 -.12417 -.28427 -.30610 -.17171 .05031" }}{PARA 6 "" 1 "" {TEXT 287 89 " D/h = .32225 .51276 .50335 .29771 - .00697 -.26214 -.34386 -.21833" }}{PARA 6 "" 1 "" {TEXT 288 88 " \+ 32 .06250 .10246 .15856 .04445 -.14342 -.28357 -.28870 - .14896 .06483" }}{PARA 6 "" 1 "" {TEXT 289 89 " D/h = . 30743 .49086 .48300 .28742 -.00280 -.24586 -.32370 -.20437" }}{PARA 6 "" 1 "" {TEXT 290 88 " 64 .03125 .09264 .14291 . 02906 -.15255 -.28343 -.28079 -.13856 .07141" }}{PARA 6 "" 1 "" {TEXT 291 89 " D/h = .30074 .48088 .47371 .28268 - .00098 -.23854 -.31460 -.19806" }}{PARA 6 "" 1 "" {TEXT 292 88 " \+ 128 .01563 .08789 .13532 .02159 -.15700 -.28340 -.27701 - .13357 .07455" }}{PARA 6 "" 1 "" {TEXT 293 89 " D/h = . 29754 .47611 .46926 .28041 -.00013 -.23506 -.31027 -.19505" }}{PARA 6 "" 1 "" {TEXT 294 88 " 256 .00781 .08556 .13158 . 01791 -.15920 -.28339 -.27516 -.13113 .07609" }}{PARA 6 "" 1 "" {TEXT 295 89 " D/h = .29599 .47378 .46708 .27930 \+ .00028 -.23336 -.30816 -.19358" }}{PARA 6 "" 1 "" {TEXT 296 88 " \+ 512 .00391 .08440 .12973 .01608 -.16029 -.28339 -.27424 - .12992 .07685" }}{PARA 6 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Choose a function f(x,y) and initial values (x0,y0) (not from t he" }}{PARA 0 "" 0 "" {TEXT -1 66 "book or this assignment!) such that the solution of the IVP has an" }}{PARA 0 "" 0 "" {TEXT -1 65 "intere sting but not too complicated a trajectory. Find the actual" }}{PARA 0 "" 0 "" {TEXT -1 70 "solution by hand or using dsolve. Fix a medium \+ small h. Solve your IVP" }}{PARA 0 "" 0 "" {TEXT -1 65 "using Eulers M ethod. Tabulate your solution for several values of" }}{PARA 0 "" 0 " " {TEXT -1 70 "x[i]. Graph the numerical solution and the actual solut ion on the same" }}{PARA 0 "" 0 "" {TEXT -1 6 "graph." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "2. For the same f, IC \+ and h as in Prob. 1, solve the IVP using the" }}{PARA 0 "" 0 "" {TEXT -1 69 "improved Euler's Method. Tabulate your solution for several val ues of" }}{PARA 0 "" 0 "" {TEXT -1 70 "x[i]. Graph the numerical solut ion and the actual solution on the same" }}{PARA 0 "" 0 "" {TEXT -1 6 "graph." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "3. For the same f, IC and h as in Prob. 1, solve the IVP using the " }}{PARA 0 "" 0 "" {TEXT -1 70 "Runge-Kutta Method. Tabulate your sol ution for several values of x[i]." }}{PARA 0 "" 0 "" {TEXT -1 64 "Grap h the numerical solution and the actual solution on the same" }}{PARA 0 "" 0 "" {TEXT -1 6 "graph." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "4. Compare the three methods using your tables \+ and graphs. Compare the" }}{PARA 0 "" 0 "" {TEXT -1 56 "amount of comp utation versus the accuracy of the answer." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "5. Choose another initial value problem (not from the book or this" }}{PARA 0 "" 0 "" {TEXT -1 58 "as signment!) For one of the two methods, Improved Euler or" }}{PARA 0 " " 0 "" {TEXT -1 64 "Runge-Kutta, compute the solution of the IVP with \+ at least eight" }}{PARA 0 "" 0 "" {TEXT -1 68 "different h's. Tabulat e the approximate solution at eight different" }}{PARA 0 "" 0 "" {TEXT -1 69 "x[i]'s. Compute the errors and the ratio error/h^2 for im proved Euler" }}{PARA 0 "" 0 "" {TEXT -1 70 "or error/h^4 for Runge-Ku tta. discuss the evidence that Improved Euler" }}{PARA 0 "" 0 "" {TEXT -1 65 "is second order accurate or Runge-Kutta is fourth order a ccurate." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "6. Now consider an unstable problem with a spout-type instability \+ (see" }}{PARA 0 "" 0 "" {TEXT -1 65 "page 134.) For example, you may a nalyze f:= (x,y)-> cos(3*x)+2*y;" }}{PARA 0 "" 0 "" {TEXT -1 63 "x0:=0 .0; y0:= -2/13; x1:= 2.0. Draw the graphs and tabulate the" }}{PARA 0 "" 0 "" {TEXT -1 69 "numerical approximations. Show that the numerica l methods don't do a" }}{PARA 0 "" 0 "" {TEXT -1 58 "very good job of \+ approximating the solution in this case. " }}}}{MARK "18 1 0" 41 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }