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{SECT 0 {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 52 "A Separable Fredh
olm Integral Equation of 2nd kind\nA" }{TEXT -1 44 "nimation of soluti
on parametrized by lambda." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "restart:\nint(k(x,y)*u(y),y=a..b) -
 lambda*u(x) = F(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%$intG6$*&
-%\"kG6$%\"xG%\"yG\"\"\"-%\"uG6#F-F./F-;%\"aG%\"bGF.*&%'lambdaGF.-F06#
F,F.!\"\"-%\"FGF9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "k:=(x,
y)->1-3*x*y;\na:=0; b:=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kGj+6
$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),&\"\"\"F.*(\"\"$F.9$F.9%F.!\"\"F
)F)F)6$\"\"!F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG\"\"!" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG\"\"\"" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 133 "alpha[1]:=x->1;    beta[1]:=y->1;\nalpha[2]:=x-
>-3*x; beta[2]:=y->y;\ncheck_k:=(x,y)->add(alpha[n](x)*beta[n](y),n=1.
.2):\ncheck_k(x,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&alphaG6#\"
\"\"j+6#%\"xG6\"6$%)operatorG%&arrowGF+F'F+F+F+6#\"$p\"" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>&%%betaG6#\"\"\"j+6#%\"yG6\"6$%)operatorG%&arro
wGF+F'F+F+F+6#\"#l" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&alphaG6#\"\"
#j+6#%\"xG6\"6$%)operatorG%&arrowGF+,$*&\"\"$\"\"\"9$F2!\"\"F+F+F+6#\"
$H\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%betaG6#\"\"#j+6#%\"yG6\"6$
%)operatorG%&arrowGF+9$F+F+F+6#\"#l" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#,&\"\"\"F$*(\"\"$F$%\"xGF$%\"yGF$!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 183 "A:=matrix(2,2);\nfor i from 1 to 2 do\n  for j from \+
1 to 2 do\n     A[i,j]:=int(beta[i](x)*alpha[j](x),x=a..b);\n  od; una
ssign('n');\nod; unassign('m');\nA=evalm(A);\nlinalg[eigenvects](A);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%&arrayG6%;\"\"\"\"\"#F(7\"" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AGK%'matrixG6#7$7$\"\"\"#!\"$\"
\"#7$#F*F-!\"\"Q(pprint06\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7%#\"\"
\"\"\"#F%<#K%'vectorG6#7$\"\"$F%Q(pprint16\"7%#!\"\"F&F%<#KF)6#7$F%F%Q
(pprint2F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "with(linalg):
\nC:=linsolve( A-lambda*[[1,0],[0,1]],[F[1], F[2] ] );\nc[1]:=C[1]; c[
2]:=C[2];" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names
 norm and trace have been redefined and unprotected\n" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"CGK%'vectorG6#7$,$*(\"\"#\"\"\",(*(F+F,%'lambd
aGF,&%\"FG6#F,F,F,*&\"\"$F,&F16#F+F,!\"\"*&F+F,F0F,F,F,,&F,F7*&\"\"%F,
)F/F+F,F,F7F7,$*(F+F,,(*&F+F,F5F,F7F0F,*(F+F,F/F,F5F,F,F,F9F7F7Q(pprin
t36\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"cG6#\"\"\",$*(\"\"#F',(*
(F*F'%'lambdaGF'&%\"FGF&F'F'*&\"\"$F'&F/6#F*F'!\"\"*&F*F'F.F'F'F',&F'F
4*&\"\"%F')F-F*F'F'F4F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"cG6#\"
\"#,$*(F'\"\"\",(*&F'F*&%\"FGF&F*!\"\"&F.6#F*F**(F'F*%'lambdaGF*F-F*F*
F*,&F*F/*&\"\"%F*)F3F'F*F*F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 59 "u:=x->(1/lambda)*(add(c[n]*alpha[n](x),n=1..2)-f(x));\nu(x);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uGj+6#%\"xG6\"6$%)operatorG%&arr
owGF(*&%'lambdaG!\"\",&-%$addG6$*&&%\"cG6#%\"nG\"\"\"-&%&alphaGF66#9$F
8/F7;F8\"\"#F8-%\"fGF<F.F8F(F(F(6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#*&%'lambdaG!\"\",(*(\"\"#\"\"\",(*(F(F)F$F)&%\"FG6#F)F)F)*&\"\"$
F)&F-6#F(F)F%*&F(F)F,F)F)F),&F)F%*&\"\"%F))F$F(F)F)F%F%**\"\"'F),(*&F(
F)F1F)F%F,F)*(F(F)F$F)F1F)F)F)F4F%%\"xGF)F)-%\"fG6#F=F%F)" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "for m from 1 to 2 do \n   F[m]:=int
(f(x)*beta[m](x),x=a..b);\nod; unassign('m');" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%\"FG6#\"\"\"-%$intG6$-%\"fG6#%\"xG/F.;\"\"!F'" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"FG6#\"\"#-%$intG6$*&-%\"fG6#%\"xG
\"\"\"F/F0/F/;\"\"!F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 715 "f
:=x->cos(3*x);\ncomponents_of_F:=seq(F[i],i=1..2);\nunassign('lambda')
;\nu=evalf(u(x));\nid:=z->z:\nnlambda:=101: lambda_min:=-1: lambda_max
:=+1:\ndlambda:=(lambda_max-lambda_min)/nlambda:\n# (next two lines ju
st to scan slower through lambda=0)\n   bump:=(x0,delta)->(x->exp(-5*(
(x-x0)/delta)^2)):\n   slower:=(x0,delta,h)->h-bump(x0,delta)*(D(h)(x0
)*unapply(x-x0,x)):\n#  plot(slower(0,(lambda_max-lambda_min)/2,id)(la
mbda), \n#          lambda=lambda_min..lambda_max);\nfor i from 0 to n
lambda do\n  lambda:=evalf(slower(0,(lambda_max-lambda_min)/2,id)(lamb
da_min+i*dlambda));\n  p[i]:=plot(u(x),x=a-0.1..b+0.1,-10..10,title=co
nvert(lambda,string)):\nod: unassign('i'):\nplots[display](seq(p[i],i=
0..nlambda),insequence=true);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"fGj+6#%\"xG6\"6$%)operatorG%&arrowGF(-%$cosG6#,$*&\"\"$\"\"\"9$F2F2F
(F(F(6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0components_of_FG6$,
$*&#\"\"\"\"\"$F)-%$sinG6#F*F)F),(*&#F)\"\"*F)-%$cosGF-F)F)*&F(F)F+F)F
)#F)F1!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"uG*&%'lambdaG!\"\",
(*($\"\"#\"\"!\"\"\",&*&$\"+S0+3%*!#6F-F&F-F-$\"+%H3H;'!#5F-F-,&$F-F,F
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-F&F-F'F-F6F'%\"xGF-F-*&$F-F,F--%$cosG6#,$*&$\"\"$F,F-FEF-F-F-F'F-" }}
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