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restart; u d a$sin Pi$ A d 4 Pi$ π 4 C u := a sin π A := 4 π π 4 C de d diff u, $4 Cu =: bca d u =:bcb d DDu =: bcc d u =:bcd d DDu =: s2 d dsolve de, bca, bcb, bcc, bcd, u : plot A$sin Pi$, rhs s2, =.. ; () (2).2..8.6.4.2.2.4.6.8 P3 a) restart; N d 3:h d N : f d 2 $3
f := 2 3 de dkdiff u, $2 = f : bca d u =:bcb d u =: ue d dsolve de, bca, bcb, u : for i from to N C do i d i K $h od: p d piecewise % % 2, K 2 K 2, : for i from 2 to N do p i d piecewise i K %! i, K ic % ic,, od: ik ic pnc d piecewise N % % NC, K N NC K N plot p inde $inde =..NC, =.. NC ; K ik ik ik, :, i % (3).8.6.4.2.2.4.6.8 uh d sum u n $p n, n =..N C : duhd d diff uh, : for i from to N C do v i d pi od: for i from to N C do dvd i d diff v i, od:
eq d u =: for i from 2 to N do eq i d int duhd$dvd i, = ik.. ic = int f$v i, = i K.. ic od: eq N C d unc =: for i from to N C do eq i od: sol d solve eq inde $inde =..NC, u inde $inde =..NC : evalf sol ; u =., u 2 =.329287, u 3 =.5349794239, u 4 =. assign % ; plot uh, rhs ue, =.. ; (4).5.4.3.2. b).2.4.6.8 e d uh Krhs ue : Ee d 2 $int diff e, Ee :=.2335997 Ene :=.48329422 2, =.. ; Ene d evalf sqrt Ee ; (5)
restart; d :2 d 4 : 3 d 2 : 4 d 3 4 : 5 d :f d 2$3 : p d piecewise % % 3, p2 d piecewise % % 3, p3 d piecewise % % 3, : K3 $ K2 K3 $ K2, : K $ K3 2 K $ 2 K3, : K $ K2 3 K $ 3 K2, 3 % % 5, K5 $ K4 3 K5 $ 3 K4, p4 d piecewise 3 % % 5, K3 $ K5 4 K3 $ 4 K5, : p5 d piecewise 3 % % 5, K3 $ K4 5 K3 $ 5 K4, : plot p, p2, p3, p4, p5, =.. ;.8.6.4.2.2.4.6.8 uh d u$p Cu2$p2 Cu3$p3 Cu4$p4 Cu5$p5 : duhd d diff uh, : v2 d p2 : v3 d p3 : v4 d p4 : dv2d d diff v2, : dv3d d diff v3, : dv4d d diff v4, :
eq d u =: eq2 d int duhd$dv2d, =..3 = int f$v2, =..3 : eq3 d int duhd$dv3d, =..5 = int f$v3, =..5 : eq4 d int duhd$dv4d, = 3..5 = int f$v4, = 3..5 : eq5 d u5 =: s d solve eq, eq2, eq3, eq4, eq5, u, u2, u3, u4, u5 : assign % ; ue d K 5 : plot uh, ue, =..5 ;.5.4.3.2..2.4.6.8 P4 a e d uh Kue : Ee d 2 $int diff e, 2, =..5 ; Ene d evalf sqrt Ee ; Ee := 2 468 Ene :=.6245339 restart; N d 2:h d N : f d 2 (6)
f := 2 de dkdiff u, $2 = f : bca d D u =:bcb d u =: ue d dsolve de, bca, bcb, u : for i from to N C do i d i K $h od: K 2 p d piecewise % % 2,, : K 2 for i from 2 to N do p i d piecewise i K %! i, K ic % ic,, od: ik ic pnc d piecewise N % % NC, K N NC K N plot p inde $inde =..NC, =.. NC ; K ik ik ik, :, i % (7).8.6.4.2.2.4.6.8 uh d sum u n $p n, n =..N C : duhd d diff uh, : for i from to N C do v i d pi od: for i from to N C do dvd i d diff v i, od:
eq d int duhd$dvd, =.. 2 = int f$v, =.. 2 : for i from 2 to N do eq i d int duhd$dvd i, = ik.. ic = int f$v i, = i K.. ic od: eq N C d unc =: for i from to N C do eq i od: sol d solve eq inde $inde =..NC, u inde $inde =..NC : evalf sol ; u =., u 2 = 75., u 3 =. assign % ; plot uh, rhs ue, =.. ; (8) 8 6 4 2 b.2.4.6.8 e d uh Krhs ue : Ene d evalf sqrt 2 $int diff e, 2, =.. ; Ene := 2.424452 restart; (9)
d :2 d 2 : 3 d : f d 2 : p d piecewise % % 3, p2 d piecewise % % 3, p3 d piecewise % % 3, plot p, p3, p2, =..3 ; K3 $ K2 K3 $ K2, : K $ K3 2 K $ 2 K3, : K $ K2 3 K $ 3 K2, :.8.6.4.2.2.4.6.8 uh d u$p Cu2$p2 Cu3$p3 : duhd d diff uh, : v d p : dvd d diff v, : v2 d p2 : v3 d p3 : dv2d d diff v2, : dv3d d diff v3, : eq d int duhd$dvd, =..3 = int f$v, =..3 : eq2 d int duhd$dv2d, =..3 = int f$v2, =..3 : eq3 d u3 =: s d solve eq, eq2, eq3, u, u2, u3 ;
s := u =, u2 = 75, u3 = assign % ; ue d K$ 2 : plot uh, ue, =..3 ; () 8 6 4 2.2.4.6.8 e d uh Kue : Ee d 2 $int diff e, 2, =..3 ; Ene d evalf sqrt Ee ; Ee := Ene :=. () P5 restart; N d 2:h d N : f d f := de dkdiff u, $2 C$ diff u, =f:bcad u =:bcb d u =: ue d dsolve de, bca, bcb, u : for i from to N C do i d i K $h od: (2)
K 2 p d piecewise % % 2,, : K 2 for i from 2 to N do pi d piecewise i K %! i, K ic % ic,, od: ik ic pnc d piecewise N % % NC, K N NC K N plot p inde $inde =..NC, =.. NC ; K ik ik ik, :, i %.8.6.4.2.2.4.6.8 uh d sum u n $p n, n =..N C ; 2 % and! 2 uh := u K2 % and % 2 otherwise Cu 2 2 K2 2 % and % otherwise Cu 3 (3)
K C2 2 % and % otherwise duhd d diff uh, : for i from to N C do vi d pi od: for i from to N C do dvd i d diff v i, od: eq d u =: for i from 2 to N do eq i d int duhd$dvd i, = ik.. ic C $ int duhd $v i, = ik.. ic = int f$v i, = ik.. ic od; eq 2 := K7 u C4 u 2 C3 u 3 =5 eq N C d unc =: for i from to N C do eq i od: sol d solve eq inde $inde =..NC, u inde $inde =..NC ; sol := u =,u 2 = 5 4, u 3 = (4) (5) evalf sol ; u =., u 2 =.25, u 3 =. assign % ; plot uh, rhs ue, =.. ; (6)
.2.8.6.4.2.2.4.6.8 e d uh Krhs ue : Ene d evalf sqrt 2 $int diff e, 2, =.. ; Ene :=.63549375 restart; f d f := de dkdiff u, $2 C$ diff u, =f:bcad u =:bcb d u =: ue d dsolve de, bca, bcb, u : d :2 d 2 : 3 d 3 := p d piecewise % % 3, p2 d piecewise % % 3, K3 $ K2 K3 $ K2, : K $ K3 2 K $ 2 K3, : (7) (8) (9)
p3 d piecewise % % 3, plot p, p3, p2, =..3 ; K $ K2 3 K $ 3 K2, :.8.6.4.2.2.4.6.8 uh d u$p Cu2$p2 Cu3$p3 : duhd d diff uh, : v d p : dvd d diff v, : v2 d p2 : v3 d p3 : dv2d d diff v2, : dv3d d diff v3, : eq d u =: eq2 d int duhd$dv2d, =..3 C$ int duhd$v2, =..3 = int f$v2, =..3 ; eq2 := K 28 3 u C 6 3 u2 C4 u3 = 2 3 (2) eq3 d u3 =: s d solve eq, eq2, eq3, u, u2, u3 ; s := u =,u2 = 5, u3 = 4 (2) assign % ;
plot uh, rhs ue, =.. ;.2.8.6.4.2.2.4.6.8 e d uh Krhs ue : Ene d evalf sqrt 2 $int diff e, 2, =.. ; Ene :=.478833 (22)