Cauchy Duhamel Cauchy Cauchy Poisson Cauchy 1. Cauchy Cauchy ( Duhamel ) u 1 (t, x) u tt c 2 u xx = f 1 (t, x) u 2 u tt c 2 u xx = f 2 (
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1 Cauchy Duhamel Cauchy CauchyPoisson Cauchy 1. Cauchy Cauchy ( Duhamel) u 1 (t, x) u tt c 2 u xx = f 1 (t, x) u 2 u tt c 2 u xx = f 2 (t, x) 1
2 C 1 C 2 u(t, x) = C 1 u 1 (t, x) + C 2 u 2 (t, x) u tt c 2 u xx = C 1 f 1 (t, x) + C 2 f 2 (t, x) Bernoulli Fourier 1.2 Cauchy u tt c 2 u xx = f(t, x), (1.1) t = : u =, u t =, (1.2) c > f(t, x) t x (1.1) Cauchy(1.1)-(1.2) w tt c 2 w xx =, (1.3) t = τ : w =, w t = f(τ, x). (1.4) Cauchy(1.3)-(1.4) w = w(t, x; τ), (1.5) 1.1 w = w(t, x; τ) Cauchy(1.3)-(1.4) τcauchy (1.1)-(1.2) u(t, x) = w(t, x; τ)dτ. (1.6) 1.1 Duhamel (1.6)u(t, x)(1.2) 2
3 (1.6) u(, x) =. (1.7) (1.6) u t (t, x) = w(t, x; t) + w t (t, x; τ)dτ, (1.4) u t t= = w(, x; ) =. (1.8) (1.7) (1.8)(1.2) (1.6)u = u(t, x)(1.1) (1.6) (1.4) u t (t, x) = w(t, x; t) + u tt = w t (t, x; t) + u tt c 2 u xx = = w t (t, x; τ)dτ = w tt (t, x; τ)dτ = u xx = w t (t, x; τ)dτ. w tt (t, x; τ)dτ + f(t, x). (1.9) w xx (t, x; τ)dτ. (1.1) w tt (t, x; τ)dτ + f(t, x) c 2 w xx (t, x; τ)dτ [ wtt (t, x; τ) c 2 w xx (t, x; τ) ] dτ + f(t, x) (1.11) = f(t, x). (1.3) (1.11)u(t, x)(1.1) 1.1 f(t, x) t x u t [, t] t i = t i+1 t i (i = 1, 2,, n), t i f(t, x) t f(t i, x) f(t i, x) = F (t i, x) (F (t i, x)ρ ) t i ρ f(t i, x) t i t i 3
4 Cauchy w tt c 2 w xx =, (1.12) t = t i : w =, w t = f(t i, x) t i. (1.13) Cauchy(1.12)-(1.13) w = w(t, x; t i, t i )f(t, x) Cauchy(1.1) - (1.2)u = u(t, x) u(t, x) = lim t i n w(t, x; t i, t i ). (1.14) i=1 (1.12) w t i w(t, x; τ) Cauchy w tt c 2 w xx = (t > τ), t = τ : w =, w t = f(τ, x) Cauchy(1.1)-(1.2) u(t, x) = lim t i n i=1 (1.15) w(t, x; t i, t i ) = t i w(t, x; t i ). (1.16) w(t, x; t i, t i ) = lim t i 1.1 n w(t, x; t i ) t i = i=1 w(t, x; τ)dτ. w(t, x; τ) Cauchy(1.15) t = t τ (1.15) w t t c2 w xx = ( t > ), t = : w =, w t = f(τ, x) (1.17) D Alembert (4.13) Cauchy(1.17) w(t, x; τ) = 1 2c x+c t x c t f(τ, ξ)dξ = 1 2c (1.6) Cauchy(1.1)-(1.2) u(t, x) = 1 2c x+c(t τ) x c(t τ) f(τ, ξ)dξdτ = 1 2c 4 x+c(t τ) x c(t τ) Ω f(τ, ξ)dξ. (1.18) f(τ, ξ)dξdτ, (1.19)
5 Ω (τ, ξ) (t, x)ξ ( 1.1) τ (t, x) ξ x = c(τ t) Ω ξ x = c(τ t) ξ 1.1. Ω Cauchy(1.1)-(1.2)(1.19) Cauchy(1.1)-(1.2) f C 1 (1.19) u t = 1 2 [f(τ, x + c(t τ)) + f(τ, x c(t τ))] dτ, (1.2) u tt = f(t, x) + c 2 u x = 1 2c u xx = 1 2c [f x (τ, x + c(t τ)) f x (τ, x c(t τ))] dτ, (1.21) [f(τ, x + c(t τ)) f(τ, x c(t τ))] dτ, (1.22) [f x (τ, x + c(t τ)) f x (τ, x c(t τ))] dτ. (1.23) u tt c 2 u xx = f(t, x), u(t, x)(1.1) (1.19)(1.2) u t= =, u t t= =, u(t, x)(1.2)(1.19)u(t, x) Cauchy (1.1)-(1.2) 5
6 Cauchy u tt c 2 u xx = f(t, x) (t >, x R), t = : u = ϕ(x), u t = ψ(x) (x R) (1.24) 1.2 Cauchy 1. [ (1 x ] u )2 = 1 x h x a (1 x 2 u 2 h )2 t 2 (h > ) u = F (x at) + G(x + at), h x F, G t = : u = ϕ(x), u t = ψ(x). 2. ϕ(x) ψ(x) 3. (Goursat) 2 u t = 2 u 2 a2 x, 2 u x at= = ϕ(x), u x+at= = ψ(x) (ϕ() = ψ()). 4. (1.24) f(x, t) 1x[x 1, x 2 ] [x 1, x 2 ] 2x [x 1, x 2 ] [x 1, x 2 ] 6
7 5. u tt a 2 u xx =, x >, t >, u t= = ϕ(x), u t t= =, u x ku t x= =, k 6. u tt u xx =, < t < kx, k > 1, u t= = ϕ (x), x, u t t= = ϕ 1 (x), x, u t t=kx = ψ(x), ϕ () = ψ() 7. u tt u xx =, < t < f(x), u t=x = ϕ(x), u t=f(x) = ψ(x), ϕ() = ψ(), t = f(x) x = t x = t x f (x) u t 2 u = t sin x, 2 x2 u u t= =, t = sin x. t= 9. u tt = a 2 u xx + u t= =, u t t= = x. 2 tx (1 + x 2 ) 2, 7
8 u tt c 2 u xx = f(t, x) (2.1) u(t, x) u(t, x). ( ) ( ).... (2.1) u(t, x) x = x = L u(t, ) = u(t, L) =. (2.2) (2.2)(2.1) t = u(t, x) u(, x) = ϕ(x), t = ψ(x) (x [, L]), (2.3) t= (2.3) (2.1) (2.3) (2.1) (2.2)- u tt c 2 u xx = f(t, x) (t >, x (, L)), (2.4) t = : u = ϕ(x), u t = ψ(x) (x [, L]), (2.5) x = : u = (t > ), (2.6) 8
9 (2.1) x = L : u = (t > ). (2.7) U = {(t, x) t, x L} (2.8) (2.4)-(2.7)u = u(t, x) {(t, x) t >, < x < L} (2.4) x [, L] (2.5) x =, L (2.6) (2.7) (2.2) Dirichlet x = x T (x) xx = T () u x = x u x =. (2.9) x= u x = γ(t), (2.1) x= γ(t) t Neumann (2.9)Neumann(2.1) Neumann Hooke u = u x = T u x Hooke T u x = ku x=, x= k ( σu) u x x= + =, (2.11) 9
10 σ = k/t ( σu) u x x= + = µ(t), (2.12) µ(t) t (2.1)(4.1) (2.1)u f(t, x) f(t, x) (2.6)-(2.7) u x= = α(t), u x=l = β(t) (2.3) ϕ ψ (2.3) (1) (2) (3)
11 2.2 Cauchy u tt c 2 u xx = f(t, x), t = : u = ϕ(x), u t = ψ(x), (2.13) x = : u =, x = L : u =. (2.13) v tt c 2 v xx =, t = : v = ϕ(x), v t = ψ(x), x = : v =, x = L : v = (2.14) w tt c 2 w xx = f(t, x), t = : w =, w t =, x = : w =, x = L : w =. 11 (2.15)
12 u = v + w. (2.16) (2.13)(2.14)(2.15) (2.14) (2.14)(2.15) (2.14) u(t, x) = T (t)x(x) (2.17).. (2.14) u tt c 2 u xx =, (2.18) t = : u = ϕ(x), u t = ψ(x), (2.19) x = : u =, (2.2) x = L : u =. (2.21) (2.18) (2.2)-(2.21) u(t, x) = T (t)x(x), (2.22) T (t), X(x) t x (2.22)(2.18) X(x)T (t) c 2 X (x)t (t) =, T (t) c 2 T (t) = X (x) X(x). (2.23) 12
13 (2.23) t x λ T (t) + c 2 λt (t) = (2.24) X (x) + λx(x) =. (2.25) (2.23)(2.24) (2.25) (2.24) t (2.25) x (2.24) (2.25) (2.22) (2.22)(2.2) (2.21) (2.25)X(x) X() =, X(L) =. (2.26) X = X(x)(2.25)-(2.26) λ 1 λ < λ < (2.25) X(x) = C 1 e λx + C 2 e λx. (2.27) (2.26) C 1 + C 2 =, C 1 e λl + C 2 e λl =. (2.28) (2.28) C 1 C e λl e λl, C 1 = C 2 =, λ < 2 λ = (2.25) X(x) = C 1 + C 2 x. (2.29) (2.26) X(x) 13
14 3 λ > λ > (2.25) X(x) = C 1 cos λx + C 2 sin λx. (2.3) X() = C 1 =. (2.31) X(L) = C 2 sin λl =. C 2 sin λl =. λ = λ k = k2 π 2 L 2 (k = 1, 2, ). (2.32) X k (x) = C k sin( kπ x) L (k = 1, 2, ). (2.33) (2.33) sin( kπ L x) (2.25)(2.26) λ k = k 2 π 2 /L 2 λ k (2.24) A k, B k T k (t) = A k cos kπc L t + B k sin kπc t (k = 1, 2, ), (2.34) L (2.18)(2.2)-(2.21) u k (t, x) = T k (t)x k (x) = ( A k cos kπc L t + B k sin kπc ) L t sin kπ x (k = 1, 2, ). L (2.18)-(2.21) u(t, x) = u k (t, x) = (2.19)A k B k. ( A k cos kπc L t + B k sin kπc ) L t sin kπ x, (2.35) L 14
15 u(t, x) ( kπc = A k sin kπc t L L t + B k cos kπc ) L t sin kπ L x. (2.19) ϕ(x) = ψ(x) = A k sin kπ L x, B k kπc L sin kπ L x. A k kπc L B k ϕ(x) ψ(x) [, L]Fourier A k = 2 L ϕ(η) sin kπη L L dη, B k = 2 kπc L ψ(η) sin kπη L dη. (2.36) (2.35)(2.18)-(2.21) (2.36) ϕ(x) ψ(x) (2.35) (2.35) (2.18) ϕ(x) ψ(x) (2.35) (2.35) (2.18) (2.19)-(2.21) Fourier 2.1 f(x) [, l] m m + 1 f (i) () = f (i) (l) = (i =, 2,, 2[ m 2 ]). f(x) [, l]fourier f(x) a k sin kπx, l a k km a k f(x) m + 1f (m+1) (x) [, l] m f (m+1) (x) 15 a (m+1) k sin kπx, l
16 m f (m+1) (x) a(m+1) 2 + a (m+1) k cos kπx. l Parseval (a (m+1) ) (a (m+1) k ) 2 = 2 l l [ f (m+1) (x) ] 2 dx <. a (m+1) k. m a (m+1) k l = 2 f (m+1) (ξ) sin kπξ dξ l l [ ] = 2 l f (m) (ξ) sin kπξ l 2 kπ l f (m) (ξ) cos kπξ dξ l l l l [ ] = 2 kπ l l f (m 1) (ξ) cos kπξ l 2( kπ l l l )2 l f (m 1) (ξ) sin kπξ dξ l = 2( kπ l l )2 l f (m 1) (ξ) sin kπξ dξ, l f (m 1) (x) x = x = l m a (m+1) k = ( 1) m+1 2 ( kπ l )m+1 a k. a (m+1) k = ( 1) m 2 ( kπ l )m+1 a k. (a (m+1) k ) 2 <, k 2m+2 a 2 k <. Cauchy k m a k k 2m+2 a k 2 1 k 2 <
17 2.1 ϕ(x) C 3, ψ(x) C 2 ϕ() = ϕ(l) = ϕ () = ϕ (L) = ψ() = ψ(l) =, (2.37) A k, B k (2.36) k 2 A k k 2 B k 2.1 (2.35) x t u(2.35) u 2.1 ϕ(x) C 3, ψ(x) C 2 (2.37) (2.18)-(2.21) (2.35)A k B k (2.36) (2.37) (2.22)(2.18)-(2.21) Fourier Fourier Fourier ϕ(x) ψ(x) 2.1 ϕ(x) ψ(x) ϕ(x) ψ(x) ϕ n (x) = n A k sin kπ L x, ψ n(x) = n B k kπc L sin kπ L x ϕ n (x) ψ n (x) (2.18) u n (t, x) = n ( A k cos kπc L t + B k sin kπc ) L t sin kπ x. (2.38) L n u n (t, x) (2.35) u(t, x) {u n }u n (t, x) ϕ(x) ψ(x) n u n (t, x)... 17
18 u(t, x) u n (t, x) (2.35)(2.18)-(2.21) ( u k (t, x) = A k cos kπc L t + B k sin kπc ) L t sin kπ L x = N k cos(ω k t + Q k ) sin kπ L x (2.39) N k = A 2 k + B2 k, ω k = kπc L, sin Q k = B k N k, cos Q k = A k N k. N k ω k Q k c kω k u k (t, x) = N k sin kπ L x cos(ω kt + Q k ) N k sin kπ L x xx = ml/k (m =, 1,, k) (2.35) ω 1 = πc L ω k ω 1 x x = x = 1x = a a (, 1) h h 1 u(t, x)(2.18)-(2.21) h ϕ(x) = a x, x [, a], h (2.4) 1 a (1 x), x [a, 1], ψ(x). (2.41) 18
19 u(t, x) (2.35) (2.41) B k. (2.4) A k = 2 1 ϕ(η) sin(kπη)dη = 2 a h η sin(kπη)dη a a = 2h π 2 a(1 a)k 2 sin(kπa). h (1 η) sin(kπη)dη 1 a u(t, x) = 2h π 2 a(1 a) 1 sin(kπa) sin(kπx) cos(kπct). k u tt c 2 u xx = f(t, x), (2.42) t = : u =, u t =, (2.43) x = : u =, (2.44) x = L : u =. (2.45) Cauchy w = w(t, x; τ) w tt c 2 w xx = (t > τ), t = τ : w =, w t = f(τ, x), x = : w =, x = L : w = (2.46) τ u(t, x) = w(t, x; τ)dτ (2.47) (2.42)-(2.45) 19
20 s = t τ, (2.46) w ss c 2 w xx = (s > ), s = : w =, w s = f(τ, x), x = : w =, x = L : w =. (2.48) (2.35)-(2.36) w = w(s, x; τ) = B k (τ) sin kπc L s sin kπ L x = B k (τ) sin kπc L (t τ) sin kπ x, (2.49) L B k (τ) = 2 kπc L f(τ, ξ) sin kπ ξdξ. (2.5) L (2.49)(2.47)(2.42)-(2.45) u(t, x) = w(t, x; τ)dτ = sin kπ L x 2.1 f(t, x) C 2 B k (τ) sin kπc (t τ)dτ. (2.51) L f(t, ) = f(t, L) = (2.51)(2.42)-(2.45) 2.4 u tt c 2 u xx = f(t, x), (2.52) t = : u = ϕ(x), u t = ψ(x), (2.53) x = : u = γ 1 (t), (2.54) x = L : u = γ 2 (t). (2.55) 2
21 ϕ(x), ψ(x), f(x) γ 1 (t), γ 2 (t) γ 1 () = γ 2 () =. (2.52)-(2.55) (2.14) (2.15) z tt c 2 z xx =, (2.56) t = : z =, z t =, (2.57) x = : z = γ 1 (t), (2.58) x = L : z = γ 2 (t). (2.59) (2.52)-(2.55) u = v + w + z. (2.56)-(2.59) (2.56)-(2.59) (2.14) (2.15) Z(t, x) = γ 1 (t) + x L (γ 2(t) γ 1 (t)). (2.6) Z = Z(t, x) (2.58) (2.59) U U = U(t, x) U(t, x) = z(t, x) Z(t, x) (2.61) U tt c 2 U xx = γ 1 (t) x L (γ 2 (t) γ 1 (t)) (2.62) t = : U = z(, x) Z(, x) = γ 1 () x L (γ 2() γ 1 ()), U t = z t (, x) Z t (, x) = γ 1() x L (γ 2() γ 1()). (2.63) UU (2.61)(2.56) -(2.59) z(t, x) = U(t, x) + Z(t, x). (2.64) 21
22 1. 2 u t 2 u 2 c2 x =, 2 (1) u t= = sin 3πx L, u t t= = x(l x), x = : u =, x = L : u = ; 2 u t 2 u 2 c2 x =, 2 u u(, t) =, (L, t) =, (2) x u(x, ) = h L x, u (x, ) =. t 2. 2 u t = 2 u 2 c2 x 2 u(, t) =, u(l, t) = A sin ωt, u(x, ) = u (x, ) =. t 3. (1) u x= = u x x=l = u tt c 2 u xx =, < x < l, t > u t= = sin 3 2l πx, u t t= = sin 5 2l πx. (2) u x x= = u x x=l =, u t= = x, u t t= =. 4. u tt c 2 u xx = g, < x < l, t >, u x= = u x x=l =, u t= =, u t t= = sin πx 2l, 22
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