{VERSION 2 3 "APPLE_PPC_MAC" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 256 "" 0 "" {TEXT -1 32 "Chapter 1 Section 4. K is Smal l" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 146 " \+ In this section we give the first of two concepts for \"K is small. \" This notion leads to a method for approximating solutions by itera tion." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 12 "Exercise 1.4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "Exercise 1." }}{PARA 0 "" 0 "" {TEXT -1 63 "We define K a nd ask if K is small in the sense of this section." }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 32 "k:=(x,t)->piecewise(t<=x,x-t,0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "assume(x<1);additionally(x>0);\nint (k(x,t),t=0..1);\nx:='x';" }}}{PARA 0 "" 0 "" {TEXT -1 101 "We see tha t on the interval [0,1], K fits within the bounds to be small as descr ibed in this section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "(b) We compute iterates in the process for approximati ng the solution y(x) = " }{TEXT 256 1 "K" }{TEXT -1 11 "[y](x) + x." } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "phi[0]:=x->x;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "assume(x<1);additionally(x>0);\nint (k(x,t)*phi[0](t),t=0..1)+x;\nphi[1]:=unapply(\",x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "int(k(x,t)*phi[1](t),t=0..1)+x;\nphi[2]:= unapply(\",x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "int(k(x,t )*phi[2](t),t=0..1)+x;\nphi[3]:=unapply(\",x);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 20 "taylor(sinh(x),x=0);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "(c) According to the coro llary in this section the difference between the solution y and " } {XPPEDIT 18 0 "phi[3]" "&%$phiG6#\"\"$" }{TEXT -1 16 " does not exceed " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "r:=1/2;\nr^4/(1-r)*1;" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "6 0" 11 } {VIEWOPTS 1 1 0 1 1 1803 }