{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 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 0 1 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 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 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 49 "Chapter 1 Section 6. Neither S mall not Separable" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 " In this section, we solve the equation" }}{PARA 0 " " 0 "" {TEXT -1 42 " y = K y + f" }} {PARA 0 "" 0 "" {TEXT -1 37 "where K and f are square integrable. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "Here is t he method broken into steps." }}{PARA 0 "" 0 "" {TEXT -1 15 "(1) Write K as " }{XPPEDIT 18 0 "K=K[n]+G" "/%\"KG,&&F#6#%\"nG\"\"\"%\"GGF(" } {TEXT -1 7 " where " }{XPPEDIT 18 0 "K[n]" "&%\"KG6#%\"nG" }{TEXT -1 30 " is separable and G is small ." }}{PARA 0 "" 0 "" {TEXT -1 31 "(2) Form the resolvent R for G." }}{PARA 0 "" 0 "" {TEXT -1 22 "(3) Make \+ z = (1+R) f. " }}{PARA 0 "" 0 "" {TEXT -1 12 "(4) Compute " }{XPPEDIT 18 0 "(1+R)*K[n]" "*&,&\"\"\"F$%\"RGF$F$&%\"KG6#%\"nGF$" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 14 "(4) Solve y = " }{XPPEDIT 18 0 "(1+R )*K[n]" "*&,&\"\"\"F$%\"RGF$F$&%\"KG6#%\"nGF$" }{TEXT -1 7 " y + z." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 8 "Exercis e" }}{PARA 0 "" 0 "" {TEXT -1 39 "Here is the function K for this exam ple" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "K:=(x,y)->piecewise(t < x,x*t+x-t,x*t);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 76 "We solve the integral equation y = Ky + f by the techni ques of this section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 7 "Step 1:" }{TEXT -1 9 " Choose " }{XPPEDIT 18 0 "K[n] " "&%\"KG6#%\"nG" }{TEXT -1 18 " and G as follows." }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 49 "Kn:=(x,t)->x*t;\nG:=(x,t)->piecewise(t < x,x -t,0);" }}}{PARA 0 "" 0 "" {TEXT -1 23 "Verify that G is small." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "assume(0 " 0 "" {MPLTEXT 1 0 37 "R:=(x,t)->piecewise(t " 0 "" {MPLTEXT 1 0 45 "assume(t " 0 "" {MPLTEXT 1 0 10 "expand(\"); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "combine(\",exp);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "convert(\",trig);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "t:='t': x:='x':" }}}{PARA 0 "" 0 "" {TEXT -1 41 "Step 3: We compute z(x) = f(x) + R[f](x)." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "assume(x>0): additionally(x< 1):\nx+int(R(x,s)*s,s=0..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "expand(\");combine(\",exp);convert(\",trig);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "z:=x->sinh(x);" }}}{PARA 0 "" 0 "" {TEXT -1 16 "Step 4: Compute " }{XPPEDIT 18 0 "(1+R)*K[n]" "*&,&\"\"\"F$%\"R GF$F$&%\"KG6#%\"nGF$" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "x*t+int(R(x,s)*s*t,s=0..x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "expand(\");combine(\",exp);convert(\",trig);simp lify(\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "x:='x';" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "H:='H';" }}}{PARA 0 "" 0 "" {TEXT -1 18 "Step 5: Solve y = " }{XPPEDIT 18 0 "(1+R)*K[n]" "*&,&\"\" \"F$%\"RGF$F$&%\"KG6#%\"nGF$" }{TEXT -1 7 " y + z." }}{PARA 0 "" 0 "" {TEXT -1 29 "This is a separable equation." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 67 "Int(H(x,s)*y(s),s=0..1)+sinh(x)=sinh(x)*int(s*y(s), s=0..1)+sinh(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "eq:=a*i nt(t*sinh(t),t=0..1) + 1=a;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "A:=solve(eq,a);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "yy:= x->A*sinh(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "x*int(t*yy (t),t=0..1) + int((x-t)*yy(t),t=0..x) + x;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 12 "simplify(\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "expand(\");combine(\",exp);convert(\",trig);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "combine(\",trig);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "denom(\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "convert(\",exp);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 11 "Development" }}{PARA 0 "" 0 "" {TEXT -1 10 "Compute G2" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "assume(x " 0 "" {MPLTEXT 1 0 104 "assume(t0):\nint(G(x, s)*G(s,t),s=0..1);\nfactor(\");\nx:='x': t:='t';" }}}{PARA 0 "" 0 "" {TEXT -1 26 "Here is the result for G2." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "G2:=(x,t)->piecewise(t < x,(x-t)^3/3!,0);" }}}{PARA 0 "" 0 "" {TEXT -1 11 "Compute G3." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "assume(t0):\n int(G(x,s)*G2(s,t),s=0..1);\nfactor(\");" }}}{PARA 0 "" 0 "" {TEXT -1 34 "I recognize the developing series." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "series(sinh(z),z=0,9);" }}}}{MARK "0 0" 49 } {VIEWOPTS 1 1 0 1 1 1803 }