Sturm-Liouville Green KEN ZOU Hermite Legendre Laguerre L L [p(x) d2 dx 2 + q(x) d ] dx + r(x) u(x) = Lu(x) = 0 (1) L = p(x) d2 dx

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Sturm-Liouville Green KEN ZOU 2006 4 23 1 Hermite Legendre Lguerre 1 1.1 2 L L p(x) d2 2 + q(x) d + r(x) u(x) = Lu(x) = 0 (1) L = p(x) d2 2 + q(x) d + r(x) (2) L = d2 2 p(x) d q(x) + r(x) (3) 2 (Self-Adjoint opertor) L = L Lu(x) = Lu(x) = d p(x) du(x) + q(x)u(x) = 0 (4) p (x) = q(x) L Lu(x) = d2 2 p(x)u(x) d q(x)u(x) + r(x)u(x) = d p (x)u(x) + p(x) d u(x) q (x)u(x) q(x) d u(x) + r(x)u(x) = p (x)u(x) + 2p (x) d d2 u(x) + p(x) 2 u(x) q (x)u(x) q d u(x) + r(x)u(x) = p(x) d2 2 u(x) + { 2p (x) q(x) } d u(x) + { p (x) q (x) + r(x) } u(x) 1 J.C.F.Sturm 1803-1855 Switzerlnd, J.Liouville 1809-1882) Frnce Sturm-Liouville 1836-1837 2 1

(1) (1.1) 2p (x) q(x) = q(x) p (x) = q(x) p (x) q (x) + r(x) = r(x) p (x) = q (x) Lu(x) = Lu(x) L = L p(x) d2 2 + q(x) d + r(x) u(x) = Lu(x) = 0 1/p(x)exp x q(t)/p(t)dt 1 x p(x) exp q(t) p(t) dt Lu(x) = 1 x q(t) = exp p(t) dt u (x) + q(x) p(x) exp x p(x) exp q(t) x q(t) p(t) dt p(x) p(t) dt d2 2 + q(x) d u (x) + r(x) p(x) exp + r(x) x q(t) p(t) dt d x q(t) x exp p(t) dt u q(t) (x) = exp p(t) dt u (x) + q(x) x p(x) exp q(t) p(t) dt u (x) 1 x p(x) exp q(t) p(t) dt Lu(x) = d = d { x } q(t) du(x) exp p(t) dt P (x) du(x) + Q(x)u(x) u(x) u(x) + r(x) { x } p(x) exp q(t) p(t) dt u(x) x 3 y xy + 2y = 0 x 3 x exp x 3 = exp 1 x y x x 3 y + 2 x 3 y = 0 e 1 x y e 1/x x 2 y + 2e 1/x x 3 y = 0 d e d e 1/x = e 1/x x 2 1/x dy + 2e 1/x x 3 y = 0 2

1.2 L Lu(x) = d p(x) du(x) q(x)u(x) = λρ(x)u(x) (5) ρ(x) (weighting function) 3 ρ(x) > 0 4 ρ(x) = 1 λ Lu i (x) = λ i ρ(x)u i (x) (i = 1, 2,, ) (6) λ i L u i (x) λ i 2 A. ( ) b u(t) u() = 0, u(b) = 0 B. ( ) b 5 u () = 0, u (b) = 0 C. ( ) u() = 0, u (b) = 0 (5) MEMO ( ) 1 u() + 2 u () = 0 b 1 u(b) + b 2 u (7) (b) = 0 1, 2, b 1, b 2 2 = b 2 = 0 1 = b 1 = 0 2 = b 1 = 0 3 2.1 4 1.4 3

Lu, v = u, Lv Lu, v + u, Lv = + d ( p(x) dv(x) ) u(x) d ( p(x) du(x) ) v(x) ( q(x)u(x)v(x) + u(x)q(x)v(x)) = p(x) dv b ( du u(x) p(x)du v(x) dv + p(x) dv ) du (8) = p(x) dv b u(x) p(x)du v(x) = p(b)v (b)u(b) v(b)u (b) p()v ()u() v()u () αv() + βv () = 0 αu() + βu () = 0 (9) (7) (non-trivil solution) v() v () u() u () = 0 v()u () v ()u() = 0 v()u () v ()u() = 0 (8) Lu, v = u, Lv 1.3 (Sturm-Liouville) p(x) q(x) r(x) (Legendre) (Bessel) (Hermite) (Lguerre) L Lu(x) = d p(x) du(x) q(x)u(x) = λρ(x)u(x) (10) 4

(1) p(x) = 1, q(x) = 0, ρ(x) = 1 ( ) d d u(x) = λu(x) d 2 u(x) = λu(x) 2 (11) (2) (Legendre) p(x) = 1 x 2, q(x) = 0, ρ(x) = 1, ( 1 x 1) d (1 x 2 ) du(x) + λu(x) = 0 (1 x 2 ) d2 u(x) 2 2x du(x) + λu(x) = 0 (12) λ = n(n + 1) (3) (Bessel) p(x) = x, q(x) = x, ρ(x) = 1/x, (x 0) d x du(x) + xu(x) = λ x u(x) d 2 u(x) 2 + 1 du(x) x + (1 + λx 2 ) u(x) = 0 (13) λ = n 2 (4) (Hermitel) p(x) = e x2, q(x) = 0, ρ(x) = e x2 x, λ = 2n, (n = 0, 1, 2, ) d e d 2 u(x) 2 (5) (Lguerre) x2 du(x) + 2ne x2 u(x) = 0 2x du(x) + 2nu(x) = 0 p(x) = xe x, q(x) = 0, ρ(x) = e x, λ = n, (n = 0, 1, 2, ) d xe x d2 u(x) 2 x du(x) + ne x u(x) = 0 + (1 x) du(x) + nu(x) = 0 (14) (15) (6) Chrbyshev) 5

p(x) = 1 x 2, q(x) = 0, ρ(x) = 1/ 1 x 2, λ = n 2 d 1 x 2 du(x) + (1 x 2 ) d2 u(x) 2 n 2 1 x 2 u(x) = 0 x du(x) + n2 u(x) = 0 (16) 1.3.1 d 2 u(x) = λu(x) 2 2 u(x) = e tx 6 t 2 + λ = 0, t = ±i λ u(x) = C 1 e i λx + C 2 e i λx λ u(x) C 1 e αx + C 2 e αx λ < 0, λ = α 2 (α > 0) u(x) = C 1 x + C 2 λ = 0 C 1 cosβx + C 2 sinβx (λ > 0, λ = β 2 (β > 0) (17) (1) u(0) = 0, u(l) = 0 λ 0 C 1 = C 2 = 0 u(x) 0 λ > 0 u(0) = 0 C 1 = 0 u(l) = 0 C 2 sinβl = 0 C 2 0 β = nπ n λ L λ n λ n = β 2 = n2 π 2 ( nπ ) L 2, u n(x) = sin L x (2) u(0) = u(l), u (0) = u (L) λ < 0 C 1 = C 2 = 0 u(x) 0 λ = 0 u(0) = u(l)0 C 1 = 0. u(x) = C 2 C 2 C 2 = 1 λ = 0, u(x) = 1 λ > 0 (C 2 1 + C2 2 )sinβl = 0 β = nπ L 6 ( ) 6

( nπ λ n = β 2 = L ) 2 ( nπ ) 2 ( nπ ) 2, un (x) = C 1 cos L x + C2 sin L x (n = 1, 2, ) 1.4 L Lu(x) = d p(x) du(x) q(x)u(x) = λρ(x)u(x) (18) Lu i (x) = λ i ρ(x)u i (x) (i = 1, 2,, ) (19) ρ(x) λ i 7 u i (x) λ i 1. (20) {u n (x)}(n = 1, 2, ) λ i λ j u i (x) u j (x) ρ(x) ρ(x)u i (x)u j (x) = 0 (i j) (20) 2. ui (x)u i (x) = 1 u i (x) {u n (x)} {u n (x)} 3. {u i (x)} δ(x ξ) = u i (x)u i (ξ) (21) i=1 {u i (x)} 4. EPILOG Sturm Liouville 7 λ i 7

2 2.1 b Lu(t) = d dt p(t) du(t) q(t)u(t) = f(t) (22) dt p(t) > 0 p(t) q(t) b (22) α 1 u() + α 2 u () = 0 β 1 u() + β 2 u () = 0 (23) L G(t, ξ) = δ(t ξ) (24) G(t, ξ) (23) (24) f(ξ) b ξ L G(t, ξ)f(ξ)dξ = L G(t, ξ)f(ξ)dξ = f(ξ)δ(t ξ)dξ = f(t) (25) L (25) (22) u(t) u(t) = G(t, ξ)f(ξ)dξ (26) L L 1 G(t, ξ) = L 1 δ(t ξ) t = ξ δ(t ξ) L 1 G(t, ξ) δ(t ξ) f(t) L 1 G(t, ξ) u(t) = G(t, ξ)f(ξ)dξ (26) f(ξ) (24) G(t, ξ) 8

2.2 - Lu(x) = d p(x) du(x) q(x)u(x) = f(x) (27) b 4 8 ( ) ( ) 1 2 (1) 2.2.1 Lu(x) = 0 LG(x, ξ) = δ(x ξ) G(x, ξ) u(x) G(x, ξ) + u(x) LG(x, ξ) = δ(x ξ) 9-1 L = d 2 / 2 G(x, ξ) d 2 G(x, ξ) = δ(x ξ) 2 1 d G(x, ξ) = θ(x ξ) = 0 x < ξ 1 x > ξ 8 ( 9 9

0 x < ξ G(x, ξ) = 1 x > ξ G(x, ξ) = x ξ + x ξ 2 2 x G(x, ξ) = x ξ 2 10 y y = x ξ y 1 ξ x ξ 1 x -2 u(0) = u(1) d2 G(x, ξ) = δ(x ξ) 2 1 2 d 2 u(x)/ 2 = 0 u = x + b b 0 ξ 1 G(x, ξ) = 1 x ξ + x + b 2 G(0, ξ) = 1 2 ξ + b = 0, G(1, ξ) = 1 1 ξ + = 0 2 G(x, ξ) = 1 ( 2 x ξ + ξ 1 ) x 1 2 2 ξ ξ(x 1) x > ξ = x(ξ 1) x < ξ 10 δ(x) = d θ(x) 10

-3 u(0) = u(1) d2 u(x) 2 + x 2 = 0 (26) u(x) = = 1 0 x 0 = (x 1) G(x, ξ)f(ξ)dξ ξ(x 1)f(ξ)dξ + x 0 = 1 12 x(x3 1) ξ 3 dξ + x 1 x 1 x x(ξ 1)f(ξ)dξ (ξ 1)ξ 2 dξ 1 0 = x 0 + 1 x -2 x < ξ x > ξ x 0 ξ 0 < ξ < x G(x, ξ) = ξ(x 1) 2.2.2 2 x = Lu 1 (x) = 0, x = b Lu 2 (x) = 0 2 A(ξ)u 1 (x) ( x < ξ) G(x, ξ) = (28) B(ξ)u 2 (x) (ξ < x b) G(x, ξ) G(x, ξ) x = ξ 11 A(ξ)u 1 (ξ) = B(ξ)u 2 (ξ) (29) LG(x, ξ) = δ(x ξ) x = ξ 0 x = ξ + 0 ξ+0 ξ 0 d dt p(x) du(x) ξ+0 ξ 0 q(x)u(x) = ξ+0 ξ 0 δ(x ξ) (30) p(x) d ξ+0 G(x, ξ) = 1 ξ 0 ( ) dg x=ξ+0 ( ) dg = 1 x=ξ 0 p(ξ) 11 b 11

A(ξ) B(ξ) A(ξ)u 1(ξ) B(ξ)u 2(ξ) = 1 p(ξ) (29) (31) A(ξ) B(ξ) (28) A(ξ) B(ξ) (31) u 2 (ξ) A(ξ) = p(ξ)(u 2 (ξ)u 1 (ξ) u 1(ξ)u 2 (ξ)) = u 2(ξ) p(ξ) (ξ) u 1 (ξ) B(ξ) = p(ξ)(u 2 (ξ)u 1 (ξ) u 1(ξ)u 2 (ξ)) = u 1(ξ) p(ξ) (ξ) (32) (ξ) ( ) u 1 (ξ) (ξ) = u 1 (ξ) u 2 (ξ) u 2 (ξ) = u 1(ξ)u 2(ξ) u 2 (ξ)u 1(ξ) (33) u 1 (ξ) u 2 (ξ) (ξ) 0 u 1 u 2 Lu = 0 (pu 1) qu 1 = 0 (pu 2) qu 2 = 0 1 u 2 2 u 1 u 1 (pu 2) u 2 (pu 1) = 0 ( 1 2 6 ) u 1 (pu 2) u 2 (pu 1) = p u 1 u 2 + p u 1u 2 + p u 1 u 2 (p u 1u 2 + pu 1u 2 + pu 1u 2) = p u 1 u 2 p u 1u 2 = d p(u1 u 2 u 1u 2 ) = 0 p(u 1 u 2 u 1u 2 ) = p(x) (x) = 0 (32) p(x) (x) = p(ξ) (ξ) = = p(0) (0) A(ξ) = u 2(ξ) p(0) (0) B(ξ) = u 1(ξ) p(0) (0) (34) 12

A(ξ)u 1 (x) ( x < ξ) G(x, ξ) = B(ξ)u 2 (x) (ξ < x b) (35) (34)(35) G(x, ξ) = G(ξ, x) -4 0, l d2 u = f(x) u(0) = u(l) = 0 2 p(x) = 1 u = 0 c 1 c 2 u(x) = c 1 x + c 2 u(0) = 0 u 1 (x) = c 1 x u(l) = 0 u 2 = c 1 (x l) A(ξ)u 1 (x) ( x < ξ) G(x, ξ) = B(ξ)u 2 (x) (ξ < x b) (36) (0) u 1 (0) u 2 (0) (0) = u 1 (0) u 2 (0) = u 1(0)u 2(0) u 2 (0)u 1(0) = c 2 1l (37) A(ξ) B(ξ) (34) A(ξ) = u 2(ξ) p(0) (0) = ξ l c 1 l B(ξ) = u 1(ξ) p(0) (0) = ξ c 1 l (38) (36) x(ξ l) G(x, ξ) = l ξ(x l) l (0 x < ξ) (ξ < x l) (39) 13

(26) u(x) = l 0 G(x, ξ)f(ξ)dξ l ξ(x l) = dξ + l 0 l 0 (40) x(ξ l) dξ l -5 G(0, ξ) = G(1, ξ) = 0 ( ) d 2 2 + k2 G(x, ξ) = δ(x ξ) u + k 2 u = 0 u(0) = 0 u 1 (x) = c 1 sinkx u(1) = 0 u 2 = c 2 seck sink(1 x) (0) u 1 (0) u 2 (0) (0) = u 1 (0) u 2 (0) = u 1(0)u 2(0) u 2 (0)u 1(0) = c 1 c 2 k tn k (41) A(ξ) B(ξ) (34) A(ξ) = u 2(ξ) p(0) (0) B(ξ) = u 1(ξ) p(0) (0) = sink(1 ξ) c 1 k sink = cotk sin kξ c 2 k (42) (36) sinkxsink(1 ξ) G(x, ξ) = ksink sink(1 x)sinkξ ksink (0 x < ξ) (43) (ξ < x 1) 1.2 0 u(x) 14

Lu(x) = f(x) LG(x, ξ) = δ(x ξ) (44) 1 G(x, ξ) 2 u(x) G(x, ξ)lu(x) u(x)lg(x, ξ) = G(x, ξ)f(x) u(x)δ(x ξ) x b MEMO -2 G(x, ξ)p(x) du(x) ξ) b u(x)p(x)dg(x, = dξ G(x, ξ)f(x) u(ξ) x ξ G(x, ξ) = G(ξ, x) u(x) = G(x, ξ)f(ξ)dξ + dg(x, ξ) u(x)p(x) G(x, ξ)p(x) du(x) b (45) dξ (26) (45) 2 u(x) 1.2 u(x) 12 A. b u(t) (45) u () u (b) G(x, ) = G(x, b) = 0 u () u (b) u(x) = G(x, ξ)f(ξ)dξ + dg(x, ξ) b u(ξ)p(ξ) (46) dξ B. u () u (b) ( ) (45) u() u(b) 2 G (x, ) = G (x, b) = 0 u() u(b) u(x) = G(x, ξ)f(ξ)dξ G(x, ξ)p(ξ) du(ξ) b (47) C. x = b Au + Bu = C 3 AG BG = 0 15

Gu G u = C B G u(x) = G(x, ξ)f(ξ)dξ + G(x, ξ) C(ξ) b B(ξ) p(ξ) (48) -6 u(0) = 1 u(1) = 1 d 2 u(x) 2 + k 2 u(x) + f(x) = 0 x = 0 1 1 5 (49) u(x) = 1 0 G(x, ξ)f(ξ)dξ + dg(x, ξ) ξ=1 u(ξ) (49) dξ ξ=0 (43) sinkxsink(1 ξ) G(x, ξ) = ksink sink(1 x)sinkξ ksink (0 x < ξ) (ξ < x 1) (50) ξ ξ = 0 1 sinkxcosk(1 ξ) dg(x, ξ) (0 x < ξ) ξ = 1 : sinkx sink sink = dξ sink(1 x)coskξ sink(1 x) (ξ < x 1) ξ = 0 : sink sink (51) sink(1 x) u(x) = ksink sinkx sink x 0 sinkξf(ξ)dξ + sinkx ksink sink(1 x) u(1) u(0) sink 1 x sink(ξ 1)f(ξ)dξ (52) 2.2.3 3. Lu(x) + λρu(x) = 0 (53) Lu(x) + kρ(x)u(x) = f(x) (54) 16

(53) λ n (n = 0, 1, 2, ) u n Lu n (x) + λ n ρ(x)u(x) = 0 (55) u(x) u(x) = η n u n (x) (56) η n η n = n=0 (54) u n (x) (55) u(x) u(x)u n (x)ρ(x) (57) Luu n + kρuu n (Lu n u + λ n ρu n u) = fu n Luu n Lu n u + (k λ n )ρuu n = fu n (58) b L (58) (57) (Luu n Lu n u) = 0 (k λ n ) (k λ n )η n = ρ(x)u(x)u n (x) = f(x)u n (x) f(x)u n (x) (59) λ n k η n η n = 1 k λ n f(ξ)u n (ξ)dξ (60) (56) u(x) = = = ( 1 b ) f(ξ)u n (ξ)dξ u n (x) k λ n ( ) 1 u n (ξ)u n (x) f(ξ)dξ k λ n n=0 n=0 G(x, ξ)f(ξ)dξ (61) G(x, ξ) = n=0 1 k λ n u n (ξ)u n (x) (62) 17

(54) f(x) = δ(x ξ) (61) Lu(x) + kρ(x)u(x) = δ(x ξ) u(x) = = G(x, ξ) G(x, α)f(α)dα = G(x, α)δ(α ξ)dα (63) G(x, ξ) k = 0 G(x, ξ) = n=0 1 λ n u n (ξ)u n (x) (64) -7 G(0, ξ) = G(l, ξ) = 0 ( ) d 2 2 + k2 G(x, ξ) = δ(x ξ) d2 u 2 + k2 = 0, u(0) = u(l) = 0 c 1 c 2 u(x) = c 1 coskx + c 2 sinkx u(0) = 0 : u(l) = 0 : u(0) = c 1 = 0 u(x) = c 2 sinkx u(l) = c 2 sinkl = 0 k n = nπ l (n = 1, 2, ) (65) 13 λ n sin nπ l x ( nπ λ n = k2 2 = l ) 2 l 0 sin nπ l x 2 = 2 u n 2 u n (x) = sinnπ l x u n (x) u n (x)u n (ξ) = δ(x ξ) 13 n = 0 sinnx 0 18 n

G(x, ξ) = = 2 n=1 u n (x)u n (ξ) k 2 λ n sin nπ x sinnπ ξ (66) ( nπ ) 2 k 2 l n=1 2.2.4 4 d 2 G s dt 2 µ 2 G s = δ(t ξ) (µ > 0) (67) G s G s g(ω) = G s (t, ξ) = 1 2π G s (t, ξ)e iω(t ξ) dt g(ω)e iω(t ξ) dω (68) G s dg s dt = 1 2π d 2 G s dt 2 = 1 2π iωg(ω)e iω(t ξ) dω ω 2 g(ω)e iω(t ξ) dω (69) δ 1 = δ(t ξ) = 1 2π δ(t ξ)e iω(t ξ) dt 1 e iω(t ξ) dω (70) (67) ( ω 2 µ 2 )g(ω)e iω(t ξ) dω = g(ω) = 1 ω 2 + µ 2 1 e iω(t ξ) dω (71) 19

(68) G s (t, ξ) = 1 2π = 1 2π 1 ω 2 + µ 2 eiω(t ξ) dω 1 (ω iµ)(ω + iµ) eiω(t ξ) dω (72) G s (t, ξ) ±iµ C 1 14 G s (t, ξ) = 2πiResG s, iµ = 2πi lim (ω iµ) 1 ω iµ (ω iµ)(ω + iµ) eiω(t ξ) dω = 1 (73) 2µ e µ(t ξ) C 2 G s (t, ξ) = 2πiResG s, iµ = 2πi lim (ω + iµ) 1 ω iµ (ω iµ)(ω + iµ) eiω(t ξ) dω = 1 2µ eµ(t ξ) 1 2µ e µ(t ξ) (t > ξ) G s (t, ξ) = 1 2µ eµ(t ξ) (t < ξ) (74) (75) Im R 0 iµ iµ C 1 R ω Re C 2 (67) 15 G(t, ξ) G s (t, ξ) d2 u 2 µ2 u = 0 G(t, ξ) = G s (t, ξ) + k 1 e µt + k 2 e µt (76) G(0, ξ) = G(l, ξ) = 0 (0 < ξ < l) (77) 14 15 L(y) = R(x) L(y) = R(x) L(y) = 0 20

0 = G s (0, ξ) + k 1 + k 2 0 = G s (l, ξ) + k 1 e µt + k 2 e µt (78) k 1 k 2 ( 1 k 1 = 2µ e µ(t+ξ) 1 )/ 2µ e µ(t ξ) ( 1 k 2 = 2µ e µ(t ξ) 1 )/ 2µ e µ(t ξ) (79) = e µt e µt G(t, ξ) = G s (t, ξ) + 1 { e µt e µ(ξ t) + e µ(tξ)} e µ(l ξ t) e µ( l+ξ+t) (80) 2µ ***************************** ********************************* See you gin nd Goo Luck!! 1 2000.11 ) 2 1978.06 ( 3 2005.6 21

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