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Equations - Convergence" }{TEXT 284 0 "" }} {PARA 256 "" 0 "" {TEXT 285 0 "" }}{PARA 257 "" 0 "" {TEXT 286 25 "Nat han Collier, Autar Kaw" }{TEXT 286 0 "" }}{PARA 257 "" 0 "" {TEXT 286 27 "University of South Florida" }{TEXT 286 0 "" }}{PARA 257 "" 0 "" {TEXT 286 24 "United States of America" }{TEXT 286 0 "" }}{PARA 257 "" 0 "" {TEXT 286 15 "kaw@eng.usf.edu" }{TEXT 286 0 "" }}}{SECT 0 {PARA 258 "" 0 "" {TEXT 287 12 "Introduction" }{TEXT 287 0 "" }}{PARA 259 "" 0 "" {TEXT 288 136 "This worksheet demonstrates the use of Maple to i llustrate the convergence of Euler's method of solving ordinary differ ential equations." }{TEXT 288 0 "" }}{PARA 259 "" 0 "" {TEXT 288 0 "" }}{PARA 259 "" 0 "" {TEXT 288 197 "Euler's method of solving ordinary \+ differential equations uses the derivative and value of the function a t the initial condition to project the location and value of the next \+ point on the solution." }{TEXT 288 0 "" }}}{SECT 0 {PARA 258 "" 0 "" {TEXT 287 14 "Initialization" }{TEXT 287 0 "" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 289 8 "restart;" }{MPLTEXT 1 289 0 "" }{MPLTEXT 1 289 13 "\nwith(plots):" }{MPLTEXT 1 289 0 "" }}{PARA 261 "" 1 "" {TEXT 290 49 "Warning, the name changecoords has been redefined" }{TEXT 290 0 "" }}}}{SECT 0 {PARA 262 "" 0 "" {TEXT 291 16 "Section 1: Input" } {TEXT 291 0 "" }}{PARA 263 "" 0 "" {TEXT 292 258 "The following simula tion illustrates the convergence of Euler's method of solving ordinary differential equations (ODEs). This section is the only section where the user interacts with the program. The user enters ordinary differ ential equation of the form " }{XPPEDIT 2 0 "diff(y(x), x) = f(x, y)" "6#/-I%diffGI*protectedGF&6$-I\"yG6\"6#I\"xGF*F,-I\"fGF*6$F,F)" } {TEXT 293 1 "," }{TEXT 292 42 " the initial conditions, and the value \+ of " }{TEXT 293 1 "x" }{TEXT 292 211 " at which the solution is desire d. By entering this data, the program will calculate the exact (Maple \+ numerical value if it is not exact) value of the solution, followed by the results using Euler's method with " }{TEXT 293 16 "1, 2, 4, 8 ... n" }{TEXT 292 265 " steps. The program will also display the true err or, the absolute relative percentage true error, the approximate error , the absolute relative aprroximate percentage error, and the least nu mber of significant digits correct all as a function of number of segm ents." }{TEXT 292 0 "" }}{EXCHG {PARA 263 "" 0 "" {TEXT 292 43 "Ordina ry differential equation of the form " }{XPPEDIT 2 0 "diff(y(x), x) = \+ f(x, y)" "6#/-I%diffGI*protectedGF&6$-I\"yG6\"6#I\"xGF*F,-I\"fGF*6$F,F )" }{TEXT 292 0 "" }}}{EXCHG {PARA 264 "> " 0 "" {MPLTEXT 1 294 26 "f: =(x,y)->y(x)*x-1.2*y(x);" }{MPLTEXT 1 294 0 "" }}{PARA 265 "" 1 "" {XPPMATH 20 "6#>I\"fG6\"f*6$I\"xGF%I\"yGF%F%6$I)operatorGF%I&arrowGF%F %,&*&-9%6#9$\"\"\"F2F3F3*&$\"#7!\"\"F3F/F3F7F%F%F%" }{TEXT 20 0 "" }}} {EXCHG {PARA 263 "" 0 "" {TEXT 292 23 "Boundary condition for " } {TEXT 293 1 "x" }{TEXT 292 0 "" }}}{EXCHG {PARA 264 "> " 0 "" {MPLTEXT 1 294 8 "x0:=0.0;" }{MPLTEXT 1 294 0 "" }}{PARA 265 "" 1 "" {XPPMATH 20 "6#>I#x0G6\"$\"\"!F'" }{TEXT 20 0 "" }}}{EXCHG {PARA 263 " " 0 "" {TEXT 292 8 "Boundary" }{TEXT 292 15 " condition for " }{TEXT 293 1 "y" }{TEXT 292 0 "" }}}{EXCHG {PARA 264 "> " 0 "" {MPLTEXT 1 294 8 "y0:=1.0;" }{MPLTEXT 1 294 0 "" }}{PARA 265 "" 1 "" {XPPMATH 20 "6#>I#y0G6\"$\"#5!\"\"" }{TEXT 20 0 "" }}}{EXCHG {PARA 263 "" 0 "" {TEXT 292 9 "Value of " }{TEXT 293 1 "x" }{TEXT 292 33 " at which the \+ solution is desired" }{TEXT 292 0 "" }}}{EXCHG {PARA 264 "> " 0 "" {MPLTEXT 1 294 8 "xf:=2.0;" }{MPLTEXT 1 294 0 "" }}{PARA 265 "" 1 "" {XPPMATH 20 "6#>I#xfG6\"$\"#?!\"\"" }{TEXT 20 0 "" }}}{EXCHG {PARA 263 "" 0 "" {TEXT 292 25 "Maximum number of steps, " }{TEXT 293 1 "n" }{TEXT 292 19 ". Valid values are " }{TEXT 293 18 "1, 2, 4, 8 ... 128" }{TEXT 292 1 "." }{TEXT 292 0 "" }}}{EXCHG {PARA 264 "> " 0 "" {MPLTEXT 1 294 7 "n:=128;" }{MPLTEXT 1 294 0 "" }}{PARA 265 "" 1 "" {XPPMATH 20 "6#>I\"nG6\"\"$G\"" }{TEXT 20 0 "" }}}{EXCHG {PARA 266 "" 0 "" {TEXT 295 134 "This is the end of the user's section. All informa tion must be entered before proceeding to the next section. Re-execut e the program." }{TEXT 295 0 "" }}}}{SECT 0 {PARA 262 "" 0 "" {TEXT 291 20 "Section 2: Procedure" }{TEXT 291 0 "" }}{PARA 263 "" 0 "" {TEXT 292 93 "The following procedure estimates the solution of ordina ry differential equations at a point " }{TEXT 293 2 "xf" }{TEXT 292 1 "." }{TEXT 292 0 "" }}{PARA 263 "" 0 "" {TEXT 293 1 "n" }{TEXT 292 18 " = number of steps" }{TEXT 292 0 "" }}{PARA 263 "" 0 "" {TEXT 293 2 " x0" }{TEXT 292 27 " = boundary condition for x" }{TEXT 292 0 "" }} {PARA 263 "" 0 "" {TEXT 293 2 "y0" }{TEXT 292 27 " = boundary conditio n for y" }{TEXT 292 0 "" }}{PARA 263 "" 0 "" {TEXT 293 2 "xf" }{TEXT 292 37 " = value at which solution is desired" }{TEXT 292 0 "" }} {PARA 263 "" 0 "" {TEXT 293 1 "f" }{TEXT 292 37 " = differential equat ion in the form " }{XPPEDIT 2 0 "diff(y(x), x)" "6#-I%diffGI*protected GF%6$-I\"yG6\"6#I\"xGF)F+" }{TEXT 292 0 "" }}{EXCHG {PARA 267 "> " 0 " " {MPLTEXT 1 296 25 "Euler:=proc(n,x0,y0,xf,f)" }{MPLTEXT 1 296 0 "" } {MPLTEXT 1 296 13 "\nlocal Y,h,i:" }{MPLTEXT 1 296 0 "" }{MPLTEXT 1 296 14 "\nh:=(xf-x0)/n:" }{MPLTEXT 1 296 0 "" }{MPLTEXT 1 296 7 "\nY:= y0:" }{MPLTEXT 1 296 0 "" }{MPLTEXT 1 296 28 "\nfor i from 0 by 1 to n -1 do" }{MPLTEXT 1 296 0 "" }{MPLTEXT 1 296 20 "\nY:=Y+f(x0+i*h,Y)*h:" }{MPLTEXT 1 296 0 "" }{MPLTEXT 1 296 8 "\nend do:" }{MPLTEXT 1 296 0 "" }{MPLTEXT 1 296 12 "\nreturn (Y):" }{MPLTEXT 1 296 0 "" }{MPLTEXT 1 296 10 "\nend proc:" }{MPLTEXT 1 296 0 "" }}}}{SECT 0 {PARA 262 "" 0 "" {TEXT 291 22 "Section 3: Calculation" }{TEXT 291 0 "" }}{EXCHG {PARA 268 "" 0 "" {TEXT 297 33 "The exact value of the ODE (EV) :" } {TEXT 297 0 "" }}}{EXCHG {PARA 267 "> " 0 "" {MPLTEXT 1 296 25 "ODE:=d iff(y(x),x)=f(x,y);" }{MPLTEXT 1 296 0 "" }}{PARA 265 "" 1 "" {XPPMATH 20 "6#>I$ODEG6\"/-I%diffGI*protectedGF)6$-I\"yGF%6#I\"xGF%F., &*&F+\"\"\"F.F1F1F+$!#7!\"\"" }{TEXT 20 0 "" }}}{EXCHG {PARA 269 "> " 0 "" {MPLTEXT 1 298 44 "soln:=(dsolve(\{ODE,y(x0)=y0\})):assign(soln): " }{MPLTEXT 1 298 0 "" }{MPLTEXT 1 298 6 "\ny(x);" }{MPLTEXT 1 298 0 " " }}{PARA 265 "" 1 "" {XPPMATH 20 "6#-I$expG6$I*protectedGF&I(_syslibG 6\"6#,$*&I\"xGF(\"\"\",&F,\"\"&!#7F-F-#F-\"#5" }{TEXT 20 0 "" }}} {EXCHG {PARA 269 "> " 0 "" {MPLTEXT 1 298 27 "EV:=evalf(subs(x=xf,y(x) ));" }{MPLTEXT 1 298 0 "" }}{PARA 265 "" 1 "" {XPPMATH 20 "6#>I#EVG6\" $\"+g/?.n!#5" }{TEXT 20 0 "" }}}{EXCHG {PARA 266 "" 0 "" {TEXT 295 39 "This loop here calculates the following" }{TEXT 295 0 "" }}{PARA 266 "" 0 "" {TEXT 295 87 "AV = approximate value of the ODE using Euler's \+ Method by calling the procedure \"Euler\"" }{TEXT 295 0 "" }}{PARA 266 "" 0 "" {TEXT 295 15 "Et = true error" }{TEXT 295 0 "" }{TEXT 295 38 "\nabs_et = absolute relative true error" }{TEXT 295 0 "" }{TEXT 295 23 "\nEa = approximate error" }{TEXT 295 0 "" }}{PARA 266 "" 0 "" {TEXT 295 40 "ea = absolute relative approximate error" }{TEXT 295 0 " " }}{PARA 266 "" 0 "" {TEXT 295 68 "sig = least number of significant \+ digits correct in an approximation" }{TEXT 295 0 "" }}}{EXCHG {PARA 267 "> " 0 "" {MPLTEXT 1 296 26 "nth:=floor(log(n)/log(2)):" } {MPLTEXT 1 296 0 "" }{MPLTEXT 1 296 28 "\nfor i from 0 by 1 to nth do" }{MPLTEXT 1 296 0 "" }{MPLTEXT 1 296 11 "\nN[i]:=2^i:" }{MPLTEXT 1 296 0 "" }{MPLTEXT 1 296 20 "\nH[i]:=(xf-x0)/N[i]:" }{MPLTEXT 1 296 0 "" }{MPLTEXT 1 296 30 "\nAV[i]:=Euler(2^i,x0,y0,xf,f):" }{MPLTEXT 1 296 0 "" }{MPLTEXT 1 296 17 "\nEt[i]:=EV-AV[i]:" }{MPLTEXT 1 296 0 "" }{MPLTEXT 1 296 32 "\nabs_et[i]:=abs(Et[i]/EV)*100.0:" }{MPLTEXT 1 296 0 "" }{MPLTEXT 1 296 14 "\nif (i>0) then" }{MPLTEXT 1 296 0 "" } {MPLTEXT 1 296 22 "\nEa[i]:=AV[i]-AV[i-1]:" }{MPLTEXT 1 296 0 "" } {MPLTEXT 1 296 31 "\nea[i]:=abs(Ea[i]/AV[i])*100.0:" }{MPLTEXT 1 296 0 "" }{MPLTEXT 1 296 37 "\nsig[i]:=floor((2-log10(ea[i]/0.5))):" } {MPLTEXT 1 296 0 "" }{MPLTEXT 1 296 18 "\nif sig[i]<0 then " } {MPLTEXT 1 296 0 "" }{MPLTEXT 1 296 11 "\nsig[i]:=0:" }{MPLTEXT 1 296 0 "" }{MPLTEXT 1 296 8 "\nend if:" }{MPLTEXT 1 296 0 "" }{MPLTEXT 1 296 8 "\nend if:" }{MPLTEXT 1 296 0 "" }{MPLTEXT 1 296 8 "\nend do:" } {MPLTEXT 1 296 0 "" }}}}{SECT 0 {PARA 262 "" 0 "" {TEXT 291 22 "Sectio n 4: Spreadsheet" }{TEXT 291 0 "" }}{EXCHG {PARA 266 "" 0 "" {TEXT 295 171 "This table shows the approximate value, true error, absolute \+ relative true error, approximate error and relative approximate error \+ as a function of the number of segments." }{TEXT 295 0 "" }}}{EXCHG {PARA 270 "> " 0 "" {MPLTEXT 1 299 61 "with( Spread ):EvaluateSpreadsh eet(Euler_Method_Convergence):" }{MPLTEXT 1 299 0 "" }}}{PARA 256 "" 0 "" {TEXT 285 0 "" }}{EXCHG {PARA 271 "" 0 "" {SPREADSHEET {NAME "Eule r_Method_Convergence" } {ROWHEIGHTS 1 57 2 89 3 89 4 89 5 89 6 89 7 89 8 89 9 39 10 39 11 39 12 39 13 39 14 39 15 39 } {COLWIDTHS 1 180 2 180 3 180 4 180 5 180 6 220 7 180 8 270 9 180 10 245 11 180 } {SSOPTS {CELLOPTS 2 10 4 2 1 255 255 255 }1 }939 485 485 {CELL 1 1 {CELLOPTS 0 -1 -1 1 0 255 255 255 }{R5MATHOBJ "The*number*of*segments" 20 "6#**I$TheG6\"\"\"\"I'numberGF%F&I#ofGF%F&I)segmentsGF%F&" 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