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# chpp19 - 19 C A^ K — C ~ ^ C k Laplace C Fourior C 19.1...

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Unformatted text preview: 19 C A^ K , { — C . ~ ^ C k Laplace C Fourior C . 19.1 Laplace C Laplace C ~ ^u m K . u X t ' , C C ' 5 ~ . ` 5 , ' N . , , , 7 L , U K . Example 19.1 ^ Laplace C { K ∂ 2 u ∂t 2 = a 2 ∂ 2 u ∂x 2 ≤ x ≤ l (1a) u t =0 = 0 ∂u ∂t t =0 = 0 (1b) u x =0 = 0 ∂u ∂x x = l = E sin ωt (1c) Solution 1. Laplace C L { u ( x, t ) } = U ( x, p ) (2) u L ∂ 2 u ∂t 2 = p 2 U ( x, p ) (3) L ∂ 2 u ∂x 2 = d 2 U ( x, p ) d x 2 (4) p , 3 (3) , | ^ ^ . K C a 2 d 2 U ( x, p ) d x 2- p 2 U ( x, p ) = 0 (5a) U ( x, p ) x =0 = 0 d U ( x, p ) d x x =0 = E ω p 2 + ω 2 (5b) 5 C C C x ~ ( , C p ). U ( x, p ) = C 1 cosh px a + C 2 sinh px a (6) d > . ^ C 1 = 0 C 2 = Eωa p ( p 2 + ω 2 ) cosh pl a U ( x, p ) = Eωa p ( p 2 + ω 2 ) sinh px a cosh pl a (7) 2. 1 d Laplace C H u ( x, t ) = 1 2 π i Z s +i ∞ s- i ∞ Eωa e pt p ( p 2 + ω 2 ) sinh px a cosh pl a d p (8) k { : p j =0 , ± i ω, ± i πa l k- 1 2 ≡ ± i ω k ( k = 1 , 2 , 3 , ... ) E k u ( x, t ) = X j res    Eωae pt p ( p 2 + ω 2 ) sinh px a cosh pl a    p = p j (9) : ? 3 , u ( x, t ) = 0 + Ea 2i ω sin ωx a cos ωl a e i ωt- Ea 2i ω sin ωx a cos ωl a e- i ωt + ∞ X k =1   Eωa i ω k ( ω 2 k- ω 2 ) a sin ω k x a (- 1) k l e i ωt- Eωa i ω k ( ω 2 k- ω 2 ) a sin ω k x a (- 1) k l e- i ωt   u ( x, t ) = Ea ω sin ωx a cos ωl a sin ωt + 2 Eωa 2 l ∞ X k =1 (- 1) k sin ω k x a ω k ( ω 2 k- ω 2 ) sin ωt (10) ~ K g ~ K 14.4, k g > . ^ . ^ l C { , I k > . ^ g z . ^ Laplace C { , I k > . ^ g z ; l C { @ d ^ X , ^ L Laplace C ? U ( x, p ) . Example 19.2 . 9 D K ∂u ∂t- κ ∂ 2 u ∂x 2 = f ( x, t ) ,- ∞ < x < ∞ , t > 0; (11a) u t =0 = 0 ,- ∞ < x < ∞ . (11b) Solution 3 . m K , > . ^ . S , . m n : f ( x, t ) . 3 t 3 x ? m 9 ( L z A , > 6 , ). n , . K A u 3 k m S , = f ( x, t ) A v f ( ∞ , t ) = 0 f (-∞ , t ) = 0 2 u , ? A C . X J / K { , K Ak > . ^ u ±∞ → (12) y 3 Laplace C .- L { u ( x, t ) } = U ( x, p ) (13) L { f ( x, t ) } = F ( x, p ) (14) K C pU ( x, p )- κ d 2 U ( x, p ) d x 2 = F ( x, p ) (15) > . ^ K U ( ±∞ , p ) = 0 (16) 5 g , ^ ~ C { . A g...
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chpp19 - 19 C A^ K — C ~ ^ C k Laplace C Fourior C 19.1...

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