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y²L§©¤nÚµ Äky²‡©•§ÐŠ¯K(1) duÈ©•§ y(x) = y0+ Z x x0 f (t, y(t))dt (2) Ùg^PicardÅg%C{y²) •35" • y²) •˜5"Picard•3•˜5½n y²
y²L§©¤nÚµ Äky²‡©•§ÐŠ¯K(1) duÈ©•§ y(x) = y0+ Z x x0 f (t, y(t))dt (2) Ùg^PicardÅg%C{y²) •35" • y²) •˜5"Picard•3•˜5½n y²
y²L§©¤nÚµ Äky²‡©•§ÐŠ¯K(1) duÈ©•§ y(x) = y0+ Z x x0 f (t, y(t))dt (2) Ùg^PicardÅg%C{y²) •35" • y²) •˜5"Picard•3•˜5½n y²
y²L§©¤nÚµ Äky²‡©•§ÐŠ¯K(1) duÈ©•§ y(x) = y0+ Z x x0 f (t, y(t))dt (2) Ùg^PicardÅg%C{y²) •35" • y²) •˜5"5
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y x O R B A C D E 0 y b y0 b y0 a x0 x0h x0 x0h x0a ) , (x0 [0 ) , (x0 K0 G [0 0 [ ) ( :y x l [ ) ( : ~ x y l K : : w D ã1µPicard•3•˜5½nëêh AÛ¿Â (þ°ã²ŒÆA^êÆX) ~‡©•§ 1nÙ April 19, 2010 8 / 425
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3¢S¦^¥§Lipschitz^‡ Ju § ~^f(x, y)3Rþk éy ëY ê5O“"¢SþO“ ^‡'Lipschitz^‡•î ‚§•´‰1å5 ••B"XJ3Rþ∂f ∂y•3…ëY§@ o∂f ∂y3Rþk." 3Rþ ∂f ∂y ≤ L§ K∀(x, y1), (x, y2) ∈ R, 0 < θ < 1§k |f (x, y1) − f (x, y2)| = ∂f (x, y2+ θ(y1− y2)) ∂y
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b ü‡‰½•§ dy dx = f1(x, y), y(x0) = ξ0 (8) Ú dy dx = f2(x, y), y(x0) = η0 (9) Ù¥f1, f2 ∈ C[a, b] × (−∞, ∞)§…©O÷véy Lipschitz^‡" q éx ?˜«m[c, d] ⊂ (a, b)§•3ëY¼êδ(x)§¦Ø ª|f1(x, y) − f2(x, y)| ≤ δ(x), x ∈ [c, d] (10) 阃y¤á"2 y = ξ(x)´•§(9) ²L:(x0, ξ0) )-‚¶y = η(x)´•§(10) ²L:(x0, η0) )-‚§Ù
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½n3.7£)'u•§mà¼ê ëY5½n¤ y = ξ(x), x ∈ [x0, X], X < b´•§ (12) ²L(x0, ξ0) )"K∀ε > 0§∃δ(x) ≥ 0§δ(x) ∈ C[x0, X]§ é?Û•§ dy dx = g(x, y) (14) •‡g(x, y)3«•D¥ëY§÷vÛÜLipschitz^‡±9±eØ ª |f (x, y) − g(x, y)| ≤ δ(x), x0 ≤ x ≤ X, −∞ < y < ∞ @o(14) ²L(x0, ξ0) È©-‚y = η(x)•73[x0, X]þk½ §¿…3[x0, X]þ÷vØ ª(13)")éëê
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éu•§mà¹këêλ ‡©•§ dy dx = f (x, y; λ) (15) P Dλ = {(x, y, λ)|(x, y) ∈ D, α < λ < β} (16) f (x, y; λ)3DλSëY§…'uy÷vÛÜLipschitz^‡§ ÙLipschitz~êL†ëêλÃ'"Kæ^aq½n3.6 y²•{§ Œ± ±e )éëê ëY5½n")éëê
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˜„§éu‡©•§½)¯K ( dy dx = f (x, y; λ) y|x=x0 = y0 (18) ¥ Њ(x0, y0)Úëêλ§·‚?ØbX(x0, y0, λ)CħKƒA Њ¯K(18) )‘ƒXÛ?1CÄ"•Ò´`§ÐŠ¯K )Ø==•6ugCþx§Óž••6uЊ(x0, y0)Úëêλ" ½n3.9£)éЊÚëê ëYŒ‡5½n¤ •§(18)¥ f(x, y, λ) (x, y) ∈ D§λ ∈ (α, β) = I´ëY¼ ꧅'ux, y, λkëY ê"K•§(18) )y(x, x0, y0, λ)k 'ux0§y0Úλ ëY ê"Sturm-Liouville¯K
3¦)êÆÔn Nõ¯Kž§~~¬- ¦)Xe/ª ~‡ ©•§ A Š¯K£¡•Sturm-Liouville¯K¤ d dx k(x)dy dx − q(x)y(x) + λρ(x)y(x) = 0, (a < x < b) (19) −α1dy dx + β1y x=a = 0, α2 dy dx + β2y x=b = 0 (20)Ù¥λ•A Š§éA š")•A ¼ê"(19)¥ k(x)!q(x)Úρ(x)•x3(a, b)¥¿©1w ¼ê§ …
x ∈ (a, b)ž§k(x) > 0, ρ(x) > 0, q(x) ≥ 0"ex = a•k(x) ˜ ":§K‡¦A ¼êy(x)3x = aC k.¶ex = b•k(x) ˜ ":§K‡¦y(x)3x = bC k."ùp ρ(x)¤• ¼
Sturm-Liouville¯K¥•¹ 錘a>.^
‡
A
Š¯K
(20)¥•¹ 錘a>.^‡ A Š¯K"~X§ eα1 = 0§K3x = a?•1˜a>.^‡¶eβ1 = 0§K 3x = a?•1 a>.^‡¶eα1 6= 0!α2 6= 0§K3x = a? •1na>.^‡"éux = b? >.^‡•kaq ?Ø" ,§¯K(20)vkr±Ï>.^‡•)? "éu±Ï>.^ ‡¯K§•kaq ?اùpÒØ?Ø "Sturm-Liouville¯K¥•¹ 錘a~‡©
•§
A
Š¯K
(19)•¹ 錘a~‡©•§ A Š¯K"~X§ k(x) = x, q(x) = νx2, ρ(x) = x (0 < x < l)ž§(19)Òz •ν Bessel•§ d dx xdy dx −ν 2 xy + λxy = 0 qX§ k(x) = 1−x1 2, q(x) ≡ 0, ρ(x) ≡ 1 (−1 < x < 1)ž§(19)Òz •Legendre•§ d dx (1 − x2)dy dx + λy = 0 2X§(19)Òz•·‚õg-Sturm-Liouville¯K¥•¹ 錘a~‡©
•§
A
Š¯K
(19)•¹ 錘a~‡©•§ A Š¯K"~X§ k(x) = x, q(x) = νx2, ρ(x) = x (0 < x < l)ž§(19)Òz •ν Bessel•§ d dx xdy dx −ν 2 xy + λxy = 0 qX§ k(x) = 1−x1 2, q(x) ≡ 0, ρ(x) ≡ 1 (−1 < x < 1)ž§(19)Òz •Legendre•§ d dx (1 − x2)dy dx + λy = 0 2X§ k(x) ≡ 1, q(x) ≡ 0, ρ(x) ≡ 1ž§ (19)Òz•·‚õg-~‡©•§ y00(x) + λy(x) = 0Sturm-Liouville¯K¥•¹ 錘a~‡©
•§
A
Š¯K
(19)•¹ 錘a~‡©•§ A Š¯K"~X§ k(x) = x, q(x) = νx2, ρ(x) = x (0 < x < l)ž§(19)Òz •ν Bessel•§ d dx xdy dx −ν 2 xy + λxy = 0 qX§ k(x) = 1−x1 2, q(x) ≡ 0, ρ(x) ≡ 1 (−1 < x < 1)ž§(19)Òz •Legendre•§ d dx (1 − x2)dy dx + λy = 0 2X§(19)Òz•·‚õg-Sturm-Liouville¯K¥•¹ 錘a~‡©
•§
A
Š¯K
(19)•¹ 錘a~‡©•§ A Š¯K"~X§ k(x) = x, q(x) = νx2, ρ(x) = x (0 < x < l)ž§(19)Òz •ν Bessel•§ d dx xdy dx −ν 2 xy + λxy = 0 qX§ k(x) = 1−x1 2, q(x) ≡ 0, ρ(x) ≡ 1 (−1 < x < 1)ž§(19)Òz •Legendre•§ d dx (1 − x2)dy dx + λy = 0 2X§ k(x) ≡ 1, q(x) ≡ 0, ρ(x) ≡ 1ž§ (19)Òz•·‚õg-~‡©•§ y00(x) + λy(x) = 0Sturm-Liouville¯K) 5Ÿ
5Ÿ1 A Šλ ≥ 0"AO § β1+ β2 > 0£=üàØÓž•1 a>Š¯K¤ž§¯K(19)!(20) ¤kA Šλ > 0 5Ÿ2 kŒ áõ‡šKA Šλ1, λ2, · · · ÷v 0 ≤ λ1 ≤ λ2 ≤ · · · , lim n→∞λn= +∞ 5Ÿ3 éAuØÓA Š A ¼ê3[a, b]¥´‘ "Sturm-Liouville¯K) 5Ÿ
5Ÿ1 A Šλ ≥ 0"AO § β1+ β2 > 0£=üàØÓž•1 a>Š¯K¤ž§¯K(19)!(20) ¤kA Šλ > 0 5Ÿ2 kŒ áõ‡šKA Šλ1, λ2, · · · ÷v 0 ≤ λ1 ≤ λ2 ≤ · · · , lim n→∞λn= +∞ 5Ÿ3 éAuØÓA Š A ¼ê3[a, b]¥´‘ "Sturm-Liouville¯K) 5Ÿ
5Ÿ1 A Šλ ≥ 0"AO § β1+ β2 > 0£=üàØÓž•1 a>Š¯K¤ž§¯K(19)!(20) ¤kA Šλ > 0 5Ÿ2 kŒ áõ‡šKA Šλ1, λ2, · · · ÷v 0 ≤ λ1 ≤ λ2 ≤ · · · , lim n→∞λn= +∞ 5Ÿ3 éAuØÓA Š A ¼ê3[a, b]¥´‘ "Sturm-Liouville¯K) 5Ÿ
5Ÿ1 A Šλ ≥ 0"AO § β1+ β2 > 0£=üàØÓž•1 a>Š¯K¤ž§¯K(19)!(20) ¤kA Šλ > 0 5Ÿ2 kŒ áõ‡šKA Šλ1, λ2, · · · ÷v 0 ≤ λ1 ≤ λ2 ≤ · · · , lim n→∞λn= +∞ 5Ÿ3 éAuØÓA Š A ¼ê3[a, b]¥´‘ "Sturm-Liouville¯K) 5Ÿ
5Ÿ4 éuÓ˜A Š§éA A ¼ê•õ•kk•‡"e,A ŠéA ‚5Ã'A ¼ êØŽ˜‡§|^ z•{§Œ¦ù A ¼êpƒ‘ ρ(x) "duéAØÓA Š A
¼ê´‘ ρ(x) §ù B [a, b]þ ‘ ρ(x) A ¼ê X{yn(x), n = 1, 2, · · · }§¦é[a, b]þ?˜²•ŒÈ¼êf (x)§ÑŒ±UdA ¼êX?
1FourierÐm§ f (x) = ∞ X n=1 cnyn(x) (21) Ù¥ cn= 1 σn Z b a f (x)yn(x)ρ(x)dx, n = 1, 2, · · · (22) (22)¥ σn• σn= Zb a ρ(x)[yn(x)]2dx, n = 1, 2, · · · (23) (21)¥ Âñ´3L2[a, b] ‰ê¿Âƒeµ lim Z b f (x) − n X c y (x) 2 dx 1 2 = 0 (24)
