u = u(t, x 1, x 2,, x n ) u t = k u kn = 1 n = 3 n = 3 Cauchy ()Fourier Li-Yau Hanarck tcauchy F. JohnPartial Differential Equations, Springer-Verlag, 1982. 1. 1.1 Du(t, x, y, z)d(x, y, z) t Fourier dtn dsdqds u n dq = k(x, y, z) u dsdt, (1.1) n k(x, y, z)(x, y, z)(1.1) dq u n 1
DS Ω(1.1)t 1 t 2 Q = t2 t 1 { S k(x, y, z) u } n ds dt, (1.2) u us n n (t 1, t 2 )u(t 1, x, y, z) u(t 2, x, y, z) Ω ν(x, y, z)ρ(x, y, z)[u(t 2, x, y, z) u(t 1, x, y, z)]dxdydz, νρ t2 t 1 S k u n dsdt = νρ[u(t 2, x, y, z) u(t 1, x, y, z)]dxdydz. (1.3) Ω u x, y, z t Green(1.3) t2 [ ( k u ) + ( k u ) + ( k u )] dxdydzdt t 1 Ω x x y y z z ( t2 ) u = νρ t dt dxdydz, Ω t 1 t2 [ νρ u t ( k u ) ( k u ) ( k u )] dxdydzdt =. (1.4) x x y y z z t 1 Ω t 1 t 2 Ω νρ u t = ( k u ) + ( k u ) + ( k u ). (1.5) x x y y z z (1.5) kνρ k/νρ = c 2 ( ) u 2 t = u c2 x + 2 u 2 y + 2 u. (1.6) 2 z 2 () (1.5) 2
F (t, x, y, z) t2 t 1 = Ω S k u t2 n dsdt + F (t, x, y, z)dxdydzdt t 1 Ω νρ[u(t 2, x, y, z) u(t 1, x, y, z)]dxdydz. (1.5) νρ u t = ( k u ) + ( k u ) + ( k u ) + F (t, x, y, z). (1.7) x x y y z z (1.6) ( ) u 2 t = u c2 x + 2 u 2 y + 2 u + f(t, x, y, z), (1.8) 2 z 2 (1.6) f(t, x, y, z) = (1.8) F (t, x, y, z). (1.9) νρ Fourier (1.1)(1.3) t2 t 1 S dm = γ(x, y, z) U dsdt, (1.1) n γ U n dsdt = [U(t 2, x, y, z) U(t 1, x, y, z)]dxdydz, (1.11) Ω Udmdtn dsγ(x, y, z) (1.1)(1.3) 3
(1.1)(1.11)(1.1)(1.3) QukmUγ(1.3) νρ1 U t = ( γ U ) + ( γ U ) + x x y y z ( γ U ). (1.12) z γγ = c 2 (1.12)(1.6) 1.2 ( ) u(, x, y, z) = ϕ(x, y, z), (1.13) ϕ(x, y, z)t = u(t, x, y, z) (x,y,z) S = g(t, x, y, z), (1.14) S g(t, x, y, z)[, T ] S T Dirichlet QFourier dq = k u n dsdt u u n = g(t, x, y, z), (1.15) (x,y,z) S 4
u us ng(t, x, y, z)[, T ] n S Neumann 1 1u 1 u u 1 1 γ dq = γ(u u 1 )dsdt, (1.16) S Fourier1 k u n dsdt = γ(u u 1)dSdt, γu + k u n = γu 1. γk ( σu) u n (x,y,z) S + = g(t, x, y, z), (1.17) u us ng(t, x, y, z)[, T ] n S σ Cauchy u(, x, y, z) = ϕ(x, y, z) ( < x, y, z < ). (1.18) 5
uxt u t = 2 u c2 x. (1.19) 2 ( ) u 2 t = u c2 x + 2 u. (1.2) 2 y 2 Cauchy 1. L dq = γ(u u 1 )dsdt. ρνku 2. 3. Q(t)Q dq dt = βqβνρ ku 6
2. Cauchy Fourier Fourier Cauchy Cauchy 2.1 Fourier (, )f(x)fourier F [f](λ) g(λ) = f(x)e iλx dx. (2.1) f(x)(, )Fourier (, )g(λ) Fourier f(x) = 1 g(λ)e iλx dλ. (2.2) 2π g(λ)f(x) Fourier F [f](λ) f(λ) f(x)g(λ)fourier F 1 [g](x) f(x)(, )FourierF [f](λ) Fourier f(x) F 1 [F [f]] = f. FourierFourier FourierCauchy Fourier Fourier Fourier 2.1 Fourierα 1 α 2 f 1 f 2 F [α 1 f 1 + α 2 f 2 ] = α 1 F [f 1 ] + α 2 F [f 2 ]. (2.3) 2.1 f 1 (x)f 2 (x)x R f(x) = f 1 (x y)f 2 (y)dy (2.4) 1
f(x)f 1 (x)f 2 (x) f 1 (x)f 2 (x)f(x)f 1 f 2 Fourier f 1 f 2 = f 2 f 1, (2.5) 2.2 f 1 (x)f 2 (x)fourierf 1 (x)f 2 (x)fourier F [f 1 f 2 ] = F [f 1 ] F [f 2 ]. (2.6) F [f 1 f 2 ] = e iλx dx f 1 (x y)f 2 (y)dy f 1 f 2 (, ) F [f 1 f 2 ] = = = (2.6) f 2 (y)dy f 2 (y)dy e iλy f 2 (y)dy = F [f 1 ] F [f 2 ]. e iλx f 1 (x y)dx f 1 (ξ)e iλ(y+ξ) dξ f 1 (ξ)e iλξ dξ 2.3 g 1 (λ)g 2 (λ)fourierg 1 (λ)g 2 (λ)fourier 2π 2.2 F 1 [g 1 g 2 ] = 1 2π = 1 2π = 1 2π = 1 2π F 1 [g 1 g 2 ] = 2πF 1 [g 1 ] F 1 [g 2 ]. (2.7) e iλx dλ g 2 (η)dη g 2 (η)dη e iηx g 2 (η)dη = 2πF 1 [g 1 ] F 1 [g 2 ]. 2 g 1 (λ η)g 2 (η)dη g 1 (λ η)e iλx dλ g 1 (ξ)e i(η+ξ)x dξ e iξx g 1 (ξ)dξ
(2.7) 2.3 2.4 f 1 (x)f 2 (x) Fourier f 1 (x)f 2 (x)fourier (2π) 1 (2.7) F [f 1 f 2 ] = 1 2π F [f 1] F [f 2 ]. (2.8) g 1 = F [f 1 ], g 2 = F [f 2 ]. F 1 [g 1 g 2 ] = 2πF 1 [g 1 ] F 1 [g 2 ] = 2πF 1 [F [f 1 ]] F 1 [F [f 2 ]] = 2πf 1 f 2. Fourier g 1 g 2 = 2πF [f 1 f 2 ]. (2.8) 2.5 f(x)f (x)fourier x f(x) F [f ](λ) = F [f ] = iλf [f]. (2.9) e iλx f (x)dx = { e iλx f(x) } x= x= + = iλ e iλx f(x)dx = iλf [f](λ). 2.6 f(x)xf(x)fourier iλe iλx f(x)dx F [ ixf(x)] = d F [f]. (2.1) dλ F [ ixf(x)] = = d dλ ( ix)f(x)e iλx dx f(x)e iλx dx = d dλ F [f]. 3
Fourier F [f](λ) = g(λ 1,, λ n ) = Fourier f(x 1,, x n ) = (2π) n e i P n k=1 λ kx k f(x 1,, x n )dx 1 dx n. (2.11) e i P n k=1 λ kx k g(λ 1,, λ n )dλ 1 dλ n. (2.11a) Fourier2.1 2.6 2.1 f(x)(, )Fourier(, ) cos λxλsin λxλ F [f] = F [f](λ) = = e iλx f(x)dx (cos λx + i sin λx) f(x)dx f(x) cos λxdx. 2.2 Fourier Fourier Fourier 2.2 Cauchy FourierCauchy u t = c 2 u xx + f(t, x), (2.12) tcauchy u(, x) = ϕ(x). (2.13) u t = c 2 u xx, (2.14) 4
u(, x) = ϕ(x). (2.15) xfourier F [u(t, x)] = ũ(t, λ), F [ϕ(x)] = ϕ(λ). (2.16) (2.14)xFourier2.5 (2.15) dũ dt = c2 λ 2 ũ, (2.17) ũ(, λ) = ϕ(λ). (2.18) (2.17)(2.18)λ ũ(t, λ) = ϕ(λ)e c2 λ 2t. (2.19) e c2 λ 2t Fourier F 1 [e c2 λ 2t ] = 1 e (c2 λ 2t iλx) dλ = 1 e c2 t(λ ix 2c 2π 2π 2 t )2 dλ e x2 4c 2 t. e c2 t(λ ix 2c 2 t )2 dλ = e c2 tλ 2 dλ = 1 c e y2 dy = t F 1 [e c2 λ 2t ] = 1 2c πt e x 2 4c 2 t. π c t, 2.2(2.19)Cauchy(2.14)-(2.15) u(t, x) = 1 2c ϕ(ξ)e (x ξ)2 4c 2 t dξ. (2.2) πt Cauchy u t = c 2 u xx + f(t, x), (2.21) u(, x) =. (2.22) Cauchy(2.21)-(2.22) u(t, x) = t 5 w(t, x; τ)dτ, (2.23)
w(t, x; τ)cauchy w t = c 2 w xx (t > τ), (2.24) (2.2)Cauchy(2.21)-(2.22) u(t, x) = 1 2c π w(τ, x) = f(τ, x). (2.25) t f(τ, ξ) e (x ξ) 2 4c 2 (t τ) dξdτ. (2.26) t τ (2.2)(2.26)Cauchy(2.12)-(2.13) u(t, x) = 1 2c πt + 1 2c π ϕ(ξ)e (x ξ)2 4c 2 t dξ t f(τ, ξ) e (x ξ) 2 (2.27) 4c 2 (t τ) dξdτ. (t τ) 2.3 ufourier (2.27) Cauchy(2.12)-(2.13) ϕ(x) (2.2)Cauchy(2.14)-(2.15)(2.2) Poisson ϕ(x) Poisson ϕ(x) M. (2.28) (2.2) u(t, x) M 1 π = M 1 π e ζ2 dξ = π, 1 2c (x ξ) 2 π e 4c 2 t dξ e ζ2 dξ = M ( ζ = ξ x ) 2c. t (2.2)u(t, x)u(t, x) 6
t > (2.2)u(t, x)(2.14) 1 2c (x ξ) 2 πt e 4c 2 t (2.29) tx(ξ)t > (2.14) t > (2.14)(2.2) (2.2) x (2.2)x 1 2c π (x ξ)ϕ(ξ) 2c 2 t 3 2 e (x ξ)2 4c 2 t dξ, t t > (t )t > u(t, x) x = 1 2c π (x ξ)ϕ(ξ) 2c 2 t 3 2 e (x ξ)2 4c 2 t dξ, x(2.2) t > (2.2)u(t, x)(2.14) (2.2)u(t, x)(2.15) x t, x x u(t, x) ϕ(x )ε > δ > x x δ, t δ (2.2)ζ = ξ x 2c t ϕ(x ) u(t, x) = 1 π u(t, x) ϕ(x ) ε. ϕ(x ) = 1 π ϕ(x + 2c tζ)e ζ2 dζ. (2.3) ϕ(x )e ζ2 dζ. (2.31) u(t, x) ϕ(x ) = 1 [ϕ(x + 2c tζ) ϕ(x )]e ζ2 dζ. (2.32) π ε > N > 1 e ζ2 dζ π N ε 6M, 1 N π 7 e ζ2 dζ ε 6M. (2.33)
Nϕ(x)δ > x x δ, < t δ ϕ(x + 2c tζ) ϕ(x ) ε 3 ( N ζ N). (2.34) (2.28)(2.33)(2.34)(2.32) u(t, x) ϕ(x ) = 1 [ϕ(x + 2c tζ) ϕ(x )]e ζ2 dζ π 1 π N N + 2M π N ε 3 π N N [ϕ(x + 2c tζ) ϕ(x )]e ζ2 dζ e ζ2 dζ + 2M e ζ2 dζ π e ζ2 dζ + 4M ε 6M ε 3 + 2ε 3 = ε. N (2.35) Poisson(2.2)u(t, x)cauchy (2.14)- (2.15) (2.27)(2.12)(2.13) 1. Fourier (1) e ηx2 (η > ); (2) e a x (a > ); (3) x (a 2 + x 2 ), 1 (a >, k). k (a 2 + x 2 ) k 2. f(x)(, )F [f] 3. FourierCauchy u t = c 2 (u xx + u yy + u zz ), u t= = ϕ(x, y, z). 4. (2.27)(2.12)(2.13) 5. (2.14)Cauchy (1) u t= = sin x, 8
(2) (2.14) u(, x) = ϕ(x) ( < x < ), 6. (t, x, y) u(t, ) =. v(t, x, y; τ, ξ, η) = 1 (x ξ) 2 +(y η) 2 4πc 2 (t τ) e 4c 2 (t τ) v t = c 2 (v xx + v yy ), (τ, ξ, η) v t + c 2 (v ξξ + v ηη ) =. 7. u 1 (t, x)u 2 (t, x)cauchy u(t, x, y) = u 1 (t, x)u 2 (t, y)cauchy u 1 t = 2 u 1 c2 x, 2 u 1 t= = ϕ 1 (x); u 2 t = 2 u 2 c2 y, 2 u 2 t= = ϕ 2 (y), ( 2 u x + 2 u 2 y 2 u t = c2 u t= = ϕ 1 (x)ϕ 2 (y) 8. Cauchy ( 2 u x + 2 u 2 y 2 u t = c2 u t= = ϕ(x, y) u(t, x, y) = 1 4πc 2 t ), ), ϕ(ξ, η)e (x ξ)2 +(y η) 2 4c 2 t dξdη. 9
3. u t c 2 u xx = (t >, < x < l), (3.1) t = : u = ϕ(x), (3.2) x = : u =, (3.3) x = l : u x + σu =, (3.4) σ u(t, x) = T (t)x(x), (3.5) T (t)x(x)tx(3.5)(3.1) XT = c 2 X T, T c 2 T = X X. λ T + λc 2 T =, (3.6) X + λx =. (3.7) 1
(3.7)(3.3(3.4)X(x) X() =, X (l) + σx(l) =. (3.8) (3.7)(3.8) 2 (i) λ X (ii) λ > X(x) = A cos λx + B sin λx. (3.9) X() = A =. (3.8) B( λ cos λl + σ sin λl) =. (3.1) X(x)λ λ cos λl + σ sin λl =, (3.11) λ (3.12) tan λl = λ σ. (3.12) w = λl, (3.13) tan w = w lσ. (3.14) (3.1)3.1(3.14) w k > (k = 1, 2, )w k (k 1 2 )π < w k < kπ f f=tan w w 1 w 2 w f = w lσ 3.1. (3.14) 2
(3.7)(3.8) λ = λ k (3.6) λ k = ( wk l ) 2 (k = 1, 2, ) (3.15) X k (x) = B k sin λ k x = B k sin w k x (k = 1, 2, ). (3.16) l T k (t) = C k e c2 λ k t (k = 1, 2, ). (3.17) u k (t, x) = A k e c2 λ k t sin λ k x (k = 1, 2, ). (3.18) (3.1)(3.3)(3.4) u(t, x) = A k e c2 λ k t sin λ k x. (3.19) A k (3.19)(3.2) (3.3)t = u(t, x)ϕ(x) k=1 ϕ(x) = A k sin λ k x. (3.2) k=1 A k {X k } = {sin λ k x}[, l] X n X m λ n λ m X n + λ n X n =, X m + λ m X m =. X m X n [, l]x n X m (3.3)(3.4) (λ n λ m ) t λ n λ m l X n X m dx = X n X m dx = (X n X m X m X n) l =. l sin λ n x sin λ m xdx = (m n). (3.21) 3
l l M k = sin 2 1 cos 2 λ k x λ k xdx = dx 2 = l 2 sin 2 λ k l 4 = l λ k 2 tan λk l 2 λ k (1 + tan 2 λ k l) = l 2 + σ 2(σ 2 + λ k ). (3.22) (3.2)sin λ k x(3.21) A k = 1 l ϕ(ξ) sin λ k ξdξ. (3.23) M k (3.19)(3.1)-(3.4) u(t, x) = 1 l M k k=1 ϕ(ξ) sin λ k ξdξ e c2 λ k t sin λ k x. (3.24) (3.24)(3.1)-(3.4) ϕ(x) (3.24) t > tx (3.1) (3.3)(3.4) (3.24) e c2 λ k t δ > t δp > k=1 λp k e c2 λ k t ϕ ϕ(x) M (M), (3.25) (3.22) l ϕ(ξ) sin λ k ξdξ Ml 1 2 M k l. (3.26) (3.24)t > tx (3.1)(3.3)(3.4)(3.24) t > t x [, l](3.24)ϕ(x) ϕ(x)ϕ(x) C 1 ([, l])ϕ() = ϕ (l) + σϕ(l) = (3.24)(3.1)-(3.4) 4
1. u t = c 2 u xx (t >, < x < 1), u(t, ) = u x (t, 1) = (t > ), u(, x) = f(x) ( < x < 1). 2. u t = u xx (t >, < x < 1), x, < x 1 u(, x) = 2 1 1 x, 2 < x < 1, u(t, ) = u(t, 1) = (t > ). (3.27) (3.28) 3. lx =, x = l u(, x) = f(x)f(x)u u(t, x) = u. 5