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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Unit 9: Calculus of Variat
ions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "C
hapter 46: Basic Formalisms" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 29 "Section 46.2:  direct methods" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Copyright" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Co
pyright * 2001 by Addison Wesley Longman, Inc." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 302 "All rights reserved.  No
 part of this publication may be reproduced, stored in a retrieval sys
tem, or transmitted, in any form or by any means, electronic, mechanic
al, photocopying, recording, or otherwise, without the prior written p
ermission of the publisher. Printed in the United States of America." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 15 "Initializations" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 
-1 12 "Introduction" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 57 "In this section we present two methods for approx
imating " }{XPPEDIT 18 0 "y(x)" "6#-%\"yG6#%\"xG" }{TEXT -1 54 ", the \+
extremal which satisfies the endpoint conditions" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 272 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y(a)=
A" "6#/-%\"yG6#%\"aG%\"AG" }{TEXT -1 1 " " }}{PARA 273 "" 0 "" {TEXT 
-1 1 " " }{XPPEDIT 18 0 "y(b)=B" "6#/-%\"yG6#%\"bG%\"BG" }{TEXT -1 1 "
 " }}{PARA 0 "" 0 "" {TEXT -1 38 "and which stationarizes the function
al" }}{PARA 271 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "J=Int(f(x,y(x),
`y'`(x)),x=a..b)" "6#/%\"JG-%$IntG6$-%\"fG6%%\"xG-%\"yG6#F+-%#y'G6#F+/
F+;%\"aG%\"bG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 134 "First, we examine a finite difference te
chnique of Euler, and then, we consider the Rayleigh-Ritz procedure.  \+
Such methods are called " }{TEXT 256 14 "direct methods" }{TEXT -1 20 
", as opposed to the " }{TEXT 257 15 "indirect method" }{TEXT -1 105 "
 of forming and solving the Euler-Lagrange differential equation, whic
h will be seen in the next section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 
"" {TEXT -1 36 "Euler's Method of Finite Differences" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "Euler's method o
f finite differences begins by discretizing the interval " }{XPPEDIT 
18 0 "[a,b]" "6#7$%\"aG%\"bG" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "n+1
" "6#,&%\"nG\"\"\"F%F%" }{TEXT -1 18 " equispaced nodes " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 274 "" 0 "" {XPPEDIT 18 0 "x[k]=a+k*Delta
 x,k=0,`...`,n" "6&/&%\"xG6#%\"kG,&%\"aG\"\"\"*(F'F*%&DeltaGF*F%F*F*/F
'\"\"!%$...G%\"nG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "where
 " }{XPPEDIT 18 0 "Delta x=(b-a)/n" "6#/*&%&DeltaG\"\"\"%\"xGF&*&,&%\"
bGF&%\"aG!\"\"F&%\"nGF," }{TEXT -1 23 ".  The function-values " }
{XPPEDIT 18 0 "y(x[k])" "6#-%\"yG6#&%\"xG6#%\"kG" }{TEXT -1 32 " are d
esignated as the unknowns " }{XPPEDIT 18 0 "y[k]" "6#&%\"yG6#%\"kG" }
{TEXT -1 21 ".  The derivatives y'" }{XPPEDIT 18 0 "``(x[k])" "6#-%!G6
#&%\"xG6#%\"kG" }{TEXT -1 32 " are replaced with the quotients" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 275 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "(y(x[k+1])-y(x[k]))/Delta/x=(y[k+1]-y[k])/Delta/x" "6#/
*(,&-%\"yG6#&%\"xG6#,&%\"kG\"\"\"F.F.F.-F'6#&F*6#F-!\"\"F.%&DeltaGF3F*
F3*(,&&F'6#,&F-F.F.F.F.&F'6#F-F3F.F4F3F*F3" }{TEXT -1 1 " " }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "The integral is t
hen approximated by the finite sum" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "phi(y[1],`...`,y[n-1]
)=Sum(f(x[k],y[k],(y[k+1]-y[k])/Delta/x)*Delta*x,k=0..n-1)" "6#/-%$phi
G6%&%\"yG6#\"\"\"%$...G&F(6#,&%\"nGF*F*!\"\"-%$SumG6$*(-%\"fG6%&%\"xG6
#%\"kG&F(6#F;*(,&&F(6#,&F;F*F*F*F*&F(6#F;F0F*%&DeltaGF0F9F0F*FEF*F9F*/
F;;\"\"!,&F/F*F*F0" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 32 "which becomes a function of the " }
{XPPEDIT 18 0 "n-1" "6#,&%\"nG\"\"\"F%!\"\"" }{TEXT -1 10 " unknowns \+
" }{XPPEDIT 18 0 "y[1],`...`,y[n-1]" "6%&%\"yG6#\"\"\"%$...G&F$6#,&%\"
nGF&F&!\"\"" }{TEXT -1 14 ".  Of course, " }{XPPEDIT 18 0 "y[0]=y(a)" 
"6#/&%\"yG6#\"\"!-F%6#%\"aG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "A" "6#%
\"AG" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "y[n]=y(b)" "6#/&%\"yG6#%\"n
G-F%6#%\"bG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "B" "6#%\"BG" }{TEXT -1 
1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "T
he variables " }{XPPEDIT 18 0 "y[1],`...`,y[n-1]" "6%&%\"yG6#\"\"\"%$.
..G&F$6#,&%\"nGF&F&!\"\"" }{TEXT -1 51 " are then determined by the te
chniques of calculus." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 12 "Example 46.1" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Consider the funct
ion" }}{PARA 259 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x,y,`y'`)=y^
2+``(`y'`)^2+2*y*exp(x)" "6#/-%\"fG6%%\"xG%\"yG%#y'G,(*$F(\"\"#\"\"\"*
$-%!G6#F)F,F-*(F,F-F(F--%$expG6#F'F-F-" }{TEXT -1 1 " " }}{PARA 0 "" 
0 "" {TEXT -1 23 "implemented in Maple as" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "f := (x,y,yp) -> y^2
+yp^2+2*y*exp(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "where the symbol " }{XPPEDIT 18 0 
"yp" "6#%#ypG" }{TEXT -1 37 " is used to represent the derivative " }
{XPPEDIT 18 0 "y" "6#%\"yG" }{TEXT -1 2 "'." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "Let the functional " }{XPPEDIT 
18 0 "J" "6#%\"JG" }{TEXT -1 12 " be given by" }}{PARA 260 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "J=Int(f(x,y,`y'`),x=0..1)" "6#/%\"JG-%$
IntG6$-%\"fG6%%\"xG%\"yG%#y'G/F+;\"\"!\"\"\"" }{TEXT -1 1 " " }}{PARA 
0 "" 0 "" {TEXT -1 34 "with the endpoint conditions being" }}{PARA 
261 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y(0)=0" "6#/-%\"yG6#\"\"!F'
" }{TEXT -1 1 " " }}{PARA 262 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y
(1)=exp(1)" "6#/-%\"yG6#\"\"\"-%$expG6#F'" }{TEXT -1 1 " " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "The exact solution
 to this calculus of variations problem is the function" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Y := 1
/2*exp(1)*csch(1)*sinh(x)+x/2*exp(x);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "a solution we
 will learn to calculate in Sections 46.3 and 46.4.  (See Example 46.7
.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "The
 graph of the exact solution is given by" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "plot(Y,x=0..1, color
=black);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 93 "That this solution satisfies the endpoint
 conditions is seen from the following calculations." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "simplify(
subs(x=0,Y));\nsimplify(subs(x=1,Y));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Since " }
{XPPEDIT 18 0 "b-a=1-0" "6#/,&%\"bG\"\"\"%\"aG!\"\",&F&F&\"\"!F(" }
{TEXT -1 14 " = 1, we have " }{XPPEDIT 18 0 "Delta x=1/n" "6#/*&%&Delt
aG\"\"\"%\"xGF&*&F&F&%\"nG!\"\"" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "
x[k]=k/n" "6#/&%\"xG6#%\"kG*&F'\"\"\"%\"nG!\"\"" }{TEXT -1 19 ".  For \+
the moment, " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 74 " is left unsp
ecified to allow for increasingly more accurate solutions as " }
{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 14 " is increased." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The function " }
{XPPEDIT 18 0 "phi" "6#%$phiG" }{TEXT -1 12 " is given by" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "phi \+
:= n -> sum(f(k/n,y[k],(y[k+1]-y[k])/(1/n))*(1/n),k=0..n-1);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 46 "where we have made it a function of the index " }
{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 25 ".  Thus, for example, if " }
{XPPEDIT 18 0 "n=2" "6#/%\"nG\"\"#" }{TEXT -1 44 ", the sum which appr
oximates the functional " }{XPPEDIT 18 0 "J" "6#%\"JG" }{TEXT -1 3 " i
s" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 7 "phi(2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "Notice that the sum includes \+
the endpoint values " }{XPPEDIT 18 0 "y[0]=y(a)" "6#/&%\"yG6#\"\"!-F%6
#%\"aG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y[n]=y[2]" "6#/&%\"yG6#%\"
nG&F%6#\"\"#" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "y(b)" "6#-%\"yG6#%\"bG
" }{TEXT -1 67 ".  These endpoint values are know, so the approximatin
g sum becomes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 37 "q := subs(y[0]=0,y[2]=exp(1),phi(2));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 38 "There is but the one unknown, namely, " }{XPPEDIT 18 0 "y
[1]" "6#&%\"yG6#\"\"\"" }{TEXT -1 19 ", corresponding to " }{XPPEDIT 
18 0 "x[1]=1/2" "6#/&%\"xG6#\"\"\"*&F'F'\"\"#!\"\"" }{TEXT -1 70 ".  I
t is determined by setting to zero the derivative with respect to " }
{XPPEDIT 18 0 "y[1]" "6#&%\"yG6#\"\"\"" }{TEXT -1 14 ".  This yields" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 31 "q1 := solve(diff(q,y[1]),y[1]);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "The extremal \+
" }{XPPEDIT 18 0 "y(x)" "6#-%\"yG6#%\"xG" }{TEXT -1 70 " is being appr
oximated by a linear spline connecting the three points " }{XPPEDIT 
18 0 "``(x[k],y[k]),k=0,1,2" "6&-%!G6$&%\"xG6#%\"kG&%\"yG6#F)/F)\"\"!
\"\"\"\"\"#" }{TEXT -1 10 ".  If the " }{XPPEDIT 18 0 "x" "6#%\"xG" }
{TEXT -1 5 " and " }{XPPEDIT 18 0 "y" "6#%\"yG" }{TEXT -1 55 " coordin
ates of these points are entered into the lists" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Lx := [0,1/
2,1];\nLy := [0,q1,exp(1)];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "we can obtain a representa
tion of the approximating spline as" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "YY := spline(Lx,Ly,x,linea
r);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 106 "A graph of the approximate extremal, in cyan, is \+
compared to the exact extremal, in black, in Figure 46.5." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "plo
t([YY,Y],x=0..1,color=[cyan,black], labels=[x,y], labelfont=[TIMES,ITA
LIC,12], xtickmarks=2, ytickmarks=3);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "There are two
 ways the value of the functional " }{XPPEDIT 18 0 "J" "6#%\"JG" }
{TEXT -1 88 " might be approximated.  First, its value can be taken as
 the value of the discrete sum " }{XPPEDIT 18 0 "phi(y[1])" "6#-%$phiG
6#&%\"yG6#\"\"\"" }{TEXT -1 24 ", obtained in Maple with" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "J1 :
= simplify(subs(y[0]=0,y[1]=q1,y[2]=exp(1),phi(2)));" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "S
ince the known values of " }{XPPEDIT 18 0 "y[0]" "6#&%\"yG6#\"\"!" }
{TEXT -1 5 " and " }{XPPEDIT 18 0 "y[n]=y[2]" "6#/&%\"yG6#%\"nG&F%6#\"
\"#" }{TEXT -1 19 " appear in the sum " }{XPPEDIT 18 0 "phi(y[1])" "6#
-%$phiG6#&%\"yG6#\"\"\"" }{TEXT -1 67 ", it is necessary to supply tho
se values when evaluating the sum.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 30 "An alternate approximation of " }
{XPPEDIT 18 0 "J" "6#%\"JG" }{TEXT -1 148 " can be computed by evaluat
ing it along the piecewise linear path just computed as the approximat
ing linear spline.  In this event, we would compute" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "J2 := int
(f(x,YY,diff(YY,x)),x=0..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "We can also obtain the v
alue of " }{XPPEDIT 18 0 "J" "6#%\"JG" }{TEXT -1 48 " along the exact \+
extremal.  This turns out to be" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "J3 := collect(int(f(x,Y,diff
(Y,x)),x=0..1),csch);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "The floating-point equivalents f
or all three values of " }{XPPEDIT 18 0 "J" "6#%\"JG" }{TEXT -1 4 " ar
e" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "evalf(J1);\nevalf(J2);\nevalf(J3);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "T
he first is least accurate.  That is because in the passage from " }
{XPPEDIT 18 0 "J" "6#%\"JG" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "phi" "6
#%$phiG" }{TEXT -1 15 ", the extremal " }{XPPEDIT 18 0 "y(x)" "6#-%\"y
G6#%\"xG" }{TEXT -1 29 " is approximated by a degree " }{TEXT 258 4 "z
ero" }{TEXT -1 20 " spline.  The value " }{XPPEDIT 18 0 "J[1]" "6#&%\"
JG6#\"\"\"" }{TEXT -1 178 " reflects this use of a lower-degree approx
imation of the extremal.  After the values of the approximating extrem
al are obtained at the nodes, a linear spline is formed.  Hence, " }
{XPPEDIT 18 0 "J[2]" "6#&%\"JG6#\"\"#" }{TEXT -1 79 " will clearly be \+
a better approximation to the exact stationary value given by " }
{XPPEDIT 18 0 "J[3]" "6#&%\"JG6#\"\"$" }{TEXT -1 9 " because " }
{XPPEDIT 18 0 "J[2]" "6#&%\"JG6#\"\"#" }{TEXT -1 113 " is computed by \+
integrating along the more accurate linear spline, and not the degree \+
zero spline used to obtain " }{XPPEDIT 18 0 "J[1]" "6#&%\"JG6#\"\"\"" 
}{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 16 "If we next take " }{XPPEDIT 18 0 "n=3" "6#/%\"nG\"\"$" }
{TEXT -1 104 ", we will have two interior nodes at which to compute th
e solution.  The approximating sum for the case " }{XPPEDIT 18 0 "n=3
" "6#/%\"nG\"\"$" }{TEXT -1 3 " is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "q := subs(y[0]=0,y[3]=exp(1
),phi(3));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 40 "where we have supplied the known values \+
" }{XPPEDIT 18 0 "y[0]=0" "6#/&%\"yG6#\"\"!F'" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "y[n]=exp(1)" "6#/&%\"yG6#%\"nG-%$expG6#\"\"\"" }{TEXT 
-1 20 ".  The two unknowns " }{XPPEDIT 18 0 "y[1]" "6#&%\"yG6#\"\"\"" 
}{TEXT -1 5 " and " }{XPPEDIT 18 0 "y[2]" "6#&%\"yG6#\"\"#" }{TEXT -1 
164 " are determined by the ordinary techniques of multivariable calcu
lus.  Thus, set to zero the derivatives with respect to each of the un
knowns, and solve, obtaining'" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "q1 := solve(\{diff(q,y[1]),d
iff(q,y[2])\},\{y[1],y[2]\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "The extremal " }
{XPPEDIT 18 0 "y(x)" "6#-%\"yG6#%\"xG" }{TEXT -1 69 " is being approxi
mated by a linear spline connecting the four points " }{XPPEDIT 18 0 "
``(x[k],y[k]),k=0,`...`,3" "6&-%!G6$&%\"xG6#%\"kG&%\"yG6#F)/F)\"\"!%$.
..G\"\"$" }{TEXT -1 10 ".  If the " }{XPPEDIT 18 0 "x" "6#%\"xG" }
{TEXT -1 5 " and " }{XPPEDIT 18 0 "y" "6#%\"yG" }{TEXT -1 55 " coordin
ates of these points are entered into the lists" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "Lx := [0,1/
3,2/3,1];\nLy := [0,op(subs(q1,[y[1],y[2]])),exp(1)];" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 78 "we can obtain the following representation of the approximating
 linear spline." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 29 "YY := spline(Lx,Ly,x,linear);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 163 "A comparison between this more accurate spline and the exact e
xtremal is given in Figure 46.6, where the spline is drawn in cyan, an
d the exact extremal, in black." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "plot([YY,Y],x=0..1,color=[c
yan,black], labels=[x,y], labelfont=[TIMES,ITALIC,12], xtickmarks=2, y
tickmarks=3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "Approximating the value of the fun
ctional " }{XPPEDIT 18 0 "J" "6#%\"JG" }{TEXT -1 23 " with the discret
e sum " }{XPPEDIT 18 0 "phi(y[1],y[2])" "6#-%$phiG6$&%\"yG6#\"\"\"&F'6
#\"\"#" }{TEXT -1 6 " gives" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "J4 := simplify(subs(y[0]=0,o
p(q1),y[3]=exp(1),phi(3)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "where again, the values " 
}{XPPEDIT 18 0 "y[0]=0" "6#/&%\"yG6#\"\"!F'" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "y[3]=exp(1)" "6#/&%\"yG6#\"\"$-%$expG6#\"\"\"" }{TEXT 
-1 24 " are needed in this sum." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 15 "If we evaluate " }{XPPEDIT 18 0 "J" "6#%
\"JG" }{TEXT -1 35 " along the linear spline, we obtain" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "J5 := \+
int(f(x,YY,diff(YY,x)),x=0..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Once again, we look at
 the floating-point equivalents" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "evalf(J4);\nevalf(J5);\neval
f(J3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 95 "to see that evaluating the functional along the
 linear spline gives a more accurate value than " }{XPPEDIT 18 0 "phi(
y[1],y[2])" "6#-%$phiG6$&%\"yG6#\"\"\"&F'6#\"\"#" }{TEXT -1 67 ", obta
ined by evaluating the functional along a degree-zero spline." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 27 "The Rayleigh-Ritz Tec
hnique" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 280 "The Rayleigh-Ritz technique for approximating extremals \+
in the calculus of variations bears some resemblance to the Rayleigh-R
itz approximation scheme (Section 15.3) for solving differential equat
ions.  Texts such as [1] point out that the name is sometimes shortene
d to just the " }{TEXT 259 4 "Ritz" }{TEXT -1 1 " " }{TEXT 260 6 "meth
od" }{TEXT -1 88 " since W. Ritz greatly generalized, in 1908 and 1909
, the earlier work of Lord Rayleigh." }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 28 "To approximate the extremal " }
{XPPEDIT 18 0 "y(x)" "6#-%\"yG6#%\"xG" }{TEXT -1 35 " which stationari
zes the functional" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 263 "" 0 "
" {TEXT -1 1 " " }{XPPEDIT 18 0 "J=Int(f(x,y(x),`y'`(x)),x=a..b)" "6#/
%\"JG-%$IntG6$-%\"fG6%%\"xG-%\"yG6#F+-%#y'G6#F+/F+;%\"aG%\"bG" }{TEXT 
-1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 48 "with respect to curves satisfy
ing the conditions" }}{PARA 264 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 
"y(a)=A" "6#/-%\"yG6#%\"aG%\"AG" }{TEXT -1 1 " " }}{PARA 265 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "y(b)=B" "6#/-%\"yG6#%\"bG%\"BG" }{TEXT 
-1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 
"replace " }{XPPEDIT 18 0 "y(x)" "6#-%\"yG6#%\"xG" }{TEXT -1 14 " with
 the sum " }{XPPEDIT 18 0 "Phi(x)=phi[0](x)" "6#/-%$PhiG6#%\"xG-&%$phi
G6#\"\"!6#F'" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "Sum(c[k]*phi[k](x),k=1
..n)" "6#-%$SumG6$*&&%\"cG6#%\"kG\"\"\"-&%$phiG6#F*6#%\"xGF+/F*;F+%\"n
G" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "phi[0](x)" "6#-&%$phiG6#\"\"
!6#%\"xG" }{TEXT -1 49 " satisfies the nonhomogeneous boundary conditi
ons" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 266 "" 0 "" {TEXT -1 1 " \+
" }{XPPEDIT 18 0 "phi[0](a)=A" "6#/-&%$phiG6#\"\"!6#%\"aG%\"AG" }
{TEXT -1 1 " " }}{PARA 267 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "phi[
0](b)=B" "6#/-&%$phiG6#\"\"!6#%\"bG%\"BG" }{TEXT -1 1 " " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }{XPPEDIT 18 
0 "phi[k](a)=phi[k](b)" "6#/-&%$phiG6#%\"kG6#%\"aG-&F&6#F(6#%\"bG" }
{TEXT -1 5 " = 0," }{XPPEDIT 18 0 "k=1,`...`,n" "6%/%\"kG\"\"\"%$...G%
\"nG" }{TEXT -1 14 ".  This makes " }{XPPEDIT 18 0 "J" "6#%\"JG" }
{TEXT -1 37 " become a function of the parameters " }{XPPEDIT 18 0 "c[
k],k=1,`...`,n" "6&&%\"cG6#%\"kG/F&\"\"\"%$...G%\"nG" }{TEXT -1 104 ",
 and the ordinary techniques of multivariable calculus are used to det
ermine values which starionarize " }{XPPEDIT 18 0 "J(c[1],`...`,c[n])
" "6#-%\"JG6%&%\"cG6#\"\"\"%$...G&F'6#%\"nG" }{TEXT -1 23 ".  (Note th
at the same " }{XPPEDIT 18 0 "Phi(x)" "6#-%$PhiG6#%\"xG" }{TEXT -1 27 
" was used in Section 15.3.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 12 "Example 46.2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 62 "To approximate the extremal which station
arizes the functional" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 268 "" 
0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "J=Int(f(x,y,`y'`),x=0..1)" "6#/%\"
JG-%$IntG6$-%\"fG6%%\"xG%\"yG%#y'G/F+;\"\"!\"\"\"" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 25 "with endpoint conditions " }}{PARA 269 "
" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y(0)=0" "6#/-%\"yG6#\"\"!F'" }
{TEXT -1 1 " " }}{PARA 270 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y(1)
=exp(1)" "6#/-%\"yG6#\"\"\"-%$expG6#F'" }{TEXT -1 1 " " }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "where  " }{XPPEDIT 
18 0 "f(x,y,`y'`)=y^2+``(`y'`)^2+2*y*exp(x)" "6#/-%\"fG6%%\"xG%\"yG%#y
'G,(*$F(\"\"#\"\"\"*$-%!G6#F)F,F-*(F,F-F(F--%$expG6#F'F-F-" }{TEXT -1 
59 ", that is, the functional of the previous section, we take " }
{XPPEDIT 18 0 "Phi(x)" "6#-%$PhiG6#%\"xG" }{TEXT -1 3 " as" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Phi
 := x*exp(x)+add(c[k]*x*(x^k-1),k=1..2);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "The function \+
" }{XPPEDIT 18 0 "phi[0](x)=x*exp(x)" "6#/-&%$phiG6#\"\"!6#%\"xG*&F*\"
\"\"-%$expG6#F*F," }{TEXT -1 69 " satisfies the nonhomogeneous boundar
y conditions, and the functions " }{XPPEDIT 18 0 "phi[k](x)=x*(x^k-1),
k=1,2" "6%/-&%$phiG6#%\"kG6#%\"xG*&F*\"\"\",&)F*F(F,F,!\"\"F,/F(F,\"\"
#" }{TEXT -1 37 ", satisfy the homogeneous conditions " }{XPPEDIT 18 
0 "phi[k](0)=phi[k](1)" "6#/-&%$phiG6#%\"kG6#\"\"!-&F&6#F(6#\"\"\"" }
{TEXT -1 6 " = 0, " }{XPPEDIT 18 0 "k=1,2" "6$/%\"kG\"\"\"\"\"#" }
{TEXT -1 32 ".  This converts the functional " }{XPPEDIT 18 0 "J" "6#%
\"JG" }{TEXT -1 3 " to" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Jc := Int(f(x,Phi,diff(Phi,x)),x=0.
.1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 10 "with value" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Q := value(Jc);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 28 "To determine the parameters " }{XPPEDIT 18 0 "c[1]" "6#&%\"cG6#
\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "c[2]" "6#&%\"cG6#\"\"#" }
{TEXT -1 82 ", set to zero the derivatives with respect to each parame
ter, and solve, obtaining" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "q := solve(\{diff(Q,c[1])=0,diff(Q,
c[2])=0\},\{c[1],c[2]\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 
"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Hence, the Rayleigh-Ritz app
roximant becomes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 22 "Phi[1] := subs(q,Phi);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "A grap
h of the exact solution" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "Y;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "along with th
e approximate solution " }{XPPEDIT 18 0 "Phi[1]" "6#&%$PhiG6#\"\"\"" }
{TEXT -1 175 ", shows little difference between the two solutions.  We
 have used color (cyan for the approximate solution, and black for the
 exact) and varied the line thickness (thick for " }{XPPEDIT 18 0 "Phi
[1]" "6#&%$PhiG6#\"\"\"" }{TEXT -1 112 ") to distinguish one curve fro
m the other.  However, they are so close as to appear as virtually the
 same curve." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 132 "plot([Phi[1],Y],x=0..1, color=[cyan,black], thi
ckness=[3,1], labels=[x,y], labelfont=[TIMES,ITALIC,12], xtickmarks=2,
 ytickmarks=3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "The values of " }{XPPEDIT 18 0 "J
" "6#%\"JG" }{TEXT -1 30 " on each curve are computed by" }}{PARA 0 "
" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "Je \+
:= evalf(Int(f(x,Y,diff(Y,x)),x=0..1));\nJc := evalf(Int(f(x,Phi[1],di
ff(Phi[1],x)),x=0..1));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "The functional " }{XPPEDIT 
18 0 "J" "6#%\"JG" }{TEXT -1 98 ", computed on the approximate extrema
l, is marginally larger than the value on the exact extremal." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "References" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 102 "[1] Marvin J. \+
Forray, Variational Calculus in Science and Engineering, McGraw-Hill B
ook Company, 1968." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "1" 0 }{VIEWOPTS 1 1 0 1 1 
1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }