# chpp14 - 14 l C K n — l C 14.1 u g d Example 14.1< x<...

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Unformatted text preview: 14 l C { K ( n ) { — l C { . 14.1 u g d Example 14.1 : < x < l , t > ∂ 2 u ∂t 2- a 2 ∂ 2 u ∂x 2 = 0 (1) > . ^ : t > u | x =0 = 0 u | x = l = 0 (2) ^ : ≤ x ≤ l u | t =0 = φ ( x ) ∂u ∂t t =0 = ψ ( x ) (3) Solution 1. l C . 2. K 3. A , U . 4. | ^ 5 U X . l C A ( " : " !) k l C / : u ( x,t ) = X ( x ) T ( t ) X ( x ) T 00 ( t ) = a 2 X 00 ( x ) T ( t ) a 2 X ( t ) T ( t ) 1 a 2 T 00 ( t ) T ( t ) = X 00 ( x ) X ( x ) =- λ λ = x ' , q t ' ~ . u T 00 ( t ) + λa 2 T ( t ) = 0 X 00 ( x ) + λX ( x ) = 0 2 l C > . ^ X (0) T ( t ) = 0 X ( l ) T ( t ) = 0 T ( t ) U " ( K u ( x,t ) " ), X (0) = 0 X ( l ) = 0 X ( x ) v ~ > . ^ ; 9 T ( t ) v ~ . 1 K ~ + > . ^ X 00 ( x ) + λX ( x ) = 0 X (0) = 0 X ( l ) = 0 K . K " . λ E 1 . e λ = 0, ~ X ( x ) = Ax + B > . ^ X (0) = B = 0 X ( l ) = Al + B = 0 ⇒ A = 0 , X ( x ) ≡ " . e λ 6 = 0, K X ( x ) = A sin √ λ x + B cos √ λ x λ 6 = 0 > . ^ X (0) = B = 0 X ( l ) = A sin √ λ l = 0 A 6 = 0, K qk X ( x ) ≡ 0. 7 k √ λ l = nπ ( n > 0) λ = nπ l 2 , P λ n = nπ l 2 A = 1 2 , " X n ( x ) = sin nπ l x w K u ~ K : X J ~ K , . X J X ( x ) ≡ , + ^ ! K , λ , A , " , k " 3 ; = . λ A λ n , A " X n ( x ) . A , U u z λ n , d T 00 ( t ) + λ n a 2 T ( t ) = 0 T n ( t ) = C n sin nπ l at + D n cos nπ l at d u n ( x,t ) = C n sin nπ l at + D n cos nπ l at sin nπ l x ( n = 1 , 2 , 3 ,... ) v + > . ^ . A . du > . ^ g , A 5 U E, v + > . ^ . r k A U 5 u ( x,t ) = ∞ X n =1 C n sin nπ l at + D n cos nπ l at sin nπ l x / . 1 y , K Sturm-Liouville . K , λ U . 2 5 ' , ^ A = 1 L 2 | ^ 5 U X v > . ^ . J U X C n D n , v ^ u ( x, 0) = ∞ X n =1 D n sin nπ l x = φ ( x ) (4) ∂u ∂t ( x, 0) = ∞ X n =1 C n · nπ l a · sin nπ l x = ψ ( x ) (5) k m ' X . X n λ n X 00 n ( x ) + λ n X n ( x ) = 0 (6) X n (0) = 0 X n ( l ) = 0 (7) X m λ m X 00 m ( x ) + λ m X m ( x ) = 0 (8) X m (0) = 0 X m ( l ) = 0 (9) X m ( x ) × (6)- X n ( x ) × (8) ( X m X 00 n- X n X 00 m ) + ( λ n- λ m ) X n X m = 0 ( λ n- λ m ) X n X m = ( X n X m- X m X n ) , | ^ > . ^ (7), (9) ( λ n- λ m ) Z l X n ( x ) X m ( x )d x = ( X n ( x ) X m ( x )- X m ( x ) X n ( x )) | l = 0 n 6 = m , λ n 6 = λ m , Z l X n ( x ) X m ( x )d x = 0 n 6 = m ? , n = m , O k X n k 2 = Z l X 2 n ( x )d x = l 2 (4) sin mπ l x , Z l φ ( x ) sin mπ l x d x = Z l ∞ X n =1 D n sin nπ l x sin mπ l x d x = D m k X m k = D m l 2 D n = 2 l Z l φ ( x ) sin nπ l x d x d (5) C n = 2 nπa Z l ψ ( x ) sin nπ l x d x φ ( x ) ψ ( x ), U X C n D n , , K ....
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chpp14 - 14 l C K n — l C 14.1 u g d Example 14.1< x<...

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