5 5.1 [ ] ) d f(t) + a d f(t) + bf(t) : f(t) 1 dt dt ) u(x, t) c u(x, t) : u(x, t) t x : ( ) ) 1 : y + ay, : y + ay + by : ( ) 1 ) : y + ay, : yy + ay 3 ( ): ( ) ) : y + ay, : y + ay b [],,, [ ] au xx + bu xy + cu yy + du x + eu y + fu g(x, y) (5.1), a, b, c, d, e, f, g(x, y) (g(x, y) ), (5.1) au xx + bu xy + cu yy,,, λ aλ + bλ + c (5.) λ D b ac ( ax + bxy + cy 1,, ) (1) D > (hyperbolic type) () D (parabolic type) (3) D < (elliptic type),, u tt c u xx ( ) u t κu xx ( ) u xx + u yy (),, { u(x, t ), ut (x, t ) u(x, t), u(x, t)
5. [1 ] 1 [], u(x, t) t c u(x, t) x (5.3) ξ x + ct, η x ct (5.4),u(x, t) ξ, η u(ξ, η), ξ t,, ( u(ξ,η) ξ η u(x, t) t ) u(x, t) { ( u(ξ, η) c t ξ ξ { ( u(ξ, η) c u(ξ, η) ξ u(ξ, η) η + ξ t η t u(ξ, η) u(ξ, η) c + ( c) ξ η ( ) u(ξ, η) u(ξ, η) c ξ η ( u(ξ,η) η ξ u(ξ, η) ξ ξ η ( c u(ξ, η) u(ξ, η) ξ ξ η ) u(ξ,η) ξ η )} u(ξ, η) ξ η t + η ξ )} { ( u(ξ, η) c + c η ξ ) + u(ξ, η) η { ( u(ξ, η) c u(ξ, η) η u(ξ, η) η c, η t c )} η t )} ( c) (5.5) (5.6), ξ x 1, η x 1 u(x, t) x u(ξ, η) ξ u(ξ, η) η + ξ x η x u(ξ, η) u(ξ, η) + ξ η (5.7) u(x, t) ( ) u(ξ, η) u(ξ, η) ξ + x ξ ξ η x + ( ) u(ξ, η) u(ξ, η) η + η ξ η x ( ) ( u(ξ, η) + u(ξ, η) ) u(ξ, η) + + u(ξ, η) ξ ξ η η ξ η u(ξ, η) + u(ξ, η) + u(ξ, η) (5.8) ξ ξ η η (5.6) (5.8) (5.3) u(ξ, η) ξ η (5.9) η u(ξ, η) ξ φ 1 (ξ) (5.1)
, φ 1 (ξ) ξ ξ u(ξ, η) φ 1 (ξ) dξ + ψ(η) (5.11), ψ(η) η 1,,, φ(ξ), u(ξ, η) φ(ξ) + ψ(η) (5.1) ( P.15[ 3] ) u(x, t) φ(x + ct) + ψ(x ct) (5.13), 1 [ ( )] φ ψ { u(x, ) f(x) u t (x, ) t u(x, t) t g(x) (5.13) t t u {φ(x + ct) + ψ(x ct)} t φ ξ ξ t + ψ η η t ( φ c ξ ψ ) η (5.14) (5.13), (5.14) (t ξ, η x ) { u(x, ) φ(x) + ψ(x) f(x) u t (x, ) u(x, t) t t c( φ(ξ) ψ(η) ) ξ η t c( dφ(x) dψ(x) ) g(x) (5.15) dx dx (5.15) x c(φ(x) ψ(x)) x (5.16) c, (5.15) 1 φ(x) 1 f(x) + 1 c x g(s) ds + C (5.16) g(s) ds + C c ψ(x) 1 f(x) 1 g(s) ds C c c,, (5.13),(5.17),(5.18) x u(x, t) 1 [f(x + ct) + f(x ct)] + 1 c x+ct x ct (5.17) (5.18) g(s) ds (5.19), c 1, u t (x, ) g(x) u(x, t) 1 [f(x + t) + f(x t)]
[ 1 ()] u(, t) x u(, t) 1 [f(ct) + f( ct)] + 1 }{{} c ct g(s) ds ct }{{} f( x) f(x), g( x) g(x), f(x) g(x),, f(x) g(x) x < [ ()] u(, t), u(, t) x u(, t) 1 [f(ct) + f( ct)] + 1 }{{} c 1 ct g(s) ds ct }{{} f( x) f(x), g( x) g(x), f(x) g(x),, x u(, t) 1 [f( + ct) + f( ct)] + 1 }{{} c +ct g(s) ds ct }{{}, f(x) g(x),, x f(x) g(x)
1 1 ( ) 1, u(x, t) c u(x, t) t x u(x, t) φ(x + ct) + ψ(x ct) ( ) { u(x, ) f(x) u t (x, ) t u(x, t) t g(x) () u(x, t) 1 [f(x + ct) + f(x ct)] + 1 c x+ct x ct g(s) ds ( ), c 1, u t (x, ) g(x) u(x, t) 1 [f(x + t) + f(x t)] u(x, ) f(x), u t (x, ) g(x) () 1 u(x, ) f(x), u t (x, ) g(x) u(, t) () u(x, ) f(x), u t (x, ) g(x) u(, t) u(, t) () (u(x, t) φ(x + ct) + ψ(x ct): ) u(x, t) 1[f(x + ct) + f(x ct)] + 1 x+ct c x ct g(s) ds ( ) 1 1 [f(x + ct) + f(x ct)] + 1 x+ct c x ct g(s) ds (f(x), g(x) ) 1 [f(x + ct) + f(x ct)] + 1 x+ct c x ct g(s) ds (f(x), g(x) )
[ ()] ( ) 1 u(x, t) c u(x, t) (5.) t x, u(x,t) t : u(x, ) f(x), t g(x) : u(, t), u(, t) ( : ), f(x), g(x), (5.) () ( ) u(x, t) X(x)T (t) (5.1), X(x) x, T (t) t (5.1) (5.) X(x) T (t) t, c X(x)T (t), X(x) x, T (t) t, d c X(x) x T (t) (5.) 1 d T (t) 1 d X(x) (5.3) c T (t) dt X(x) dx (5.3) t, x, (5.3) k 1 d T (t) 1 d X(x) k (5.4) c T (t) dt X(x) dx, { d X(x) dx kx(x) d T (t) dt c kt (t) (5.5) (5.5) 1 (X (x) kx(x) ) (S1) k > X(x) Ae kx + Be kx (S) k X(x) Ax + B (S3) k < X(x) A cos kx + B sin kx (S1), (S), (S3),, (5.1) u(, t) X()T (t), u(, t) X()T (t) (5.6), T (t) X(), X() (5.7)
(1) k > (S1) (5.7) ( :, k q >, k q ) { A + B Ae (5.8) k + Be k, A B (X(x) ), k () k (S) (5.7) { B A + B (5.9), A B, k (3) k < (S3) (5.7) ( :, k p <, k p ) { A B sin (5.3) k B, sin k, sin nπ (n, 1,, ) k nπ (n, 1,, ) (5.31) n, k, X(x) k nπ (n 1,, ) (5.3) ) nπ nπ k k n ( : (n 1,, ) (5.33) d ( ) X(x) nπ X(x) + (5.34) dx X n (x) B n sin nπ x : (5.35) (5.33) k T (t) (5.5) (T (t) kc T (t) ) d ( ) T (t) cnπ T + (t) (5.36) dt (5.36) X(x) ( k ( ) nπ > ) (S3), c k ( cnπ ) > T n (t) C n cos cnπ t + D n sin cnπ t (5.37) (5.35), (5.37), (5.) (5.1) u n (x, t) T n (t)x n (x) (A n cos cnπ t + B n sin cnπ t) sin nπ x (5.38) u n (x, t) (A n cos ω n t + B n sin ω n t) sin nπ x (ω n cnπ ) (5.39)
[] (5.) u 1 u u 1 t c u 1 x (5.4) u c u (5.41) t x, (5.4), (5.41), c 1, c u 1 c 1 t + c ( u c u 1 c t 1 x + c ) u (5.4) x (5.4) t (c 1u 1 + c u ) c x (c 1u 1 + c u ) (5.43), c 1 u 1 + c u (5.), u 1, u, c 1, c, c 1 u 1 + c u, u i (i 1,, 3, ) (5.), c i u i (5.) (5.) u i1, (5.38), u(x, t) u n (x, t) T n (t)x n (x) (A n cos ω n t + B n sin ω n t) sin nπ x (ω n cnπ ) (5.44), A n, B n,, t u(x, ) A n sin nπ x (5.45) (5.45) f(x) f(x) A n sin nπ x ( : f(x) ) (5.46) (5.46), A n f(x) A n, (5.44) t f(x) sin nπ x dx (n 1,, ) (5.47) u(x, t) t ( A n ω n sin ω n t + B n ω n cos ω n t) sin nπ x (ω n cnπ ) (5.48)
, t u(x, t) t B n ω n sin nπ t x (5.49) (5.49) g(x) g(x) B n ω n sin nπ x ( : g(x) ) (5.5) g(x) ω n (5.51), B n g(x) ω n B n sin nπ x (5.51) B n g(x) sin nπ x dx (n 1,, ) (5.5) ω n, (5.), u(x, t) (A n cos ω n t + B n sin ω n t) sin nπ x (ω n cnπ ) (5.53), A n, B n, f(x), g(x), A n f(x) sin nπ x dx (n 1,, ) : f(x) (5.54) B n g(x) sin nπ g(x) x dx (n 1,, ) : (5.55) ω n ω n (1) u(x,t) t t g(x), (5.53) (B n ) u(x, t) ( ) nπ A n cos (ω n t) sin x (ω n cnπ ) (5.56) A n f(x) sin nπ x dx (n 1,, ) (5.57) () u(x, ) f(x), (5.53) (A n ) u(x, t) ( ) nπ B n sin (ω n t) sin x (ω n cnπ ) (5.58) B n g(x) sin nπ x dx (n 1,, ) (5.59) ω n
(P.139 5.) ( a) t, ( x < a) u(x, ) f(x), u t (x, ) g(x) 1 ( a x +a ( +a < x ), u(x, t) (A n cos ω n t + B n sin ω n t) sin nπ x (ω n cnπ ), f(x), A n, B n ) B n ω n ω n ω n +a nπω n nπω n a nπω n ( nπ { nπω n 4 sin nπ nπω n 4 cn π sin nπ g(x) sin nπ x dx sin nπ x dx ) [cos nπ x ] +a cos nπ ( + a { cos ( nπ + nπa { cos nπ { sin nπ a } ) cos nπ ( a ) ) cos (nπ nπa nπa cos sin nπ } nπa sin sin nπa sin nπa (ω n cnπ ) ) } sin nπa cos nπ cos nπa sin nπ } nπa sin 4 u(x, t) cn π sin nπ nπa sin sin } {{ } B n cnπ t sin nπ x f(x) g(x) 1 x +a a 1 a +a x f(x) g(x)
[ ()] u(x, ) f(x), u(x, t) u t (x, ) g(x) ( x ) u(x, t) X(x)T (t) { d X(x) kx(x) dx d T (t) kc T (t) dt x u(x, t) X(x)T (t), (5.6) X(x) ( x ) X(x) k ω (k < ) ( :, k p < ), { d X(x) + ω X(x) dx d T (t) (5.61) + c ω T (t) dt X(x) e λx ( :, X(x) e ax ) 1 λ X(x) + ω X(x) (λ + ω )X(x) X(x) e λx ( ), λ ±iω, X ω (x) Ae iωx + Be iωx () (ω cω), T ω (t) Ce icωt + De icωt (), u ω (x, t) X ω (x)t ω (t) (Ae iωx + Be iωx )(Ce icωt + De icωt ) A e iω(x+ct) + B e iω(x ct) + C e i( ω)(x+ct) + D e i( ω)(x ct) ω, u(x, t) 1 π 1 π 1 π 1 π 1 π u ω (x, t) dω { A(ω)e iω(x+ct) + B(ω)e iω(x ct)} dω + 1 π { A(ω)e iω(x+ct) + B(ω)e iω(x ct)} dω + 1 π { A(ω)e iω(x+ct) + B(ω)e iω(x ct)} dω + 1 π { A(ω)e iω(x+ct) + B(ω)e iω(x ct)} dω { C(ω)e i( ω)(x+ct) + D(ω)e i( ω)(x ct)} dω { C(ω)e iω (x+ct) + D(ω)e iω (x ct) } ( dω ) { C(ω)e iω (x+ct) + D(ω)e iω (x ct) } dω
u(x, ) f(x) 1 {A(ω) + B(ω)} e iωx dω π }{{} F (ω) u t (x, ) g(x) 1 icω {A(ω) B(ω)} e iωx dω π }{{} G(ω), F (ω) G(ω) f(x) g(x) 1 A(ω) + B(ω) F (ω) icω {A(ω) B(ω)} G(ω) A(ω) F (ω) B(ω) F (ω) + G(ω) icω G(ω) icω u(x,t) t t g(x) (G(ω) ) A(ω) B(ω) F (ω) u(x, t) 1 { F (ω) π eiω(x+ct) + F (ω) } eiω(x ct) dω
1 ( ) 1, u(x, t) c u(x, t) t x { u(x, ) f(x) u t (x, ) t u(x, t) t g(x) (): u(x, t) (A n cos cnπ t + B n sin cnπ t) sin nπ x A n B n f(x) sin nπ g(x) ) sin nπ ( cnπ x dx (n 1,, ) x dx (n 1,, ) ( ):, (: X(x) ( x )) u(x, t) {A(p)e ip(x+ct) + B(p)e ip(x ct) } dp A(p) F (p) 4π + G(p) 4πicp, F (p) B(p) 4π G(p) 4πicp, F (p) G(p) f(x) g(x) u(x, t) 1 {A(ω)e iω(x+ct) + B(ω)e iω(x ct) } dω π A(ω) F (ω) + G(ω) icω, F (ω) B(ω) G(ω) icω
5.3 ( ) [1 ( )] 1 ( ) u(x, t) κ u(x, t) (5.6) t x [ ( )] : u(x, ) f(x) : u(, t) u(, t) ( :, ), (5.6), u(x, t) u(x, t) X(x)T (t) (5.63) (5.63) (5.6) T (t) X(x) t κ X(x) x T (t) (5.64), κx(x)t (t), X(x) x, T (t) t 1 dt (t) 1 d X(x) ( k) (5.65) κt (t) dt X(x) dx (5.65) t, x, (5.65) k, { d X(x) kx(x) dx dt (t) (5.66) κkt (t) dt (5.66) 1 5. k,, 3 (S1 S3), u(, t) u(, t), T (t) X()T (t) X()T (t) (5.67) X() X() (5.68),, k < (S3) X(x) A cos kx + B sin kx (5.69) (5.68) { X() A X() A cos k + B sin k (5.7) A A cos k + B sin k (5.71) cos k 1, sin k, (5.71), cos nπ 1, sin nπ (n, 1,, ) k nπ (n, 1,, ) (5.7)
, X(x) ) nπ nπ k k n ( : (n, 1,, ) (5.73) X n (x) A n cos nπ x + B n sin nπ x : (5.74) (5.73) (5.66) dt (t) dt ( ) nπ T + κ (t) (5.75) (5.75) T n (t) C n e κ( nπ ) t (5.76) (5.74), (5.76), (5.6)(5.63) u n (x, t) X n (x)t n (t) { un (x, t) (A n cos nπ x + B n sin nπ x)e κ( nπ ) t (n 1,, ) u (x, t) A (n ) (5.77) [], (5.6) u(x, t) u n (x, t) n n T n (t)x n (x) A + (A n cos nπ x + B n sin nπ nπ x)e κ( ) t (5.78), A, A n, B n,, u(x, ) A + (A n cos nπ x + B n sin nπ x) f(x) (5.79) (5.79), A, A n, B n f(x) A n 1 B n 1 f(x) cos nπ x dx (n, 1,, ) (5.8) f(x) sin nπ x dx (n 1,, ) (5.81) (5.6) u(x, ) f(x), u(, t) u(, t), (5.78), A n, B n (5.8) (5.81)
(P.146 5.3) π t > u(x, ) f(x) (π x)x, u(x, t) A + (A n cos nπ x + B n sin nπ nπ x)e κ( ) t,, f(x) π, B n, f(x) A n, π,, π A n 1 π f(x) cos nπ x dx f(x) cos nπ x dx 4 (πx x ) cos nx dx π 4 π ( ) sin nx (πx x ) dx π n 4 ( [ ( )] sin nx π (πx x ) π n π (π x) sin nx dx πn πn πn πn π ( (π x) { [ (π x) ( ( π n 1 n 1 n (n ) 1 n π ) cos nx dx n )] π cos nx 1 n n ) π [sin nx] (π x) sin nx dx π cos nx dx } ) n A 1 4 π π 4 π ( π 3 4 π π 3 u(x, t) π 6 f(x) dx f(x) dx (πx x ) dx [ 1 πx 1 3 x3 )] π 8 π3 4 ) cos nx e κ(n) t n
[ ] u(x, ) f(x) u(x, t) ( x ) u(x, t) X(x)T (t) { d X(x) kx(x) dx dt (t) κkt (t) dt x u(x, t) X(x)T (t), (5.8) X(x) ( x ) X(x) k ω (k < ) ( :, k p < ), { d X(x) + ω X(x) dx dt (t) (5.83) + κω T (t) dt 1 X(x) e λx λ X(x) + ω X(x) (λ + ω )X(x) X(x) e λx ( ), λ ±iω, X ω (x) Ae iωx + Be iωx (), T (t) e λt, (λ + κω )T (t), T ω (t) Ce κω t, u ω (x, t) X ω (x)t ω (t) (Ae iωx + Be iωx )e κω t ω, u(x, t) 1 π 1 π 1 π 1 π 1 π 1 π 1 π u ω (x, t) dω ( A(ω)e iωx + B(ω)e iωx) e κωt dω A(ω)e iωx e κωt dω + 1 π A(ω)e iωx e κωt dω + 1 π A(ω)e iωx e κωt dω + 1 π F (ω)e iωx e κωt dω F (ω)e iωx κωt dω B(ω)e i( ω)x e κ( ω)t dω B(ω)e iω x e κω t ( dω ) B(ω)e iω x e κω t dω
t u(x, ) f(x) 1 F (ω)e iωx dω π, F (ω) f(x) F (ω) f(x)e iωx dx, u(x, t) u(x, t) 1 π [ ] f(y)e iωy dy }{{} f(x) F (ω) e iωx κωt dω
[ ()] : u(x, ) f(x) : u(, t) u(, t) ( : ),, u(x, t) u(x, t) X(x)T (t) (5.84) (5.84) T (t) X(x) t κ X(x) x T (t) (5.85), κx(x)t (t), X(x) x, T (t) t 1 dt (t) 1 d X(x) ( k) (5.86) κt (t) dt X(x) dx (5.86) t, x, (5.86) k, { d X(x) dx kx(x) dt (t) dt κkt (t) (5.87) 1 (X (x) kx(x) ) (S1) k > X(x) Ae kx + Be kx (S) k X(x) Ax + B (S3) k < X(x) A cos kx + B sin kx (S1), (S), (S3),, (5.84) (5.87) u(, t) X()T (t), u(, t) X()T (t) (5.88), T (t) X(), X() (5.89) (1) k > (S1) (5.89) { A + B Ae k + Be k (5.9), A B (X(x) ), k () k (S) (5.89) { B A + B (5.91), A B, k (3) k < (S3) (5.89) { A B sin k (5.9)
B, sin k, sin nπ (n, 1,, ) k nπ (n, 1,, ) (5.93) n, k, X(x) k nπ (n 1,, ) (5.94) ) nπ nπ k k n ( : (n 1,, ) (5.95) d ( ) X(x) nπ X(x) + (5.96) dx X n (x) B n sin nπ x : (5.97) (5.95) k T (t) (5.87) (T (t) κkt (t) ) ( ) dt (t) nπ T + κ (t) (5.98) dt (5.98) T n (t) C n e κ( nπ ) t (5.99) (5.97), (5.99), (5.84) u n (x, t) X n (x)t n (t) ( ) nπ u n (x, t) B n sin x e κ( nπ ) t (n 1,, ) (5.1) [], u(x, t) u n (x, t) T n (t)x n (x) ( ) nπ B n sin x nπ κ( e ) t (5.11), B n,, ( ) nπ u(x, ) B n sin x f(x) (5.1) (5.1), B n f(x) B n f(x) sin nπ x dx (n 1,, ) (5.13) u(x, ) f(x), u(, t) u(, t), (5.11), B n (5.13)
π t > u(x, ) f(x) (π x)x, u(x, t) B n sin nπ nπ x e κ( ) t, B n f(x), f(x), π B n 1 π π π π π π πn πn πn 4 πn f(x) sin nπ x dx f(x) sin nπ x dx (πx x ) sin nx dx ( ) (πx x cos nx ) dx n ( )] π ([ (πx x ) cos nx n (π x) cos nx dx + 1 n π ( ) sin nx (π x) dx n { [ ( )] sin nx π (π x) + n n [ ] cos nx π n 4 πn 3 {1 ( 1)n } { 8 πn 3 π π ) (π x) cos nx dx sin nx dx (n ) (n ) } u(x, t) 4 1 ( 1) n sin nx e κn t π n 3 [ 8 1 π (n 1) sin (n 1)x t 3 e κ(n 1) ]
1 ( ) 1, u(x, t) t κ u(x, t) x u(x, ) f(x) (u(, t) u(, t)): u(x, t) A + (A n cos nπ x + B n sin nπ nπ x)e κ( ) t A n 1 f(x) cos nπ x dx (n, 1,, ) B n 1 f(x) sin nπ x dx (n 1,, ) (: X(x) ( x )): u(x, t) 1 F (ω)e iωx κωt dω π F (ω) f(x)e iωx dx u(x, ) f(x) x u(x, t) ( ) 1 u(x, ) f(x) : u(, t) u(, t) ( ) u(x, ) f(x) : u(, t) u(, t) ( P.151) ( ) 3 u(x, ) f(x) : u x (, t) u x (, t) ( P.163) ( )
: u tt c u xx u(x, ) f(x), u t (x, ) g(x) ( ): (): u(, t) u(, t) X(x) ( x ) X (x) kx(x) X (x) + ω X(x) (k ω ) T (t) c kt (t) T (t) + c ω T (t) u(x, t) (A n cos cnπ t + B n sin cnπ t) sin nπ x 1 {A(ω)e iω(x+ct) + B(ω)e iω(x ct) } dω π A n B n f(x) sin nπx g(x) nπx sin ( cnπ ) F (ω) dx A(ω) + G(ω) icω F (ω) dx B(ω) G(ω) icω g(x) A n, B n : f(x), ( cnπ F (ω), G(ω) : f(x), g(x) ) : : x+ct x ct g(s) ds 1 {f(x + ct) + f(x ct)} + 1 c 1 {f(x + ct) + f(x ct)} + 1 c (f(x), g(x) ) x+ct g(s) ds x ct : u t κu xx u(x, ) f(x) : u(, t) u(, t) : X(x) ( x ) X (x) kx(x) X (x) + ω X(x) (k ω ) T (t) κkt (t) T (t) + κω T (t) u(x, t) F (ω)e iωx κωt dω A + A n 1 B n 1 (A n cos nπ x + B n sin nπ x)e κ( nπ ) t 1 π nπx f(x) cos dx F (ω) nπx f(x) sin dx A n, B n : f(x) f(x)e iωx dx u(x, ) f(x) ( P.151): ( P.163): u(, t) u(, t) u x (, t) u x (, t) X (x) kx(x) X (x) kx(x) T (t) κkt (t) T (t) κkt (t) u(x, t) 5-3 1 5 [3]
5.4 [ ] u(x, y) x + u(x, y) y (5.14) u(x, y) t,, [ ] u(, y) u(a, y) u(x, b) u(x, ) f(x), (5.14), u(x, y) u(x, y) X(x)Y (y) (5.15) (5.15) (5.14) X(x)Y (y) d X(x) Y (y) X(x) d Y (y) (5.16) dx dy 1 d X(x) 1 d Y (y) k (5.17) X(x) dx Y (y) dy (5.17) x, y, (5.17) k ( :, k p <, k p ) d X(x) kx(x) dx d Y (y) dy ky (y) (5.18) { X()Y (y) X(a)Y (y) X(x)Y (b) (5.19) X(x), Y (y) { X() X(a) Y (b) (5.11) (5.18) 1 5. k,, 3 (S1 S3), X(x) k < X(x) A cos kx + B sin kx (5.111) (5.11) 1 { X() A X(a) A cos ka + B sin ka B sin ka (5.11)
(5.11), X(x) sin ka (5.113) ka nπ (n 1,, ) (5.114) k nπ a k n ( nπ a X(x) X n (x) B n sin nπ a ) (n 1,, ) (5.115) x (n 1,, ) (5.116) (5.115) (5.18) (5.117) (5.11) d Y (y) dy ( nπ a ) Y (y) (5.117) Y (y) Ae nπ a y + Be nπ a y (5.118) Y (b) Ae nπ a b + Be nπ a b (5.119) B nπ e a b (B Ae nπ a b ) (5.1) A, n 1,,, C n Y n (y) Ae nπ a y Ae nπ a b e nπ a y Ae nπ a b (e nπ a (y b) e nπ a (y b) ) C n(e nπ a (y b) e nπ a (y b) ) C n sinh nπ a (y b) (sinh x ex e x, C n C n) (5.11) (5.116), (5.11), (5.14) (5.15) u n (x, y) X n (x)y n (y) B n sin nπ a x sinh nπ (y b) (n 1,, ) (5.1) a B n, (5.14) u(x, y) B n sin nπ a x sinh nπ (y b) (5.13) a, u(x, ) ( B n sinh nπ a b) sin nπ x f(x) (5.14) a (5.14) a B n sinh nπ a b f(x) sin nπ x dx (n 1,, ) (5.15) a a a B n a sinh nπb f(x) sin nπ x dx (n 1,, ) (5.16) a a (5.14) (5.13), B n (5.16) u(x, y) a f(x) sin nπ x dx a a sinh nπb sin nπ a x sinh nπ (b y) (5.17) a a