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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 68 "  Separable Fredholm integral equations: Equation \+
must have the form" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 4 "    " }{XPPEDIT 18 0 "u(x)+lambda*int(sum(phi(x)[i]*psi(y)
[i],i=1..N)*u(y),y=a..b)=f(x)" "/,&-%\"uG6#%\"xG\"\"\"*&%'lambdaGF(-%$
intG6$*&-%$sumG6$*&&-%$phiG6#F'6#%\"iGF(&-%$psiG6#%\"yG6#F8F(/F8;\"\"
\"%\"NGF(-F%6#F=F(/F=;%\"aG%\"bGF(F(-%\"fG6#F'" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "  Thus " }{XPPEDIT 18 0 "p
hi(x)[i] " "&-%$phiG6#%\"xG6#%\"iG" }{TEXT -1 4 "and " }{XPPEDIT 18 0 
"psi(y)[i]" "&-%$psiG6#%\"yG6#%\"iG" }{TEXT -1 14 "  (as well as " }
{XPPEDIT 18 0 "f(x),a,b,N" "6&-%\"fG6#%\"xG%\"aG%\"bG%\"NG" }{TEXT -1 
20 ") must be specified:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 "" 1 "" 
{TEXT -1 32 "Warning, new definition for norm" }}{PARA 7 "" 1 "" 
{TEXT -1 33 "Warning, new definition for trace" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 59 "phi:=x->[cos(x),cos(2*x)/2];psi:=y->[cos(y),co
s(2*y)];N:=2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$phiG:6#%\"xG6\"6$%
)operatorG%&arrowGF(7$-%$cosG6#9$,$-F.6#,$F0\"\"##\"\"\"F5F(F(" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$psiG:6#%\"yG6\"6$%)operatorG%&arrow
GF(7$-%$cosG6#9$-F.6#,$F0\"\"#F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%\"NG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "a:=0;b:=2*Pi
;f:=x->cos(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG\"\"!" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG,$%#PiG\"\"#" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"fG%$cosG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 14 "   The kernel " }{XPPEDIT 18 0 "k(x,y
)" "-%\"kG6$%\"xG%\"yG" }{TEXT -1 40 " and corresponding integral oper
ator are" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
26 "                          " }{XPPEDIT 18 0 "k(x,y) = sum(phi(x)[i]
*psi(y)[i],i = 1 .. N)" "/-%\"kG6$%\"xG%\"yG-%$sumG6$*&&-%$phiG6#F&6#%
\"iG\"\"\"&-%$psiG6#F'6#F1F2/F1;\"\"\"%\"NG" }}{PARA 0 "" 0 "" {TEXT 
-1 6 "   and" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 27 "                           " }{XPPEDIT 18 0 "Ku(x)=int(k(x,y)*u
(y),y=a..b)" "/-%#KuG6#%\"xG-%$intG6$*&-%\"kG6$F&%\"yG\"\"\"-%\"uG6#F.
F//F.;%\"aG%\"bG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "k:=(x,y)->s
um(phi(x)[i]*psi(y)[i],i=1..N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"kG:6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF)-%$sumG6$*&&-%$phiG6#9$6#%
\"iG\"\"\"&-%$psiG6#9%F6F8/F7;F8%\"NGF)F)" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 27 "K:=u->int(k(x,y)*u,y=a..b);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"KG:6#%\"uG6\"6$%)operatorG%&arrowGF(-%$intG6$*&-%\"
kG6$%\"xG%\"yG\"\"\"9$F5/F4;%\"aG%\"bGF(F(" }}}{EXCHG {PARA 11 "" 1 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "     Resulting in the i
ntegral equation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 30 "EQN:=u(x)+lambda*K(u(y))=f(x);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%$EQNG/,&-%\"uG6#%\"xG\"\"\"*&%'lambdaGF+-%$int
G6$*&,&*&-%$cosGF)F+-F56#%\"yGF+F+*&-F56#,$F*\"\"#F+-F56#,$F8F=F+#F+F=
F+-F(F7F+/F8;\"\"!,$%#PiGF=F+F+F4" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
6 "      " }}{PARA 0 "" 0 "" {TEXT -1 11 "By defining" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "              " }
{XPPEDIT 18 0 "alpha[i]=int(psi(y)[i]*u(y),y=a..b)" "/&%&alphaG6#%\"iG
-%$intG6$*&&-%$psiG6#%\"yG6#F&\"\"\"-%\"uG6#F/F1/F/;%\"aG%\"bG" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "         \+
     " }{XPPEDIT 18 0 "F[i]=int(psi(y)[i]*f(y),y=a..b)" "/&%\"FG6#%\"i
G-%$intG6$*&&-%$psiG6#%\"yG6#F&\"\"\"-%\"fG6#F/F1/F/;%\"aG%\"bG" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "         \+
     " }{XPPEDIT 18 0 "A[i,j]=int(phi(x)[j]*psi(x)[i],x=a..b)" "/&%\"A
G6$%\"iG%\"jG-%$intG6$*&&-%$phiG6#%\"xG6#F'\"\"\"&-%$psiG6#F06#F&F2/F0
;%\"aG%\"bG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 69 "then the integral equation above is equivalent to the matrix eq
uation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 
"          " }{XPPEDIT 18 0 "alpha+lambda*A*alpha=F" "/,&%&alphaG\"\"
\"*(%'lambdaGF%%\"AGF%F$F%F%%\"FG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 39 "If this system has a solution, i.e., if" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "       \+
   " }{XPPEDIT 18 0 "alpha=(I+lambda*A)^(-1)*F" "/%&alphaG*&),&%\"IG\"
\"\"*&%'lambdaGF(%\"AGF(F(,$\"\"\"!\"\"F(%\"FGF(" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "then the solution of the \+
integral equation can be found as" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 9 "         " }{XPPEDIT 18 0 "u(x)=f(x)-lambd
a*sum(alpha[i]*phi(x)[i],i=1..N)" "/-%\"uG6#%\"xG,&-%\"fG6#F&\"\"\"*&%
'lambdaGF+-%$sumG6$*&&%&alphaG6#%\"iGF+&-%$phiG6#F&6#F5F+/F5;\"\"\"%\"
NGF+!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "A:=matrix(N,N,(i,j)-
>int(phi(x)[j]*psi(x)[i],x=a..b));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%\"AG-%'MATRIXG6#7$7$%#PiG\"\"!7$F+,$F*#\"\"\"\"\"#" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 30 "ID:=array(1..N,1..N,identity):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "F:=vector(N,i->int(f(x)*psi(
x)[i],x=a..b));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG-%'VECTORG6#7
$%#PiG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "alpha:=map(simplify,evalm(inverse(I
D+lambda*A)&*F));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&alphaG-%'VECTO
RG6#7$*&,&\"\"\"F+*&%'lambdaGF+%#PiGF+F+!\"\"F.F+\"\"!" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "So, if ma
ple found the solution " }{XPPEDIT 18 0 "alpha " "I&alphaG6\"" }{TEXT 
-1 47 "above, the solution to the integral equation is" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "U:=x->
f(x)-lambda*innerprod(alpha,phi(x)):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 5 "U(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$cosG6#%
\"xG\"\"\"**%'lambdaGF(,&F(F(*&F*F(%#PiGF(F(!\"\"F-F(F$F(F." }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "As
 a check, the following should equal zero:" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "check:=simplify(U
(x)+lambda*K(U(y))-f(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&checkG
\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 " Also, the eigenvalues of
 the operator K can be found as " }{XPPEDIT 18 0 "1/lambda" "*&\"\"\"
\"\"\"%'lambdaG!\"\"" }{TEXT -1 7 " where " }{XPPEDIT 18 0 "lambda" "I
'lambdaG6\"" }}{PARA 0 "" 0 "" {TEXT -1 19 " is the solution of" }}
{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 17 "        \+
         " }{XPPEDIT 18 0 "det(I+lambda*A)=0" "/-%$detG6#,&%\"IG\"\"\"
*&%'lambdaGF(%\"AGF(F(\"\"!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Lambda:=solve(det(ID+lambda*
A)=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'LambdaG6$,$*$%#PiG!\"\"F)
,$F'!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "1 2
 1" 0 }{VIEWOPTS 1 1 0 1 1 1803 }