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e89" -1 315 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{PSTYLE "_pstyle 76" -1 275 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "_cstyle90" -1 316 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{PSTYLE "_pstyle77" -1 276 1 {CSTYLE " " -1 -1 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "_cstyle91" -1 317 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{PSTYLE "_pstyle78" -1 277 1 {CSTYLE "" -1 -1 "Times" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 }{PSTYLE "_ pstyle79" -1 278 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 }{CSTYLE "_cstyle92" -1 318 "Times" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }{PSTYLE "_pstyle80" -1 279 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 295 89 "The Runge-Kutta 2nd Ord er Method of Solving Ordinary Differential Equations - Convergence" } {TEXT 295 0 "" }}{PARA 257 "" 0 "" {TEXT 296 25 "Nathan Collier, Autar Kaw" }{TEXT 296 0 "" }}{PARA 257 "" 0 "" {TEXT 296 27 "University of \+ South Florida" }{TEXT 296 0 "" }}{PARA 257 "" 0 "" {TEXT 296 24 "Unite d States of America" }{TEXT 296 0 "" }}{PARA 257 "" 0 "" {TEXT 296 15 "kaw@eng.usf.edu" }{TEXT 296 0 "" }}}{SECT 0 {PARA 258 "" 0 "" {TEXT 297 12 "Introduction" }{TEXT 297 0 "" }}{PARA 259 "" 0 "" {TEXT 298 368 "This worksheet demonstrates the use of Maple to illustrate the co nvergence of the 2nd order Runge-Kutta method of solving ordinary diff erential equations. Runge-Kutta's 2nd order method of solving ordinary differential equations uses the derivative and value of the function \+ at the initial condition to project the location and value of the next point on the solution." }{TEXT 298 0 "" }}}{SECT 0 {PARA 260 "" 0 "" {TEXT 299 14 "Initialization" }{TEXT 299 0 "" }}{EXCHG {PARA 261 "> " 0 "" {MPLTEXT 1 300 8 "restart;" }{MPLTEXT 1 300 0 "" }{MPLTEXT 1 300 13 "\nwith(plots):" }{MPLTEXT 1 300 0 "" }}{PARA 262 "" 1 "" {TEXT 301 49 "Warning, the name changecoords has been redefined" }{TEXT 301 0 "" }}}}{SECT 0 {PARA 258 "" 0 "" {TEXT 297 16 "Section 1: Input" } {TEXT 297 0 "" }}{PARA 263 "" 0 "" {TEXT 302 276 "The following simula tion illustrates the convergence of the 2nd order Runge-Kutta method o f solving ordinary differential equations (ODEs). This section is the \+ only section where the user interacts with the program. The user ente rs ordinary 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 303 1 "," }{TEXT 302 42 " the initial conditions, a nd the value of " }{TEXT 303 1 "x" }{TEXT 302 229 " at which the solut ion is desired. 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 the 2nd order Runge-Kutta method with \+ " }{TEXT 303 16 "1, 2, 4, 8 ... n" }{TEXT 302 265 " steps. The program will also display the true error, the absolute relative percentage tr ue error, the approximate error, the absolute relative aprroximate per centage error, and the least number of significant digits correct all \+ as a function of number of segments." }{TEXT 302 0 "" }}{EXCHG {PARA 263 "" 0 "" {TEXT 302 43 "Ordinary 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 302 0 "" }}}{EXCHG {PARA 264 "> " 0 "" {MPLTEXT 1 304 28 "f:=(x,y)->y(x)*x^2-1.2*y(x);" } {MPLTEXT 1 304 0 "" }}{PARA 265 "" 1 "" {XPPMATH 20 "%$...G" }{TEXT 305 0 "" }}}{EXCHG {PARA 263 "" 0 "" {TEXT 302 23 "Boundary condition \+ for " }{TEXT 303 1 "x" }{TEXT 302 0 "" }}}{EXCHG {PARA 264 "> " 0 "" {MPLTEXT 1 304 8 "x0:=0.0:" }{MPLTEXT 1 304 0 "" }}}{EXCHG {PARA 263 " " 0 "" {TEXT 302 8 "Boundary" }{TEXT 302 15 " condition for " }{TEXT 303 1 "y" }{TEXT 302 0 "" }}}{EXCHG {PARA 264 "> " 0 "" {MPLTEXT 1 304 8 "y0:=1.0:" }{MPLTEXT 1 304 0 "" }}}{EXCHG {PARA 263 "" 0 "" {TEXT 302 9 "Value of " }{TEXT 303 1 "x" }{TEXT 302 33 " at which the \+ solution is desired" }{TEXT 302 0 "" }}}{EXCHG {PARA 264 "> " 0 "" {MPLTEXT 1 304 8 "xf:=2.0:" }{MPLTEXT 1 304 0 "" }}}{EXCHG {PARA 263 " " 0 "" {TEXT 302 19 "Select a value for " }{TEXT 303 2 "a2" }{TEXT 302 9 " between " }{TEXT 303 1 "0" }{TEXT 302 5 " and " }{TEXT 303 1 " 1" }{TEXT 302 59 ". Several values correspond to methods with specific names." }{TEXT 302 0 "" }{TEXT 302 21 "\n - Heun Method: " } {TEXT 303 8 "a2 = 1/2" }{TEXT 302 0 "" }}{PARA 263 "" 0 "" {TEXT 302 24 " - Midpoint Method: " }{TEXT 303 6 "a2 = 1" }{TEXT 302 0 "" }} {PARA 263 "" 0 "" {TEXT 302 25 " - Ralston's Method: " }{TEXT 303 8 "a2 = 2/3" }{TEXT 302 0 "" }}}{EXCHG {PARA 266 "> " 0 "" {MPLTEXT 1 306 8 "a2:=0.5:" }{MPLTEXT 1 306 0 "" }}}{EXCHG {PARA 263 "" 0 "" {TEXT 302 25 "Maximum number of steps, " }{TEXT 303 1 "n" }{TEXT 302 19 ". Valid values are " }{TEXT 303 18 "1, 2, 4, 8 ... 128" }{TEXT 302 1 "." }{TEXT 302 0 "" }}}{EXCHG {PARA 264 "> " 0 "" {MPLTEXT 1 304 7 "n:=128:" }{MPLTEXT 1 304 0 "" }}}{EXCHG {PARA 267 "" 0 "" {TEXT 307 134 "This is the end of the user's section. All information \+ must be entered before proceeding to the next section. Re-execute the program." }{TEXT 307 0 "" }}}}{SECT 0 {PARA 258 "" 0 "" {TEXT 297 20 "Section 2: Procedure" }{TEXT 297 0 "" }}{PARA 263 "" 0 "" {TEXT 302 93 "The following procedure estimates the solution of ordinary differe ntial equations at a point " }{TEXT 303 2 "xf" }{TEXT 302 1 "." } {TEXT 302 0 "" }}{PARA 263 "" 0 "" {TEXT 303 1 "n" }{TEXT 302 18 " = n umber of steps" }{TEXT 302 0 "" }}{PARA 263 "" 0 "" {TEXT 303 2 "x0" } {TEXT 302 3 " = " }{TEXT 302 23 "boundary condition for " }{TEXT 303 1 "x" }{TEXT 302 0 "" }}{PARA 263 "" 0 "" {TEXT 303 2 "y0" }{TEXT 302 3 " = " }{TEXT 302 24 "boundary condition for y" }{TEXT 302 0 "" }} {PARA 263 "" 0 "" {TEXT 303 2 "xf" }{TEXT 302 37 " = value at which so lution is desired" }{TEXT 302 0 "" }}{PARA 263 "" 0 "" {TEXT 303 1 "f" }{TEXT 302 37 " = differential equation in 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$F,F)" }{TEXT 302 0 "" }}{PARA 263 "" 0 "" {TEXT 303 2 "a2 " }{TEXT 302 26 " = input defined by method" }{TEXT 302 0 "" }}{EXCHG {PARA 268 "> " 0 "" {MPLTEXT 1 308 28 "RK2nd:=proc(n,x0,y0,xf,f,a2)" } {MPLTEXT 1 308 0 "" }{MPLTEXT 1 308 30 "\n local Y,h,i,a1,p1,q11,k1,k2 :" }{MPLTEXT 1 308 0 "" }{MPLTEXT 1 308 16 "\n h:=(xf-x0)/n:" } {MPLTEXT 1 308 0 "" }{MPLTEXT 1 308 9 "\n Y:=y0:" }{MPLTEXT 1 308 0 " " }{MPLTEXT 1 308 12 "\n a1:=1-a2:" }{MPLTEXT 1 308 0 "" }{MPLTEXT 1 308 16 "\n p1:=1/(2*a2):" }{MPLTEXT 1 308 0 "" }{MPLTEXT 1 308 11 "\n q11:=p1:" }{MPLTEXT 1 308 0 "" }{MPLTEXT 1 308 30 "\n for i from 0 \+ by 1 to n-1 do" }{MPLTEXT 1 308 0 "" }{MPLTEXT 1 308 21 "\n k1:=f(x 0+i*h,Y):" }{MPLTEXT 1 308 0 "" }{MPLTEXT 1 308 35 "\n k2:=f(x0+i*h +p1*h,Y+q11*k1*h):" }{MPLTEXT 1 308 0 "" }{MPLTEXT 1 308 26 "\n Y:= Y+(a1*k1+a2*k2)*h:" }{MPLTEXT 1 308 0 "" }{MPLTEXT 1 308 10 "\n end d o:" }{MPLTEXT 1 308 0 "" }{MPLTEXT 1 308 13 "\n return (Y):" } {MPLTEXT 1 308 0 "" }{MPLTEXT 1 308 10 "\nend proc:" }{MPLTEXT 1 308 0 "" }}}}{SECT 0 {PARA 260 "" 0 "" {TEXT 299 22 "Section 3: Calculatio n" }{TEXT 299 0 "" }}{EXCHG {PARA 269 "" 0 "" {TEXT 309 33 "The exact \+ value of the ODE (EV) :" }{TEXT 309 0 "" }}}{EXCHG {PARA 268 "> " 0 "" {MPLTEXT 1 308 25 "ODE:=diff(y(x),x)=f(x,y);" }{MPLTEXT 1 308 0 "" }} {PARA 265 "" 1 "" {XPPMATH 20 "6#>%$ODEG/-%%diffG6$-%\"yG6#%\"xGF,,&*& F)\"\"\")F,\"\"#F/F/*&$\"#7!\"\"F/F)F/F5" }{TEXT 305 0 "" }}}{EXCHG {PARA 266 "> " 0 "" {MPLTEXT 1 306 44 "soln:=(dsolve(\{ODE,y(x0)=y0\}) ):assign(soln):" }{MPLTEXT 1 306 0 "" }{MPLTEXT 1 306 6 "\ny(x);" } {MPLTEXT 1 306 0 "" }}{PARA 265 "" 1 "" {XPPMATH 20 "6#-%$expG6#,$*(\" #:!\"\"%\"xG\"\"\",&*&\"\"&F+)F*\"\"#F+F+\"#=F)F+F+" }{TEXT 305 0 "" } }}{EXCHG {PARA 266 "> " 0 "" {MPLTEXT 1 306 27 "EV:=evalf(subs(x=xf,y( x)));" }{MPLTEXT 1 306 0 "" }}{PARA 265 "" 1 "" {XPPMATH 20 "6#>%#EVG$ \"+s^g08!\"*" }{TEXT 305 0 "" }}}{EXCHG {PARA 267 "" 0 "" {TEXT 307 39 "This loop here calculates the following" }{TEXT 307 0 "" }}{PARA 267 "" 0 "" {TEXT 307 101 "AV = approximate value of the ODE using 2nd order Runge-Kutta method by calling the procedure \"RK2nd\"" }{TEXT 307 0 "" }}{PARA 267 "" 0 "" {TEXT 307 15 "Et = true error" }{TEXT 307 0 "" }{TEXT 307 38 "\nabs_et = absolute relative true error" } {TEXT 307 0 "" }{TEXT 307 23 "\nEa = approximate error" }{TEXT 307 0 " " }}{PARA 267 "" 0 "" {TEXT 307 40 "ea = absolute relative approximate error" }{TEXT 307 0 "" }}{PARA 267 "" 0 "" {TEXT 307 68 "sig = least \+ number of significant digits correct in an approximation" }{TEXT 307 0 "" }}}{EXCHG {PARA 268 "> " 0 "" {MPLTEXT 1 308 26 "nth:=floor(log(n )/log(2)):" }{MPLTEXT 1 308 0 "" }{MPLTEXT 1 308 28 "\nfor i from 0 by 1 to nth do" }{MPLTEXT 1 308 0 "" }{MPLTEXT 1 308 11 "\nN[i]:=2^i:" } {MPLTEXT 1 308 0 "" }{MPLTEXT 1 308 20 "\nH[i]:=(xf-x0)/N[i]:" } {MPLTEXT 1 308 0 "" }{MPLTEXT 1 308 33 "\nAV[i]:=RK2nd(2^i,x0,y0,xf,f, a2):" }{MPLTEXT 1 308 0 "" }{MPLTEXT 1 308 17 "\nEt[i]:=EV-AV[i]:" } {MPLTEXT 1 308 0 "" }{MPLTEXT 1 308 32 "\nabs_et[i]:=abs(Et[i]/EV)*100 .0:" }{MPLTEXT 1 308 0 "" }{MPLTEXT 1 308 14 "\nif (i>0) then" } {MPLTEXT 1 308 0 "" }{MPLTEXT 1 308 22 "\nEa[i]:=AV[i]-AV[i-1]:" } {MPLTEXT 1 308 0 "" }{MPLTEXT 1 308 31 "\nea[i]:=abs(Ea[i]/AV[i])*100. 0:" }{MPLTEXT 1 308 0 "" }{MPLTEXT 1 308 37 "\nsig[i]:=floor((2-log10( ea[i]/0.5))):" }{MPLTEXT 1 308 0 "" }{MPLTEXT 1 308 18 "\nif sig[i]<0 \+ then " }{MPLTEXT 1 308 0 "" }{MPLTEXT 1 308 11 "\nsig[i]:=0:" } {MPLTEXT 1 308 0 "" }{MPLTEXT 1 308 8 "\nend if:" }{MPLTEXT 1 308 0 "" }{MPLTEXT 1 308 8 "\nend if:" }{MPLTEXT 1 308 0 "" }{MPLTEXT 1 308 8 "\nend do:" }{MPLTEXT 1 308 0 "" }}}}{SECT 0 {PARA 258 "" 0 "" {TEXT 297 22 "Section 4: Spreadsheet" }{TEXT 297 0 "" }}{EXCHG {PARA 267 "" 0 "" {TEXT 307 171 "This table shows the approximate value, true error , absolute relative true error, approximate error and relative approxi mate error as a function of the number of segments." }{TEXT 307 0 "" } }}{EXCHG {PARA 270 "> " 0 "" {MPLTEXT 1 310 61 "with( Spread ):Evaluat 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{R5MATHOBJ "sig[5]" 20 "6#\"\"#" }0 }{CELL 8 1 {CELLOPTS 0 -1 -1 0 0 255 255 255 }{R5MATHOBJ "64" 20 "6#\"#k" }0 }{CELL 8 2 {CELLOPTS 0 -1 -1 0 0 255 255 255 }{R5MATHOBJ "H[6]" 20 "6#$\"++++DJ!#6" }0 }{CELL 8 3 {CELLOPTS 0 -1 -1 0 0 255 255 255 }{R5MATHOBJ "EV" 20 "6#$\"+s^g08! \"*" }0 }{CELL 8 4 {CELLOPTS 0 -1 -1 0 0 255 255 255 }{R5MATHOBJ "AV[6 ]" 20 "6#$\"+\">D`I\"!\"*" }0 }{CELL 8 5 {CELLOPTS 0 -1 -1 0 0 255 255 255 }{R5MATHOBJ "Et[6]" 20 "6#$\"'\")*z#!\"*" }0 }{CELL 8 6 {CELLOPTS 0 -1 -1 0 0 255 255 255 }{R5MATHOBJ "abs_et[6]" 20 "6#$\"+@Q XW@!#6" }0 }{CELL 8 7 {CELLOPTS 0 -1 -1 0 0 255 255 255 }{R5MATHOBJ "E a[6]" 20 "6#$\"'^,%)!\"*" }0 }{CELL 8 8 {CELLOPTS 0 -1 -1 0 0 255 255 255 }{R5MATHOBJ "ea[6]" 20 "6#$\"+P[LOk!#6" }0 }{CELL 8 9 {CELLOPTS 0 -1 -1 0 0 255 255 255 }{R5MATHOBJ "sig[6]" 20 "6#&I$sigG6\"6#\"\"'" } 0 }{CELL 9 1 {CELLOPTS 0 -1 -1 0 0 255 255 255 }{R5MATHOBJ "128" 20 "6 #\"$G\"" }0 }{CELL 9 2 {CELLOPTS 0 -1 -1 0 0 255 255 255 }{R5MATHOBJ " H[7]" 20 "6#$\"+++]i:!#6" }0 }{CELL 9 3 {CELLOPTS 0 -1 -1 0 0 255 255 255 }{R5MATHOBJ "EV" 20 "6#$\"+s^g08!\"*" }0 }{CELL 9 4 {CELLOPTS 0 -1 -1 0 0 255 255 255 }{R5MATHOBJ "AV[7]" 20 "6#$\"+F``08!\"*" }0 } {CELL 9 5 {CELLOPTS 0 -1 -1 0 0 255 255 255 }{R5MATHOBJ "Et[7]" 20 "6# $\"&X)p!\"*" }0 }{CELL 9 6 {CELLOPTS 0 -1 -1 0 0 255 255 255 } {R5MATHOBJ "abs_et[7]" 20 "6#$\"+![E'\\`!#7" }0 }{CELL 9 7 {CELLOPTS 0 -1 -1 0 0 255 255 255 }{R5MATHOBJ "Ea[7]" 20 "6#$\"'O,@!\"*" }0 } {CELL 9 8 {CELLOPTS 0 -1 -1 0 0 255 255 255 }{R5MATHOBJ "ea[7]" 20 "6# $\"+!Gx&4;!#6" }0 }{CELL 9 9 {CELLOPTS 0 -1 -1 0 0 255 255 255 } {R5MATHOBJ "sig[7]" 20 "6#\"\"$" }0 }}{TEXT 312 0 "" }}{PARA 263 "" 0 "" {TEXT 313 5 "NOTE:" }{TEXT 302 79 " To evaluate the spreadsheet, yo u need to right click on it and select evaluate" }{TEXT 302 0 "" }}}} {SECT 0 {PARA 258 "" 0 "" {TEXT 297 17 "Section 5: Graphs" }{TEXT 297 0 "" }}{EXCHG {PARA 263 "" 0 "" {TEXT 302 214 "The following code uses 2nd order Runge-Kutta method to calculate intermediate step values fo r the purpose of displaying the solution over the entire range specifi ed. 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