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{SECT 0 {PARA 257 "" 0 "" {TEXT -1 36 "Chapter 1, Section 3: K is sepa
rable" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 118
" In this section, we solve integral equations which have separabl
e kernels. We choose examples from the exercises." }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 12 "Exercise 1.3" }}{PARA
0 "" 0 "" {TEXT -1 12 "Exercise 1.d" }}{PARA 0 "" 0 "" {TEXT -1 88 "We
define K and, from it and the integral equation, decide what should b
e the form of y." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "K:=(x,t)
->2*x^2-3*t;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "y:=x->a+x+b
*x^2;" }}}{PARA 0 "" 0 "" {TEXT -1 56 "We compute the integral and col
lect the coefficients of " }{XPPEDIT 18 0 "x^2" "*$%\"xG\"\"#" }{TEXT
-1 27 ", x, and the constant term." }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 26 "int(K(x,t)*y(t),t=0..1)+x;" }}}{EXCHG {PARA 0 "> " 0
"" {MPLTEXT 1 0 13 "collect(\",x);" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 34 "A2:=coeff(\",x^2);\nA0:=\"\"-A2*x^2-x;" }}}{PARA 0 "
" 0 "" {TEXT -1 67 "We equate coefficients on the left and right side \+
of the equations." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "solve(
\{A2=b,A0=a\},\{a,b\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "
assign(\");" }}}{PARA 0 "" 0 "" {TEXT -1 76 "Here is the solution we h
ave found and a check that the solution is correct." }}{EXCHG {PARA 0
"> " 0 "" {MPLTEXT 1 0 5 "y(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 31 "int(K(x,t)*y(t),t=0..1)+x-y(x);" }}}{PARA 0 "" 0 "" {TEXT -1
0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "Ex
ercise 1.f" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}
{PARA 0 "" 0 "" {TEXT -1 88 "We define K and, from it and the integral
equation, decide what should be the form of y." }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 18 "K:=(x,t)->1/2+x*t;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 18 "y:=x->3*x^2+a*x+b;" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 14 "f:=x->3*x^2-1;" }}}{PARA 0 "" 0 "" {TEXT -1 56 "We co
mpute the integral and collect the coefficients of " }{XPPEDIT 18 0 "x
^2" "*$%\"xG\"\"#" }{TEXT -1 27 ", x, and the constant term." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "int(K(x,t)*y(t),t=-1..1)+f(x
);" }}}{PARA 0 "" 0 "" {TEXT -1 67 "We equate coefficients on the left
and right side of the equations." }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 27 "solve(\{2/3*a=a,b=b\},\{a,b\});" }}}{PARA 0 "" 0 ""
{TEXT -1 116 "This must have been a second alternative problem. We ask
what should be the form of the homogeneous adjoint problem?" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "y:=x->c*x+d;\nint(K(t,x)*y(t
),t=-1..1);" }}}{PARA 0 "" 0 "" {TEXT -1 62 "The constant function 1 i
s the only solution for this problem." }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 25 "int((3*x^2-1)*1,x=-1..1);" }}}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "Exercise II. This problem asks \+
what are the eigenvalues of an interal operator." }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 8 "restart;" }}}{PARA 0 "" 0 "" {TEXT -1 92 "We de
fine the kernel and guess at what the form must be for the homogeneous
adjoint problem." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "K:=(x,t
)->x^2*t+x*t^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "y:=x->a*
x^2+b*x;" }}}{PARA 0 "" 0 "" {TEXT -1 19 "We now compute y - " }{TEXT
256 2 "K*" }{TEXT -1 37 "[y] and ask that this should be zero." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "y(x)-lambda*int(K(t,x)*y(t),
t=0..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "collect(\",x);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "A2:=coeff(\",x^2);\nA1:
=coeff(\"\",x);" }}}{PARA 0 "" 0 "" {TEXT -1 133 "One solution for thi
s is a = 0, b = 0. We set the problem in the context of a matrix probl
em and ask if there are non-zero solutions." }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 59 "M:=matrix([[1-lambda/4,-lambda/3],[-lambda/5,1-lambda
/4]]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "det(M);" }}}{PARA
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "Exercise IV" }}
{PARA 0 "" 0 "" {TEXT -1 66 "The point to this problem is that the upp
er limit is not constant." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "
restart;" }}}{PARA 0 "" 0 "" {TEXT -1 27 "We read in the k and the f.
" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "k:=(x,t)-
>x-t;\nf:=x->x;" }}}{PARA 0 "" 0 "" {TEXT -1 58 "The suggestion for th
is problem is to take the derivative." }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 30 "h:=x->int(k(x,t)*y(t),t=0..x);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 13 "diff(h(x),x);" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 15 "diff(h(x),x,x);" }}}{PARA 0 "" 0 "" {TEXT -1 83 "We s
ee that the problem has changed to a differential equation with initit
al values" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "eq:=diff(y(x),x
,x)=diff(h(x),x,x)+diff(f(x),x,x);" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 48 "dsolve(\{eq,y(0)=0+f(0),D(y)(0)=0+D(f)(0)\},y(x));" }
}}{PARA 0 "" 0 "" {TEXT -1 43 "The answer can be converted to other fo
rms." }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "conve
rt(\",trig);\nsimplify(\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
0 "" }}}}{MARK "0 0" 36 }{VIEWOPTS 1 1 0 1 1 1803 }