齐次方程

àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK àg§,àg>.^‡ff½)¯K úôŒÆêÆX ÅV= February 1, 2008 úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK ©lCþ{ ¦)k.«þ£‚5 ‡©§¤½)¯KffÄfl, ~^ff{; nØÄ:µ ‚5 ‡©§ffU\�nÚSturm-Liouvilleff kŁ¯K; ̇g´µ r ‡©§=z¦)~‡©§, r™ ¼êUffk¼êÐmL«¤2ÂFourier?ê; ̇":µ 鐧ff¦)«k›§«‡¦«m§ Ý/«§�§Î§¥�" úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK ˜˜˜‘‘‘ÅÅÅÄÄА§§§ffffffgggddd���ÄÄÄ ÃÃÃ999 ffffffkkk...\\\ffffff999DDD���¯¯¯KKK ���SSSffffff...ÊÊÊ...ddd(Laplace)§§§ ˜‘ÅЧffgd�Ä `ffüàff½ffuffgd�Ä, Œ±8Be�½)¯K: ∂2u ∂t2 = a2 ∂ 2u ∂x2 , 0 < x < `, t > 0 u(x , t)|x=0 = 0, u(x , t)|x=` = 0, u(x , t)|t=0 = φ(x), ∂u∂t |t=0 = ψ(x), 0 < x < `. Ù¥φ(x)Úψ(x)©OL«uffЩ £ÚЩ„Ý" ½)¯KffêÆA:´µ§§§ÚÚÚ>>>...^^^‡‡‡ÑÑÑ´´´àààggg‚‚‚555ffffff§ kЩ^‡´šàgff" úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK ˜˜˜‘‘‘ÅÅÅÄÄА§§§ffffffgggddd���ÄÄÄ ÃÃÃ999 ffffffkkk...\\\ffffff999DDD���¯¯¯KKK ���SSSffffff...ÊÊÊ...ddd(Laplace)§§§ ˜‘ÅЧffgd�Ä 111˜˜˜ÚÚÚµ¦Ñ÷v§ ∂2u ∂t2 = a2 ∂ 2u ∂x2 Ú>.^ ‡u(x , t)|x=0 = 0, u(x , t)|x=` = 0, …Œ©)  u(x , t) = X (x)T (t)£©lCþ¤ff¤kš")" §⇒ X (x)T ′′(t) = a2X ′′(x)T (t)⇒ X ′′(x) X (x) = T ′′(t) a2T (t) = −λ, Ù¥ −λ ~ê" ⇒ X ′′(x) + λX (x) = 0, T ′′(t) + λa2T (t) = 0. r u(x , t) = X (x)T (t) “\>.^‡⇒ X (0)T (t) = 0, X (`)T (t) = 0, X (x)ÚT (t) ÷v{ X ′′(x) + λX (x) = 0 (0 < x < `) X (0) = X (`) = 0 T ′′(t) + λa2T (t) = 0. úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK ˜˜˜‘‘‘ÅÅÅÄÄА§§§ffffffgggddd���ÄÄÄ ÃÃÃ999 ffffffkkk...\\\ffffff999DDD���¯¯¯KKK ���SSSffffff...ÊÊÊ...ddd(Laplace)§§§ ˜‘ÅЧffgd�Ä 111���ÚÚÚµµµ ¦ÑffkŁ¯K X ′′(x) + λX (x) = 0 (0 < x < `), X (0) = X (`) = 0. ff¤kffkŁÚƒAffffk¼ê" ��� λ < 0 žžž§§X ′′(x) + λX (x) = 0Ï) X (x) = C1e √−λx + C2e− √−λx Ù¥ C1 Ú C2 ~ê. X (0) = X (`) = 0⇒ { C1 + C2 = 0 C1e √−λ` + C2e− √−λ` = 0. ⇒ C1 = C2 = 0 ⇒ X (x) ≡ 0 ⇒ffkŁ¯K3λ < 0‰ŒSØ3 ffkŁÚffk¼ê" úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK ˜˜˜‘‘‘ÅÅÅÄÄА§§§ffffffgggddd���ÄÄÄ ÃÃÃ999 ffffffkkk...\\\ffffff999DDD���¯¯¯KKK ���SSSffffff...ÊÊÊ...ddd(Laplace)§§§ ˜‘ÅЧffgd�Ä ffkŁ¯K { X ′′(x) + λX (x) = 0 (0 < x < `), X (0) = X (`) = 0. ��� λ = 0 žžž§§X ′′(x) + λX (x) = 0Ï) X (x) = C1x + C2 X (0) = X (`) = 0⇒ C2 = 0, C1`+ C2 = 0⇒ C1 = C2 = 0 ⇒ X (x) ≡ 0 ⇒ λ = 0Ø´ffkŁ" úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK ˜˜˜‘‘‘ÅÅÅÄÄА§§§ffffffgggddd���ÄÄÄ ÃÃÃ999 ffffffkkk...\\\ffffff999DDD���¯¯¯KKK ���SSSffffff...ÊÊÊ...ddd(Laplace)§§§ ˜‘ÅЧffgd�Ä ffkŁ¯KX ′′(x) + λX (x) = 0 (0 < x < `), X (0) = X (`) = 0. ��� λ > 0 žžž§§X ′′(x) + λX (x) = 0 Ï) X (x) = C1 cos √ λx + C2 sin √ λx X (0) = X (`) = 0⇒ C1 = 0, C1 cos √ λ`+ C2 sin √ λ` = 0. )� C1 = 0, C2 sin √ λ` = 0. �ykš")⇒ C2 6= 0 ⇒ sin√λ` = 0 ⇒ √λ` = npi, (n = 1, 2, 3, · · · )"Ïdλn = n2pi2`2 (n = 1, 2, 3, · · · ) žffkŁ¯Kkš")£ffk¼ê¤ Xn(x) = Cn sin npix ` , n = 1, 2, 3, · · · . úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK ˜˜˜‘‘‘ÅÅÅÄÄА§§§ffffffgggddd���ÄÄÄ ÃÃÃ999 ffffffkkk...\\\ffffff999DDD���¯¯¯KKK ���SSSffffff...ÊÊÊ...ddd(Laplace)§§§ ˜‘ÅЧffgd�Ä 111˜˜˜ÚÚÚÚÚÚ111���ÚÚÚnnnãããµµµ{ X ′′(x) + λX (x) = 0 (0 < x < `) X (0) = X (`) = 0, T ′′(t) + λa2T (t) = 0 λn = n2pi2 `2 (n = 1, 2, 3, · · · ) ffkŁ¯Kš")Xn(x) = Cn sin npix ` , n = 1, 2, 3, · · · . Tn(t) = an cos npiat ` + bn sin npiat ` , n = 1, 2, 3, · · · Œ©lCþff¤kš")£=÷v1˜Ú‡¦ff©lCþ )¤ un(x , t) = Xn(x)Tn(t) = ( An cos npiat ` + An sin npiat ` ) sin npix ` , Ù¥n = 1, 2, 3, · · · , An = anCn, Bn = bnCn" úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK ˜˜˜‘‘‘ÅÅÅÄÄА§§§ffffffgggddd���ÄÄÄ ÃÃÃ999 ffffffkkk...\\\ffffff999DDD���¯¯¯KKK ���SSSffffff...ÊÊÊ...ddd(Laplace)§§§ ˜‘ÅЧffgd�Ä 111nnnÚÚÚµµµ )ffU\§¦Ñ÷vÐ>Ł¯Kff)" ∂2u ∂t2 = a2 ∂ 2u ∂x2 , 0 < x < `, t > 0 u(x , t)|x=0 = 0, u(x , t)|x=` = 0, u(x , t)|t=0 = φ(x), ∂u∂t |t=0 = ψ(x), 0 < x < `. ‚5§ffU\�n>.^‡ffàg5⇒ ?꣐‡?ê ñ¤ u(x , t) = ∞∑ n=1 un(x , t) = ∞∑ n=1 ( An cos npiat ` + Bn sin npiat ` ) sin npix ` . ÷v§Ú>.^‡"À� An Ú Bn ¦�§÷vЩ^‡: úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK ˜˜˜‘‘‘ÅÅÅÄÄА§§§ffffffgggddd���ÄÄÄ ÃÃÃ999 ffffffkkk...\\\ffffff999DDD���¯¯¯KKK ���SSSffffff...ÊÊÊ...ddd(Laplace)§§§ ˜‘ÅЧffgd�Ä u(x , t)|t=0 = ∞∑ n=1 An sin npix ` = φ(x), 0 < x < ` ∂u ∂t |t=0 = ∞∑ n=1 npia ` Bn sin npix ` = ψ(x), 0 < x < `. ⇒ An = 2 ` ∫ ` 0 φ(x) sin npix ` dx ⇒ Bn = 2 npia ∫ ` 0 ψ(x) cos npix ` dx . ⇒ u(x , t) = ∞∑ n=1 ( An cos npiat ` + Bn sin npiat ` ) sin npix ` ´Ð>Ł¯Kff)" úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK ˜˜˜‘‘‘ÅÅÅÄÄА§§§ffffffgggddd���ÄÄÄ ÃÃÃ999 ffffffkkk...\\\ffffff999DDD���¯¯¯KKK ���SSSffffff...ÊÊÊ...ddd(Laplace)§§§ ˜‘ÅЧffgd�Äff˜‡~f ~~~. ^©lCþ{¦) vtt − 4vxx = 0, 0 < x < 2pi, t > 0 vx(0, t) = 0, v(2pi, t) = 0, v(x , 0) = −10 cos( x4 ), vt(x , 0) = 0. ))) 111˜˜˜ÚÚÚ: v(x , t) = X (x)T (t)“\§ ⇒ X (x)T ′′(t) = a2X ′′(x)T (t)⇒ X ′′(x) X (x) = T ′′(t) a2T (t) = −λ, “\>.^‡⇒ X ′(0)T (t) = 0, X (2pi)T (t) = 0, X (x)ÚT (t) ÷ v { X ′′(x) + λX (x) = 0 (0 < x < 2pi) X ′(0) = X (2pi) = 0 T ′′(t) + λa2T (t) = 0. úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK ˜˜˜‘‘‘ÅÅÅÄÄА§§§ffffffgggddd���ÄÄÄ ÃÃÃ999 ffffffkkk...\\\ffffff999DDD���¯¯¯KKK ���SSSffffff...ÊÊÊ...ddd(Laplace)§§§ ˜‘ÅЧffgd�Äff˜‡~f 111���ÚÚÚ: ?ØffkŁ¯K { X ′′(x) + λX (x) = 0, 0 < x < 2pi X ′(0) = 0, X (2pi) = 0 ��� λ < 0 žžž§§X ′′(x) + λX (x) = 0Ï) X (x) = C1e √−λx + C2e− √−λx Ù¥ C1 Ú C2 ~ê. X ′(0) = 0⇒ √−λC1 − √−λC2 = 0 X (2pi) = 0⇒ C1e2 √−λpi + C2e−2 √−λpi = 0. ⇒ C1 = C2 = 0 ⇒ X (x) ≡ 0 úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK ˜˜˜‘‘‘ÅÅÅÄÄА§§§ffffffgggddd���ÄÄÄ ÃÃÃ999 ffffffkkk...\\\ffffff999DDD���¯¯¯KKK ���SSSffffff...ÊÊÊ...ddd(Laplace)§§§ ˜‘ÅЧffgd�Äff˜‡~f ffkŁ¯K { X ′′(x) + λX (x) = 0 (0 < x < 2pi), X ′(0) = X (2pi) = 0. ��� λ = 0 žžž§§X ′′(x) + λX (x) = 0Ï) X (x) = C1x + C2 X ′(0) = X (2pi) = 0⇒ C1 = 0, 2piC1 + C2 = 0⇒ C1 = C2 = 0 ⇒ X (x) ≡ 0 ⇒ λ = 0Ø´ffkŁ" úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK ˜˜˜‘‘‘ÅÅÅÄÄА§§§ffffffgggddd���ÄÄÄ ÃÃÃ999 ffffffkkk...\\\ffffff999DDD���¯¯¯KKK ���SSSffffff...ÊÊÊ...ddd(Laplace)§§§ ˜‘ÅЧffgd�Äff˜‡~f ffkŁ¯KX ′′(x)+λX (x) = 0 (0 < x < 2pi), X ′(0) = X (2pi) = 0. ��� λ > 0 žžž§§X ′′(x) + λX (x) = 0 Ï) X (x) = C1 cos √ λx + C2 sin √ λx X ′(0) = X (`) = 0⇒ C2 √ λ = 0, C1 cos 2pi √ λ+ C2 sin 2pi √ λ = 0. )� C2 = 0, C1 cos 2pi √ λ = 0. �ykš")⇒ C1 6= 0 ⇒ cos 2pi√λ = 0 ⇒ 2pi√λ = (n − 1/2)pi, (n = 1, 2, 3, · · · )"Ï dλn = ( (2n−1) 4 )2 (n = 1, 2, 3, · · · ) žffkŁ¯Kkš")£ffk ¼ê¤ Xn(x) = cn cos (2n − 1)x 4 , n = 1, 2, 3, · · · . úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK ˜˜˜‘‘‘ÅÅÅÄÄА§§§ffffffgggddd���ÄÄÄ ÃÃÃ999 ffffffkkk...\\\ffffff999DDD���¯¯¯KKK ���SSSffffff...ÊÊÊ...ddd(Laplace)§§§ ˜‘ÅЧffgd�Äff˜‡~f 111˜˜˜ÚÚÚÚÚÚ111���ÚÚÚnnnãããµµµ{ X ′′(x) + λX (x) = 0 (0 < x < 2pi) X ′(0) = X (2pi) = 0, T ′′(t) + λa2T (t) = 0 λn = ( 2n−1 4 )2 ž(n = 1, 2, 3, · · · ) ffkŁ¯Kš")Xn(x) = cn cos (2n−1)x 4 , n = 1, 2, 3, · · · . Tn(t) = an cos (2n−1)t 2 + bn sin (2n−1)t 2 , n = 1, 2, 3, · · · Œ©lCþff¤kš")£=÷v1˜Ú‡¦ff©lCþ )¤ vn(x , t) = [An cos (2n − 1)t 2 +Bn sin (2n − 1)t 2 ] cos (2n − 1)x 4 , Ù¥n = 1, 2, 3, · · · , An = anCn, Bn = bnCn" úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK ˜˜˜‘‘‘ÅÅÅÄÄА§§§ffffffgggddd���ÄÄÄ ÃÃÃ999 ffffffkkk...\\\ffffff999DDD���¯¯¯KKK ���SSSffffff...ÊÊÊ...ddd(Laplace)§§§ ˜‘ÅЧffgd�Äff˜‡~f ½)¯K:  vtt − 4vxx = 0, 0 < x < 2pi, t > 0 vx(0, t) = 0, v(2pi, t) = 0, v(x , 0) = −10 cos( x4 ), vt(x , 0) = 0. 111nnnÚÚÚ: v(x , t) = ∑∞ n=1 vn(x , t) =∑∞ n=1[An cos (2n−1)t 2 + Bn sin (2n−1)t 2 ] cos (2n−1)x 4 , ÷v§±9> Ł^‡, À�An9Bn¦÷vÐŁ^‡, v(x , 0) = −10 cos(x 4 )⇒ ∞∑ n=1 An cos (2n − 1)x 4 = −10 cos x 4 , vt(x , 0) = 0⇒ ∞∑ n=1 Bn(2n − 1) 2 cos (2n − 1)x 4 = 0. )�A1 = −10, An = 0(n = 2, 3, · · · ), Bn = 0(n = 1, 2, · · · ). ⇒ v(x , t) = −10 cos( t 2 ) cos( x 4 ). úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK ˜˜˜‘‘‘ÅÅÅÄÄА§§§ffffffgggddd���ÄÄÄ ÃÃÃ999 ffffffkkk...\\\ffffff999DDD���¯¯¯KKK ���SSSffffff...ÊÊÊ...ddd(Laplace)§§§ Ã9 ffk.\ff9D�¯K Äݏ`, ý¡ý9…gfiØ�)9þ§˜à§Ýð" , ˜àý9ffk.\, §ff§Ý©ÙG¹Œ8( ∂u ∂t = a 2 ∂2u ∂x2 , 0 < x < `, t > 0 u(x , t)|x=0 = 0, ∂u(x ,t)∂x |x=` = 0, u(x , t)|t=0 = φ(x), 0 < x < `. Ù¥φ(x)´§ffЩ§Ý" úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK ˜˜˜‘‘‘ÅÅÅÄÄА§§§ffffffgggddd���ÄÄÄ ÃÃÃ999 ffffffkkk...\\\ffffff999DDD���¯¯¯KKK ���SSSffffff...ÊÊÊ...ddd(Laplace)§§§ Ã9 ffk.\ff9D�¯K ∂u ∂t = a 2 ∂2u ∂x2 , 0 < x < `, t > 0 u(x , t)|x=0 = 0, ∂u(x ,t)∂x |x=` = 0, u(x , t)|t=0 = φ(x), 0 < x < `. 111˜˜˜ÚÚÚµµµ ¦Ñ÷vàg§Úàg>.^‡ffŒ�  u(x , t) = X (x)T (t)£©lCþ¤ff¤kš" )"u(x , t) = X (x)T (t) “\àg§∂u∂t = a 2 ∂2u ∂x2 ⇒ X (x)T ′(t) = a2X ′′(x)T (t)⇒ X”(x) X (x) = T ′(t) a2T (t) = −λ, ¼ê X (x) Ú T (t) ©O÷v X ′′(x) + λX (x) = 0, T ′(t) + λa2T (t) = 0. u(x , t) = X (x)T (t) “\>.^‡u(x , t)|x=0 = 0, ∂u(x ,t)∂x |x=` = 0 ⇒ X (0)T (t) = 0, X ′(`)T (t) = 0,⇒ X (0) = X ′(`) = 0. úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK ˜˜˜‘‘‘ÅÅÅÄÄА§§§ffffffgggddd���ÄÄÄ ÃÃÃ999 ffffffkkk...\\\ffffff999DDD���¯¯¯KKK ���SSSffffff...ÊÊÊ...ddd(Laplace)§§§ Ã9 ffk.\ff9D�¯K dd¼ê X (x) Ú T (t) ©O÷v{ X ′′(x) + λX (x) = 0, X (0) = X ′(`) = 0. T ′(t) + λa2T (t) = 0. úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK ˜˜˜‘‘‘ÅÅÅÄÄА§§§ffffffgggddd���ÄÄÄ ÃÃÃ999 ffffffkkk...\\\ffffff999DDD���¯¯¯KKK ���SSSffffff...ÊÊÊ...ddd(Laplace)§§§ Ã9 ffk.\ff9D�¯K { X ′′(x) + λX (x) = 0, X (0) = X ′(`) = 0. T ′(t) + λa2T (t) = 0. 111���ÚÚÚµµµ ¦ÑffkŁ¯KX ′′(x) + λX (x) = 0, X (0) = X ′(`) = 0 ff¤kffffkŁÚffk¼ê" I � λ < 0 ž§Ï)X (x) = C1e √−λx + C2e− √−λx . X (0) = 0⇒ C1 + C2 = 0 X ′(`) = 0⇒ C1 √−λe √−λ` + C2 √−λe− √−λ` = 0. ⇒ C1 = C2 = 0 ⇒ X (x) = 0 k")" úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK ˜˜˜‘‘‘ÅÅÅÄÄА§§§ffffffgggddd���ÄÄÄ ÃÃÃ999 ffffffkkk...\\\ffffff999DDD���¯¯¯KKK ���SSSffffff...ÊÊÊ...ddd(Laplace)§§§ Ã9 ffk.\ff9D�¯K ffkŁ¯K X ′′(x) + λX (x) = 0, X (0) = X ′(`) = 0. I � λ = 0ž§Ï)X (x) = C1x + C2, X (0) = X ′(`) = 0⇒ C2 = 0, C1 = 0⇒ C1 = C2 = 0⇒ X (x) = 0 ffkŁ¯Kk")" úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK ˜˜˜‘‘‘ÅÅÅÄÄА§§§ffffffgggddd���ÄÄÄ ÃÃÃ999 ffffffkkk...\\\ffffff999DDD���¯¯¯KKK ���SSSffffff...ÊÊÊ...ddd(Laplace)§§§ Ã9 ffk.\ff9D�¯K { X ′′(x) + λX (x) = 0 X (0) = X ′(`) = 0 T ′(t) + λa2T (t) = 0 I � λ > 0 ž§Ï)X (x) = C1 cos √ λx + C2 sin √ λx X (0) = 0⇒ C1 = 0, X ′(`) = 0⇒ −C1 √ λ sin √ λ`+ C2 √ λ cos √ λ` = 0. ⇒ C1 = 0, C2 cos √ λ` = 0 (C2 6= 0) ⇒ cos √ λ` = 0 ⇒ √λ` = (n − 1/2)pi, (n = 1, 2, 3, · · · )" B λn = (n−1/2) 2pi2 `2 B Xn(x) = Cn sin (n−1/2)pix` , B Tn(t) = ane (n−1/2)2pi2a2 `2 t , úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK ˜˜˜‘‘‘ÅÅÅÄÄА§§§ffffffgggddd���ÄÄÄ ÃÃÃ999 ffffffkkk...\\\ffffff999DDD���¯¯¯KKK ���SSSffffff...ÊÊÊ...ddd(Laplace)§§§ Ã9 ffk.\ff9D�¯K B Xn(x) = Cn sin (n−1/2)pix` , B Tn(t) = ane (n−1/2)2pi2a2 `2 t , àg§Ú>.^‡{ ∂u ∂t = a 2 ∂2u ∂x2 , 0 < x < `, t > 0 u(x , t)|x=0 = 0, ∂u(x ,t)∂x |x=` = 0, ff¤kŒUffŒCþ©lffš"): B un(x , t) = Xn(x)Tn(t) = Ane (n−1/2)2pi2a2 `2 t sin (n−1/2)pix` (n = 1, 2, 3, · · · ) úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK ˜˜˜‘‘‘ÅÅÅÄÄА§§§ffffffgggddd���ÄÄÄ ÃÃÃ999 ffffffkkk...\\\ffffff999DDD���¯¯¯KKK ���SSSffffff...ÊÊÊ...ddd(Laplace)§§§ Ã9 ffk.\ff9D�¯K un(x , t) = Xn(x)Tn(t) = Ane (n−1/2)2pi2a2 `2 t sin (n−1/2)pix` 111nnnÚÚÚµµµ U\�nÚ>.^‡ffàg5Ÿ ⇒ u(x , t) = ∞∑ n=1 un(x , t) = ∞∑ n=1 Ane (n−1/2)2pi2a2 `2 t sin (n − 1/2)pix ` ÷vàg§±9>.^‡. é�·�ffAn ¦�§U÷vЩ^ ‡ u(x , t)|t=0 = φ(x)⇒ ∞∑ n=1 An sin (n − 1/2)pix ` = φ(x), ⇒ An = 2 ` ∫ ` 0 φ(x) sin (n − 1/2)pix ` dx , n = 1, 2, 3, · · · úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK ˜˜˜‘‘‘ÅÅÅÄÄА§§§ffffffgggddd���ÄÄÄ ÃÃÃ999 ffffffkkk...\\\ffffff999DDD���¯¯¯KKK ���SSSffffff...ÊÊÊ...ddd(Laplace)§§§ �Sff.Ê.d(Laplace)§ Œ» aff���§þeü¡ý9§���>�ff§Ý  ψ(x , y), SÜØ�)9þ,²L˜ãžm�§���ff§Ýˆ �²ï§¦§Ý©Ù u(x , y)? ∂2u ∂x2 + ∂ 2u ∂y2 = 0, x2 + y2 ≤ a2, |u(x , y)| < +∞, x2 + y2 ≤ a2, u(x , y) = ψ(x , y), x2 + y2 = a2. úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK ˜˜˜‘‘‘ÅÅÅÄÄА§§§ffffffgggddd���ÄÄÄ ÃÃÃ999 ffffffkkk...\\\ffffff999DDD���¯¯¯KKK ���SSSffffff...ÊÊÊ...ddd(Laplace)§§§ �Sff.Ê.d(Laplace)§ { ∂2u ∂x2 + ∂ 2u ∂y2 = 0, |u(x , y)| < +∞, x2 + y2 ≤ a2, u(x , y) = ψ(x , y), x2 + y2 = a2. Ú?4‹Ix = r cos θ, y = r sin θ, ⇒ u(r , θ) = u(x , y), φ(θ) = ψ(a cos θ, a sin θ), ⇒ ∂ 2u ∂x2 + ∂2u ∂y2 = 1 r ∂ ∂r (r ∂u ∂r ) + 1 r2 ∂2u ∂θ2 , ⇒  1 r ∂ ∂r (r ∂u ∂r ) + 1 r2 ∂2u ∂θ2 = 0, 0 < r < a, 0 ≤ θ < 2pi, |u(r , θ)| < +∞, 0 < r < a, 0 ≤ θ < 2pi, u(r , θ) = u(r , θ + 2pi), 0 < r < a, 0 ≤ θ < 2pi, u(r , θ)|r=a = φ(θ). úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK ˜˜˜‘‘‘ÅÅÅÄÄА§§§ffffffgggddd���ÄÄÄ ÃÃÃ999 ffffffkkk...\\\ffffff999DDD���¯¯¯KKK ���SSSffffff...ÊÊÊ...ddd(Laplace)§§§ �Sff.Ê.d(Laplace)§ ïÄ  1 r ∂ ∂r (r ∂u ∂r ) + 1 r2 ∂2u ∂θ2 = 0, 0 < r < a, 0 ≤ θ < 2pi, |u(r , θ)| < +∞, u(r , θ) = u(r , θ + 2pi), u(r , θ)|r=a = φ(θ). 111˜˜˜ÚÚÚµµµ ¦Ñ÷vàg§Ú^‡ |u(r , θ)| < +∞ ± 9u(r , θ) = u(r , θ + 2pi)ffŒ� u(r , θ) = R(r)H(θ)£©lC þ¤ff¤kš")" u(r , θ) = R(r)H(θ) “\àg§ ⇒ (rR ′(r) + r2R ′′(r))H(θ) + H ′′(θ)R(r) = 0 ⇒ H′′(θ)H(θ) = − rR ′(r)+r2R′′(r) R(r) = −λ, ¼ê R(r) Ú H(θ)©O÷v H ′′(θ) + λH(θ) = 0, r2R ′′(r) + rR ′(r) + λR(r) = 0. úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK ˜˜˜‘‘‘ÅÅÅÄÄА§§§ffffffgggddd���ÄÄÄ ÃÃÃ999 ffffffkkk...\\\ffffff999DDD���¯¯¯KKK ���SSSffffff...ÊÊÊ...ddd(Laplace)§§§ �Sff.Ê.d(Laplace)§ R(r) Ú H(θ)©O÷vŸo>.^‡? u(r , θ) = u(r , θ + 2pi) ⇒ R(r)H(θ) = R(r)H(θ + 2pi) ⇒ H(θ) = H(2pi + θ), |u(r , θ)| < +∞ ⇒ R(r)H(θ) < +∞⇒ |R(r)| < +∞. Ïd¼ê R(r) Ú H(θ) ©O÷v{ H ′′(θ) + λH(θ) = 0, H(θ) = H(2pi + θ), { r2R ′′(r) + rR ′(r) + λR(r) = 0, 0 ≤ r ≤ a |R(r)|k., 0 ≤ r ≤ a. úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK ˜˜˜‘‘‘ÅÅÅÄÄА§§§ffffffgggddd���ÄÄÄ ÃÃÃ999 ffffffkkk...\\\ffffff999DDD���¯¯¯KKK ���SSSffffff...ÊÊÊ...ddd(Laplace)§§§ �Sff.Ê.d(Laplace)§ { H ′′(θ) + λH(θ) = 0, H(θ) = H(2pi + θ), { r2R ′′(r) + rR ′(r) + λR(r) = 0, 0 ≤ r ≤ a |R(r)|k., 0 ≤ r ≤ a. 111���ÚÚÚµµµ ¦ÑffkŁ¯KH ′′(θ) + λH(θ) = 0, H(θ) = H(2pi + θ) ff¤kffffkŁÚffk¼ê" I � λ < 0 ž§Ï)H(θ) = C1e √−λθ + C2e− √−λθ. d^ ‡H(θ) = H(2pi + θ) ⇒ C1(1− e2pi √−λ)e √−λθ + C2(1− e−2pi √−λ)e− √−λθ = 0 ⇒ C1 = C2 = 0⇒ H(θ) = 0⇒k") úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK ˜˜˜‘‘‘ÅÅÅÄÄА§§§ffffffgggddd���ÄÄÄ ÃÃÃ999 ffffffkkk...\\\ffffff999DDD���¯¯¯KKK ���SSSffffff...ÊÊÊ...ddd(Laplace)§§§ �Sff.Ê.d(Laplace)§ { H ′′(θ) + λH(θ) = 0, H(θ) = H(2pi + θ), { r2R ′′(r) + rR ′(r) + λR(r) = 0, 0 ≤ r ≤ a |R(r)|k., 0 ≤ r ≤ a. I � λ = 0 ž§Ï)H(θ) = C1θ + C2, d^ ‡ H(θ) = H(2pi + θ) ⇒ 2piC1 = 0 ⇒ C1 = 0, C2 ?¿"ffk Ł¯KkffkŁλ0 = 0, ffk¼êH0(θ) = C0, Ù¥ C0 ?¿š "~ê" úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK ˜˜˜‘‘‘ÅÅÅÄÄА§§§ffffffgggddd���ÄÄÄ ÃÃÃ999 ffffffkkk...\\\ffffff999DDD���¯¯¯KKK ���SSSffffff...ÊÊÊ...ddd(Laplace)§§§ �Sff.Ê.d(Laplace)§ ¦ÑffkŁ¯KH ′′(θ) + λH(θ) = 0, H(θ) = H(2pi + θ)ff¤kff ffkŁÚffk¼ê" I � λ > 0 ž§Ï)H(θ) = C1 cos √ λθ + C2 sin √ λθ, d^ ‡H(θ) = H(2pi+ θ) ⇒ λ = n2 (n = 1, 2, 3, · · · ), C1 Ú C2 ?¿ š"~ê. ffkŁÚffk¼ê λ = n2, Hn(θ) = Cn cos(nθ) + Dn sin(nθ), n = 1, 2, 3, · · · . nÜ: ffkŁ¯KH ′′(θ) + λH(θ) = 0, H(θ) = H(2pi + θ)ff¤k ffffkŁÚffk¼ê: ffkŁλn = n 2(n = 0, 1, 2, 3, · · · ) ffk¼êHn(θ) = Cn cos(nθ) + Dn sin(nθ) (n = 0, 1, 2, 3, · · · ) úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK ˜˜˜‘‘‘ÅÅÅÄÄА§§§ffffffgggddd���ÄÄÄ ÃÃÃ999 ffffffkkk...\\\ffffff999DDD���¯¯¯KKK ���SSSffffff...ÊÊÊ...ddd(Laplace)§§§ �Sff.Ê.d(Laplace)§ { H ′′(θ) + λH(θ) = 0, H(θ) = H(2pi + θ), { r2R ′′(r) + rR ′(r) + λR(r) = 0, 0 ≤ r ≤ a |R(r)|k., 0 ≤ r ≤ a. éλn = n 2(n = 0, 1, 2, 3, · · · )§H(θ)kš") Hn(θ) = Cn cos(nθ) + Dn sin(nθ), ùžR(r)´ÄEkš")?{ r2R ′′(r) + rR ′(r) + n2R(r) = 0, 0 ≤ r ≤ a |R(r)|k., 0 ≤ r ≤ a. ´˜‡ îîî...§§§. ŁC†r = et , R(r) = R(t), r2R ′′(r) + rR ′(r) + n2R(r) = 0⇒ R ′′(t) + n2R(t) = 0 ⇒ Rn(t) = { a0 + b0t, n = 0 ane nt + bne −nt , n = 1, 2, · · · úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK ˜˜˜‘‘‘ÅÅÅÄÄА§§§ffffffgggddd���ÄÄÄ ÃÃÃ999 ffffffkkk...\\\ffffff999DDD���¯¯¯KKK ���SSSffffff...ÊÊÊ...ddd(Laplace)§§§ �Sff.Ê.d(Laplace)§ { H ′′(θ) + λH(θ) = 0, H(θ) = H(2pi + θ), { r2R ′′(r) + rR ′(r) + λR(r) = 0, 0 ≤ r ≤ a |R(r)|k., 0 ≤ r ≤ a. r2R ′′(r) + rR ′(r) + n2R(r) = 0⇒ R ′′(t) + n2R(t) = 0 ⇒ Rn(r) = { a0 + b0 ln r , n = 0 anr n + bnr −n, n = 1, 2, · · · d |R(r)| < +∞ ⇒ Rn(r) = anrn, n = 0, 1, 2, · · · , úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK ˜˜˜‘‘‘ÅÅÅÄÄА§§§ffffffgggddd���ÄÄÄ ÃÃÃ999 ffffffkkk...\\\ffffff999DDD���¯¯¯KKK ���SSSffffff...ÊÊÊ...ddd(Laplace)§§§ �Sff.Ê.d(Laplace)§  1 r ∂ ∂r (r ∂u ∂r ) + 1 r2 ∂2u ∂θ2 = 0, 0 < r < a, 0 ≤ θ < 2pi, |u(r , θ)| < +∞, u(r , θ) = u(r , θ + 2pi), u(r , θ)|r=a = φ(θ). �λn = n 2(n = 0, 1, 2, · · · )ž, Hn(θ) = Cn cos(nθ) + Dn sin(nθ), Rn(r) = anr n, nã: ÷vàg§Ú^‡ |u(r , θ)| < +∞± 9u(r , θ) = u(r , θ + 2pi)ff¤kŒ©lCþffš") un(r , θ) = r n(An cos(nθ) + Bn sin(nθ)), n = 0, 1, 2, 3, · · · , Ù¥ An Ú Bn ´?¿ØӞš"ff~ê" úúúôôôŒŒŒÆÆÆêêêÆÆÆXXX ÅÅÅVVV=== àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK àààggg§§§,àààggg>>>...^^^‡‡‡ffffff½½½)))¯¯¯KKK ˜˜˜‘‘‘ÅÅÅÄÄА§§§ffffffgggddd���ÄÄÄ ÃÃÃ999 ffffffkkk...\\\ffffff999DDD���¯¯¯KKK ���SSSffffff...ÊÊÊ...ddd(Laplace)§§§ �Sff.Ê.d(Laplace)§  1 r ∂ ∂r (r ∂u ∂r ) + 1 r2 ∂2u ∂θ2 = 0, 0 < r < a, 0 ≤ θ < 2pi, |u(r , θ)| < +∞, u(r , θ) = u(r , θ + 2pi), u(r , θ)|r=a = φ(θ). 111nnnÚÚÚµµµ)))ffffffUUU\\\§§§¦¦¦ÑÑÑ÷÷÷vvv>>>ŁŁŁ¯¯¯KKKffffff)))"‚5§ffU\ �n±9^‡ |u(r , θ)| < +∞±9u(r , θ) = u(r , θ+ 2pi)ffŒ\5, u(r , θ) = ∞∑ n=0 un(r , θ) = A0 + ∞∑ n=1 rn(An cos(nθ) + Bn sin(nθ)), ÷v§Ú |u(r , θ)| < +∞