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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 65 "To find estimates for eigenvalues of the Sturm-Liu
oville equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{XPPEDIT 18 0 "diff(p*diff(u,x),x)-q*u+lambda*r*u=0" "/,(-%%diffG6$*&%
\"pG\"\"\"-F%6$%\"uG%\"xGF)F-F)*&%\"qGF)F,F)!\"\"*(%'lambdaGF)%\"rGF)F
,F)F)\"\"!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 16 "input p,q,r and " }{XPPEDIT 18 0 "sigma" "I&sigmaG6\"" }{TEXT 
-1 39 " (to define boundary conditions) below:" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "p:=(x+1):q:
=x:r:=x^2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "L:=u->diff(p*
diff(u,x),x)-q*u+lambda*r*u:L(u(x))=0;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/,*-%%diffG6$-%\"uG6#%\"xGF+\"\"\"*&,&F+F,F,F,F,-F&6$F%F+F,F,*&F
+F,F(F,!\"\"*(%'lambdaGF,F+\"\"#F(F,F,\"\"!" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 57 "The interval is (a,b) . Declare boundary conditions her
e:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "a:=0:b:=1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
9 "sigma:=0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "BC:=u->diff
(u,x)+sigma*u:BC(u(x))=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG
6$-%\"uG6#%\"xGF*\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 " If u is a normalized eigenfunc
tion with respect to the inner product" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 34 "<u,v> = int(r(x)*u(x)*v(x),x=a..b)" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "with eig
envalue lambda, then J(u)=lambda where" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 7 "J(u) = " }{XPPEDIT 18 0 "int(p(x)*diff
(u(x),x)^2+q(x)*u(x)^2,x=a..b)+p(a)*sigma(a)*u(a)^2+p(b)*sigma(b)*u(b)
^2" ",(-%$intG6$,&*&-%\"pG6#%\"xG\"\"\"*$-%%diffG6$-%\"uG6#F+F+\"\"#F,
F,*&-%\"qG6#F+F,*$-F26#F+\"\"#F,F,/F+;%\"aG%\"bGF,*(-F)6#F?F,-%&sigmaG
6#F?F,-F26#F?\"\"#F,*(-F)6#F@F,-FE6#F@F,-F26#F@\"\"#F," }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "An upper bound for l
ambda can be found by evaluating J at arbitrary u which have been" }}
{PARA 0 "" 0 "" {TEXT -1 101 "normalized. The procedure below computes
 J at any u satisfying the BC. First u is normalized so that " }}
{PARA 0 "" 0 "" {TEXT -1 38 "U=u/<u,u>^(1/2) then J(U) is computed:" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 26 "J:=proc(u,p,q,r,a,b,sigma)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 
"local normu,U,JINT,JBC,JX;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "norm
u:=int(r*u*u,x=a..b)^(1/2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "U:=u
/normu;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "JINT:=int((p*diff(U,x)^2
+q*U^2),x=a..b);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "JBC:=subs(x=b,s
igma*p*U)+subs(x=a,sigma*p*U);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "J
X:=JINT+JBC;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "JX;" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 4 "end;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"JG:6)%\"uG%\"pG%\"qG%\"rG%\"aG%\"b
G%&sigmaG6'%&normuG%\"UG%%JINTG%$JBCG%#JXG6\"F4C(>8$*$-%$intG6$*&9'\"
\"\"9$\"\"#/%\"xG;9(9)#F>F@>8%*&F?F>F7!\"\">8&-F:6$,&*&9%F>-%%diffG6$F
HFBF@F>*&9&F>FHF@F>FA>8',&-%%subsG6$/FBFE*(9*F>FQF>FHF>F>-Fen6$/FBFDFh
nF>>8(,&FLF>FXF>F^oF4F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
8 "For the " }{XPPEDIT 18 0 "p,q,r,sigma,a,b " "6(%\"pG%\"qG%\"rG%&sig
maG%\"aG%\"bG" }{TEXT -1 37 " defined in this file we use u=1 and " }
{XPPEDIT 18 0 "u=cos(pi*x)" "/%\"uG-%$cosG6#*&%#piG\"\"\"%\"xGF)" }
{TEXT -1 2 " :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 28 "evalf(J(1,p,q,r,a,b,sigma));" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#$\"5+++++++++:!#>" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 36 "evalf(J(cos(x*Pi),p,q,r,a,b,sigma));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#$\"5^:=]6/ce&)R!#=" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 88 "Since J(u)>=0 for the problem stated here, we conclude th
at all eigenvalues are positive" }}{PARA 0 "" 0 "" {TEXT -1 54 "and th
at the dominant eigenvalue (smallest) - call it " }{XPPEDIT 18 0 " lam
bda[0]" "&%'lambdaG6#\"\"!" }{TEXT -1 15 " is in [0,3/2]." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 
11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}}{MARK "15 3 0" 1 }{VIEWOPTS 1 1 0 1 1 1803 }