Introduction to Numerical Analysis of Differential Equations Naoya Enomoto (Kyoto.univ.Dept.Science(math))
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- たつや かやぬま
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1 Introduction to Numerical Analysis of Differential Equations Naoya Enomoto (Kyoto.univ.Dept.Science(math)) s00x0427@ip.media.kyoto-u.ac.jp 1 1
2 Van der Pol Bergers 2 KdV 2
3 Fourier Lax Kreiss van der Pol / / Poisson
4 4.6.2 x Burgers Burgers KdV Laplace Poisson Lax Kreiss Bergers KdV
5 y(t) di y (i = 0, 1,, p) dti y (ordinary differential equation) u i u(x 1, x 2,, x n ) (i = 0, 1,, p) x j1 x j2 x ji u (partial differential equation) d2 r dt 2 = ω2 x 2 2 5
6 (1) (2) (3) 3 t R : : : : : dx dt = x d 2 x dt = x 2 dx dt = x2 + p(t)x + q(t) d 2 x dt + 1 ) dx (1 2 t dt + n2 x = 0 t 2 d 2 x dt + 2tdx 2 dt + 2nx = 0 : (1 t 2 ) d2 x dt + 2tdx + n(1 + n)x = 0 2 dt : x(1 x) d2 y + (γ (α + β + 1)x)dy dx2 dx αβy = 0 Painlevé 3 [ ] [ ] 6
7 Painlevé P : P : P : P : P : P : d 2 y = 6y 2 + t dt 2 d 2 y = 2y 3 + ty + α dt 2 d 2 y = 1 ( ) 2 dy 1 dy dt 2 y dt t dt + 1 t (αy2 + β) + γy 3 + δ y d 2 y = 1 ( ) 2 dy + 3 dt 2 2y dt 2 y3 + 4ty 2 + 2(t 2 α)y + β y ( d 2 y 1 = dt 2 2y + 1 ) ( ) 2 dy 1 dy (y 1)2 + y 1 dt t dt t 2 d 2 y = 1 ( 1 dt 2 2 y + 1 y ) ( ) 2 dy y t dt y(y 1)(y t) + t 2 (t 1) 2 ( αy + β y ) ( 1 t + 1 t ( α + β t y + γ t 1 t(t 1) + δ 2 (y 1) 2 (y t) 2 ) dy y t dt ) Newton Newton Newton : m d2 r dt 2 = F (r) : d 2 x dt 2 = ω2 x 7
8 : d 2 x dt + f(x)dx 2 dt + xg(x) = 0 van der Pol : dx 2 dt a(1 2 x2 ) dx + x = 0 (a > 0) dt : d 2 x dt + adx 2 dt + x x3 = 0 Hill : : d 2 u dt + q(t)u = 0 2 d 2 u dt + 2 (ω2 2ɛ cos t)u = 0 : dx = αx γxy dt dy = βy + δxy dt : : : dn = αn(1 λn) dt dx dt = y z dy dt = x + αy dz = b cz + xz dt dx = σ(x y) dt dy = γx y xz dt dz = bz + xy dt 8
9 3 2 3 or : [ ] : a : : b : u t = ν u u = µ u + f(x, t) t u t = ν 2 u x β u 2 x u t = ν 2 u x + 2 ku2 u t = ν 2 u + au ku2 x2 a Brown b 1930 Fisher Kolmogorov-Petrovskii-Piskov u u t = u + f(u) 9
10 : [ ] : a 1 : b : KdV : : : 2 u t = u 2 2 u = u + f(x, t) t2 [ ] 2 u t + k u 2 t 2 u c2 x 2 u t + c u x = 0 [1 ] u t + cu u x = 0 u t + cu u x ν 2 u x = 0 2 u t + 6u u x + 3 u x = 0 3 a Heaviside b Navie-Stokes Laplace Poisson Laplace a : u = 0 P oisson [Laplace ] : u = f Helmholtz b : u + k 2 u = 0 Helmholtz : u k 2 u = 0 a Laplace b Schrodinger 2 φ + Eφ = 0 2m 10
11 : ρ 2 u = µ u + (λ + µ) grad div u + ρf t 2 : µ u (λ + µ) grad div u = ρf : ρ 2 ν t = (λ + 2 µ) 2 ν + ρg : ρ 2 u t 2 = 2 x 2 ( EI 2 u x 2 ) + f Schrodinger : i ( ) t φ = 2 2m + V (r) φ div E = ρ ε div B = 0 Maxwell : rot E + B t = 0 E rot B εµ 0 t = µ 0i v Eular : t + (v )v + 1 ρ grad p = f Navie Stokes : Eular Lagrange : Hamilton : v t + (v )v + 1 grad p ν v = f ρ d L L = 0 dt q i q { i qi = H p i p i = H q i Hamilton Jacobi : H (q, q S, t) + S Klien Gordon : Dirac : ( 1c µ=1 t = 0 t m2 c γ µ φ x µ + iκφ = 0 ) φ = 0 Einstein : R i j(x) 1 2 δi jr(x) = κt i j (1 i, j 4) [ ] : R i j(x) 1 2 δi jr(x) + Λδ i j = κt i j (1 i, j 4) 11
12 2 d 2 x dt 2 = ω2 x t 0 x(t 0 ) dx dt (t 0) n n (1): (2): (3): (1):Derichlet (2):Neumann (3): 1.2 Painlevé 12
13 2 (1): (2): (3): (4):
14 [0, 1] [ ] 14
15 Lax Fourier 2.1 Step f x f(x + x) f(x) x 2 x, D ) t t D D 2 D 1 D 15
16 (1): (2): (3): (4): 2.2 df f(x + h) f(x) (x) lim dx h 0 h f x f(x + x) f(x) x f x f(x) f(x x) x 1 2 f x f(x + x) f(x x) 2 x 16
17 2.1. df/dx f/ x f(x + x) f(x), x f(x) f(x x), x f(x + x) f(x x) 2 x, f(x) x f 0 x x x x + x x f(x) Taylor f(x + x) = f(x) + ( x) df ( x)2 d 2 f ( x)3 d 3 f (x) + (x) + dx 2! dx2 3! dx + 3 O(( x)4 ) Taylor f(x + x) f(x) x = df ( x) d 2 f (x) + dx 2! dx (x) + 2 O(( x)2 ) x x 1 f(x) f(x x) x f(x + x) f(x x) 2 x = df dx x d 2 f (x) 2 = df ( x)2 (x) + dt 6 dt (x) + 2 O(( x)2 ) d 2 f dx (x) + 2 O(( x)3 ) x 1 x 2 3f(x) 4f(x x) + f(x 2 x) 2 x 17
18 3f(x) 4f(x x) + f(x 2 x) 2 x = df ( x)2 d 3 f (x) dx 3 dx (x) + 3 O(( x)3 ) x 2 2f(x + x) + 3f(x) 6f(x x) + f(x 2 x) 6 x 2f(x + x) + 3f(x) 6f(x x) + f(x 2 x) 6 x = df ( x)3 d 4 f (x) + dx 12 dx (x) + 4 O(( x)4 ) x 3 f(x ± s x) (s = 0, 1, 2, ) Remark Remark d 2 f f(x + x) 2f(x) + f(x x) dx2 ( x) 2 Taylor f(x + x) 2f(x) + f(x x) ( x) 2 = d2 f ( x)2 d 4 f (x) + dx2 12 dx + 4 O(()4 ) x 2 f(x + 2 x) + 16f(x + x) 30f(x) + 16f(x x) f(x 2δx) ( x) 2 x KdV 1 18
19 Laplace 3 Laplace 1 u t (x, t) = 2 u (x, t) x [0, 1], t [0, T ] x2 u(0, t) = u(1, t) = 0 t [0, T ] u(x, 0) = sin x x [0, 1] [0, 1] sin x 0 5 Step u (x, t) 1 t u(x, t + ) u(x, t) 2 u(x + x, t) 2u(x, t) + u(x x, t) ( x) 2 u t (x, t) = 2 u (x, t) x2 u(x, t + ) u(x, t) = u(x + x, t) 2u(x, t) + u(x x, t) ( x) 2 1 [0, 1] N = 1/ x (j, k) (j x, k) (0 j N, k = 0, 1, 2, ) x,
20 (i, j) u u(j x, k) u k j u k+1 j u k j = uk j+1 2u k j + u k j 1 ( x) 2 u k+1 j = λu k j+1 + (1 2λ)u k j + λu k j 1 λ = ( x) 2 λ u k+1 j (k + 1) 3 k u k j 1 u k j u k j+1 20
21 x 2 x 3 x 1 = N x u k j u 0 j = sin (j x) (j = 0, 1, 2,, N 1, N) u k j u k 0 = u k N = 0 (k = 0, 1, 2, ) j = 0, j = N 2 2 C++ TeX Visual 21
22 λ 1/2 λ > 1/2 λ = ()/(δx) 2 1/2 λ 1/2 sin 0 λ < 1/2 λ > 1/2 λ δx Fourier Lax Kreis Lax
23 Burgers 1 Banach Key Word Banach {u k j } max j (u k j ) = u k C k u k C u k u k λ > 1/2 k C 7 /( x) 2 1/2 / 6 [ ],[ ] 7 23
24 2.4.2 Fourier Fourier u k+1 = Su k S S = m c m T m T f(x) T (f(x)) = f(x + x) m Z c m, x u k j = g n exp(iξj x) i = 1 ξ g x, T exp(iξj x) T exp(iξj x) = T exp(iξx) (j x = x ) = exp(iξ(x + x)) (T ) = exp(iξ x) exp(iξj x) T m exp(iξj x) = exp(iξm x) exp(iξj x) u k+1 = Su k S g k+1 exp(iξj x) = m c m g m exp(iξm x) exp(iξj x) g = m c m exp(iξm x) g S 1 S x x = h() k ξ 1 24
25 0 k T n, g n C C C = exp(kt ) (1 + KT/n) n (1 + K) n von Neumann von Neumann u k+1 = Su k S g g 1 + K() K g x, g 1 + K x = h() 2 S 1 u k+1 = S 0 u k S 0 = p C (0) p T p, S 1 = q C ( q1)t q u k j = g k exp(iξj x) S = S 1 1 S 0 g = p C(0) p exp(iξp x) q C(1) q exp(iξq x) 2 c m g k g ( )γ von Neumann γ = 1 + K K 1 von Neumann 2 Fourier 25
26 2.4.3 Lax Lax S t u(x, t) u(x, t + ) = Su(x, t) + O() ν Z >0 u(x, t + ) = Su(x, t) + O(() ν+1 ) ν f t u(x, t) E(t) u(x, t) = E(t)f(x) f 0 t T E(t)f K f K T 8 0 n = t T 0 f U(x, t) u(x, t) = {S n E(t)}f 0 U(x, t) t u(t, x) nk = t 0 n 0 n Lax 8 26
27 2.3. Lax ( ) u t = P u x u = P ( t x) u Lax Fourier Kreiss Fourier Fourier ξ x A(ξ, x) x A ξ x ξ, 0 < < τ, 0 m T A(ξ, x) m K von Neumann m Kreiss 27
28 2.4. Kreiss ξ, x( = h n n A(ξ, h) F (A): C A F A m C m (R):C R z > 1 z A F (A zi) 1 C R z 1 (S):C S, C B A F S (i) S, S 1 C S (ii) B = SAS 1 B = κ 1 B12 B 13 B 1n 0 κ 2 B 23 B 2n 0 0 κ 3 B 3n κ n κ 1 κ 2 κ n κ i 1 (i = 1, 2,, n) B ij C B (1 κ j ) (H): A F Hermite H C H C 1 H I H C HI A HA H n n A, B A B A B 28
29 f(t) p (p = 1, 2, ), δx 2 t δt u k j f k k (k + 1) 1 3 Newton
30 du (t) = λu(t) dt, du u log u = λt + C (t) = λdt u(t) = Ce λt (C ) u(t) 1 (u(t + ) u(t))/ u(t + ) u(t) = λu(t) u n+1 = (1 + λ)u n u n n u(0) = u 0 = a u n = (1 + λ) n a C = a u(t) = ae λt n = t(const.) 0(n ) ( lim u n = lim a 1 + λt ) n = ae λt 0 n n n=const. n = const. 0 t n = const. 0 u(t) u(t ) = λu(t) u n+1 u n = u n+1 n u n = a (1 λ) n 30
31 n = t(const.) 0(n ) ( lim u n = lim a 1 λt ) n = ae λt 0 n n n=const. u(t + ) u(t ) 2 = λu(t) u n+1 = u n 1 + 2λ()u n n + 1 n, n 1 2 u 1, u 0 2 u 0 = a u 0 u 1 u 0 u 1 u 1 2 2f(x + x) + 3f(x) 6f(x x) + f(x 2 x) 6 x u n+1 = 1 2 [ 6()u n 3u n + 6u n 1 u n 2 ] n + 1 n 2, n 1, n 3 u 0 u 1, u 2 = 0.1 =
32 q(t) 0 t 3.3 k m d 2 q dt 2 (t) + ω2 0q(t) = 0 ω 2 0 = k m 0 Hamilton dq p(t) = m 0 (t) 2 dt dp (t) = kq(t) dt dq dt (t) = 1 p(t) m 0 HamiltonianH(t) dh dt H(t) = 1 2m 0 p(t) kq(t)2 (t) = 0 32
33 p(t + ) p(t) = kq(t) q(t + ) q(t) = 1 p(t) m 0 p m+1 p m = kq m q m+1 q m = p m m 0 q m+1, p m+1 p m+1 = p m kq m q m+1 = q m + p m m 0 0 q(0) = q 0 0 p(0) = p 0 q m+1, p m+1 q m, p m q(t) H m (t) = 1 p 2 m + 1 2m 0 2 kq2 m p q p(t + ) p(t) = kq(t) q(t) q(t ) = 1 p(t) m 0 p m+1 = p m kq m q m+1 = q m + m 0 p m+1 q q m, p m p m+1 q m ω ω 0 1 H(t) dh (t) = 0 dt H(t + ) H(t) = 0 33
34 H(t + ) H(t) H(t) = 1 2m 0 p(t) kq(t)2 = 1 [ ] [ ] p(t + ) p(t) p(t + ) + p(t) m 0 2 [ ] [ ] q(t + ) q(t) q(t + ) + q(t) +k 2 0 p(t + ) p(t) = k (q(t + ) + q(t)) 2 q(t + ) q(t) = 1 (p(t + ) + p(t)) 2m 0 { p m+1 p m = k 2 (q m+1 q m ) q m+1 q m = 1 2m 0 (p m+1 + p m ) p m+1, q m+1 ) (1 ()2 4m 0 k p m k()q m p m+1 = ( ) 1 + ()2 4m 0 k ) (1 ()2 q m+1 = 4m 0 k ( 1 + ()2 4m 0 k + m 0 p m ) 3.4 dn (t) = αn(1 λn(t)) (α, λ 0) dt N(t) dn N(t) MN(t) dt M 34
35 M = a bn(t) N 2 ( ) 1 dn 1 N 2 (t) dt = α N(t) λ ( ) 1 n(t) = λ N(t) dn dt (t) = d ( ) 1 dt N(t) λ = 1 dn N 2 (t) dt = αn(t) n(t) = Ce αt N(0) = N 0 N(t) = 1 λ + ( N 1 0 λ ) e αt N(t) 1 λ 0 t N 0 0 < N 0 < 1/λ N(t + ) N(t) = αn(t)(1 λn(t)) N n+1 = ()αn n (1 λn n ) + N n α = λ = 1 N 0 = N = N = Step
36 N(t) = g(t)/f(t) 1 dg f dt + g d dt ( 1 f ) = α g f ( 1 λ f g ) 1 dg f dt g df f 2 dt = α g ( 1 λ g ) f f f 2 f dg dt g df dt = αg(f λg) f, g g(t + ) g(t) f(t + ) f(t) f g = αg(t)(f(t) λg(t)) g(t + )f(t) g(t)f(t + ) = α()g(t)(f(t) λg(t)) f(t) f(t)h(t), g(t) g(t)h(t) {g(t + )f(t) g(t)f(t + )} h(t + )h(t) = α()g(t)(f(t) λg(t))(h(t)) 2 f, g h(t) 2 h(t + )h(t) 3 g(t + )f(t) g(t)f(t + ) = α()g(t + )(f(t) λg(t)) f, g N 2 N n+1 = N n 1 ()α(1 λn n ) g(t + )f(t) g(t)f(t + ) = α()g(t)(f(t + ) λg(t + )) N N n+1 = (1 + ()α)n n 1 + ()α(1 λn n ) 36
37 3 g(t + )f(t) g(t)f(t + ) = 1 α() [g(t + )f(t) + g(t)f(t + )] α()λg(t)g(t + ) 2 N ( N n+1 = ()α) N n 1 1α() + ()αλn 2 n N 2 N(t + ) N(t ) 2 = αn(t)(1 λn(t)) N n+1 = 2α()N n (1 λn n ) + N n 1 1 = N 1 = 1.0 = 0.7 N = 1 N = 1 N = 0 N = 1 37
38 + van der Pol 3.5 van der Pol van der Pol Poincare-Bendixson dx 2 dt a(1 2 x2 ) dx + x = 0 (a > 0) dt 1 dx/dt = p(t) dx (t) = p(t) dt dp dt (t) = a(1 x2 )p x x, p x(t + ) x(t) = p(t) p(t + ) p(t) = a(1 x(t) 2 )p(t) x(t) { x n+1 = x n + ()p n p n+1 = p n + () [a(1 x 2 n)p n x n ] 38
39 a = 0.01 x 0 = 0.01, p 0 = 0.0 = 2.0 a = 10, x 0 = 0.01, p 0 = 0.0 = 0.5 a / 2 / 2 dx = αx γxy dt dy = βy + δxy dt x(t) y(t) x, y { x n+1 = x n + (αx n γx n y n ) y N+1 = y n (βy n δx n y n ) 0 39
40 α, β, γ, δ dx dt = y z dy dt = x + αy dz = b cz + xz dt x n+1 = x n ()(y n + z n ) y n+1 = y n + ()(x n + ay n ) z n+1 = z n + ()(b cz n + x n z n ) a = 0.2, b = 0.2, c = 5.7 xz b bx dx dt = y z dy dt = x + αy dz = bx cz + xz dt x n+1 = x n ()(y n + z n ) y n+1 = y n + ()(x n + ay n ) z n+1 = z n + ()(bx n cz n + x n z n ) 9 9 [ 2] 40
41 3 10 dx = σ(x y) dt dy = γx y xz dt dz = bz + xy dt x n+1 = x n + ()σ(x n y n ) y n+1 = y n + ()(γx n y n x n z n ) z n+1 = z n ()(bz n x n y n ) σ = 10, γ = 26, b = d2 u (x) + q(x)u(x) = f(x) (0 < x < 1) dx2 3 u(0) = 0, u(1) = a Derichlet du du (0) = b, dx dx (1) = c (Neumann ) du (0) = b, u(1) = a ( ) dx [ ] 41
42 1 100 x d 2 u dt 2 u(x + x) 2u(x) + u(x x) ( x) 2 q(x) = λ, f(x) = µx d 2 u (x) = λu(x) + µx (0 < x < 1) dx2 u(x + x) 2u(x) + u(x x) ( x) 2 = λu(x) + µx u m+1 2u m + u m 1 ( x) 2 = λu m + µm x u m+1 (2 + λ( x) 2 )u m + u m 1 = µm( x) 3 Neumann u 0 = 0, u N = a Derichlet u 1 u 0 u N u N 1 = b, = c (Neumann ) x x u 1 u 0 = b, u(1) = a ( ) x Derichlet {j x} (j = 0, 1,, N). (2 + λ( x) 2 ) 1 1 (2 + λ( x) 2 ) (2 + λ( x) 2 ) u 1 u 2.. u N 1 = µ( x) 3 2µ( x) 3.. (N 1)µ( x) 3 a 42
43 0 1 Derichlet u 0, u N 2 Derichlet u 1, u N 1 3 Neumann (2 + λ( x) 2 ) 1 1 (2 + λ( x) 2 ) (2 + λ( x) 2 ) u 1 u 2. =.. u N 1 u N b xµ( x) 3 2µ( x) 3.. (N 1)µ( x) 3 c x 0 1 Neumann 2 Neumann u 1,, u N
44 1 1 1 (2 + λ( x) 2 ) 1 1 (2 + λ( x) 2 ) (2 + λ( x) 2 ) u 1 u 2... u N 1 = b x µ( x) 3 2µ( x) 3.. (N 1)µ( x) 3 a 0 1 Neumann Derichlet 1 1 x
45 4 1 Lax u(t, x) = T (t)x(x) 11 [ ] [ ] [MKI] 45
46 2 4.2 u t (x, t) = 2 u (x, t) x [0, 1], t [0, T ] x2 u(0, t) = φ(t), u(1, t) = ψ(t) t [0, T ] u(x, 0) = f(x) x [0, 1] t x = λu k j+1 + (1 2λ)u k j + λu k j 1 u k 0 = φ(k), u k N = ψ(k) u 0 j = f(j x) u k+1 j λ = ( x) 2 λ (1) (2) / (3) (4) (1) 46
47 4.2.1 Fourier u k+1 = Su k S S T S = λt 1 + (1 2λ)T 0 + λt 1 S = m c m T m g g = m c m exp(iξm x) g g = λ exp(iξ x) + (1 2λ) + λ exp( iξ x) = 1 4λ sin 2 ( ξ x 2 von Neumann 1 K 1 4λ sin 2 ( ξ x 2 K 4λ sin 2 ( ξ x 2 ) 2 + K ) 1 + K ξ x/2 π/2 ξ λ K 4 ) λ 1 2 von Neumann / 1/2 47
48 (1) u t = k u + f (t > 0, x Ω R n, n = 1, 2, 3 ) u(t, x) Ω = ψ(t, x) Ω (Dirichlet 1 ) u(0, x) = ϕ(x) x Ω 4.1. [ ] u = u(t, x) u/ t = u + f(x, t) Dirichlet (1) f f 0 u t k u u(t, x) G T Γ T (2) f f 0 u t k u u(t, x) G T Γ T f = 0 Γ T (2) (2-1) t > 0 (2-2) t < 0 Fourier (2-3) t > 0 t > 0 C 48
49 { 1 x [ 1, 1] f(x) = 0 otherwise (2) (3) (4) x 2 5 (5) [ a, a] 0 t > 0 t > 0 Fourier 49
50 Fourier (4) (5) (5 ) x 0 x t > 0 x 3 / x x P P P A A B B x / x = 1 x AB ()/( x) 2 P λ 1/2 50
51 4.2.3 k u k+1 j u k j = λu k+1 j+1 2λuk+1 j + λu k+1 j 1 u k u k+1 1 2λ λ λ1 1 2λ λ λ λ 1 2λ u k+1 1 u k u k+1 N 1 = u k 1 + λφ((k + 1)) u k 2.. u ( N 1) k + λψ((k + 1)) g = λ sin 2 ( ξ x 2 ) g 1 N = 100 x = 0.01 /( x) 2 1/2 = = (N-1) u k+1 j = u k 1 j + 2λ(u k j+1 2u k j + u k j+1) 51
52 ( ) ( ) ξ x ξ x g = 4λ sin 2 ± λ sin g > 1 u k+1 j = u k j λ(uk j+1 2u k j + u k j 1) λ(uk+1 j+1 2uk+1 j + u k+1 j 1 ) u k u k+1 u k+1 g = 1 2λ sin 2 ( ξ x λ sin 2 ( ξ x 2 ) ) λ 1 4, 3 λ 1 6 1/4 1/6 ( x) 2 ADI :Alternating Direction Implicit Method u k u k+1 2 x u k+1/2 u k+1/2 = u k λ(( x)2 u k+1/2 + y 2 u k ) y u k+1 u k+1 = u k+1/ λ(( x)2 u k+1/2 + ( y) 2 u k+1 ) 2 g 1, g 2 g g = g 1 g 2 = g 1 g 2 = 1 2λ sin 2 ( η x λ sin 2 ( ξ x 2 ) ) 1 2λ sin 2 ( ξ x λ sin 2 ( η x 2 ) ) 1 52
53 u t + c u x = 0 (c > 0) u t 2 = c 2 u x 2 ( ) ( ) u u t + c u x t c u = 0 x 2 2 f, g u(x, t) = f(x ct) + g(x + ct) f(x ct) x u t + c u x = t, x t x t x t, x u(t +, x) u(t, x) u(t, x + x) u(t, x) + c = 0 u k+1 j = u k j c ( ) u k x j+1 u k j 53
54 2 1 u k+1 j (k + 1) 2 k u k j u k j+1 g = 1 c ( (exp(iξ x) 1) = 1 + c ) c x x x exp(iξ x) λ = / x = const. g 1 + cλ cλ g g ξ 1 + K cλ cλ t x u(t +, x) u(t, x) u k+1 j u(t, x + x) u(t, x x) + c = u k j c 2 ( u k x j+1 uj 1) k 3 1 = 0 54
55 u k+1 j (k + 1) 3 u k j 1 u k j u k j+1 k g = 1 c ( ) ( ) (exp(iξ x) exp( iξ x)) = 1 ic sin(ξ x) 2 x x g = ( ) c 2 sin 2 (ξ x) x λ = / x = const. max g = 1 + c 2 λ 2 > 1 λ ρ = /( x) 2 = const. g = 1 + c 2 ( x) 2 sin2 (ξ x) = 1 + c 2 ρ sin 2 (ξ x) c2 ρ K c 2 rho/2 von Neumann / x = const. x 2 t x u(t +, x) u(t, x) u k+1 j u(t, x) u(t, x x) + c = u k j c ( u k x j uj 1) k t, x = 0 u k+1 j (k + 1) 3 k u k j 1 u k j 55
56 g = 1 c (1 exp( iξ x)) = x ( 1 c x ) + c x exp( iξ x) g 1 cλ cλ 1 cλ = c x 1 λ = / x = const. cλ cλ c > 0 c > 0 c < 0 t, x / / 1 y = x ct 1 c > 0 f(x ct) x 1/c 56
57 1 P = (x, t) P 1/c x P = (x, t) t, x (x, t) / x / x /( x) 2 57
58 4.3.4, x 1 f u(x, t) = f(x ct) f / x 1/c 58
59 λ = / x λ = 1/2 3 u k+1 j = 1 2 (uk j+1 + u k j 1) c 2 x (uk j+1 u k j 1) u k+1 j = u k 1 j c x (uk j+1 u k j 1) u k+1 j = u k j 1 ( ) (u kj+1 u kj 1) ( ) 2 + c2 (u k j+1 2u k j + u k 2 x 2 x j 1) λ 1/c u t 2 = u
60 u(t +, x) 2u(t, x) + u(t, x) () 2 u k+1 j u(t, x + x) 2u(t, x) + u(t, x x) = c ( x) 2 = λ 2 cu k j ( 1 λ 2 c ) u k j + λ 2 cu k j 1 u k 1 j 2 2 (k + 1) k (k 1) (j 1) x (j + 1) x j x u t2 = u t [0, ), x [0, 1] u(x, 0) = φ(x), t u (x, 0) = ψ(x) u(0, t) = 0, u(1, t) = 0 Neumann u k 0 = u k N = 0, u 0 j = φ(j x) u 1 u(t + ) u(t) 60 = ψx
61 u 1 j = u 0 j + ()ψ(j x) u(x, t) t Taylor u(x, t) = u(x, 0) + u ()2 2 u (x, 0) + t 2 t (x, 0) + 2 O(()3 ) t = 0 2 u t (x, 0) = u 2 c 2 x (x, 0) = 2 cu0 j+1 2u 0 j + u 0 j 1 ( x) 2 u 1 j = u 0 j + ψ(j x) + c 2 ( ) 2 (u 0 j+1 2u 0 j + u 0 x j 1) 61
62 c = 1 1/c u k+1 j = λ 2 cu k j ( 1 λ 2 c ) u k j + λ 2 cu k j 1 u k 1 j u k j = g k exp(iξj x) g 2 g 2 2g + 1 = 4g ( ) 2 ( ) ξ x sin 2 x 2 a = x sin ( ) ξ x 2 g 2 2(1 a 2 )g + 1 = 0 a 1 g = 1 2a 2 ± i2a 1 a 2 g = 1 a > 1 g g 1 g 2 g 1 g 2 = 1 g 1, g 2 1 ( ) ( ) a = ξ x x sin 2 (1) (2) x 1 a 1 g 1 = g 2 = 1 x > 1 a > 1 ξ max ( g 1, g 2 ) > 1 2 / [ ] 62
63 bababababababababababababababababababababababab 63
64 4.5 Laplace Poisson Derichlet Neumann Derichlet Neumann 64
65 Poisson Poisson 2 u x + 2 u = f(x, y) x (0, 1), y (0, 1) 2 y2 u(x, 0) = a(x) x (0, 1) u(x, 1) + u (x, 1) = 0 x (0, 1) y u(0, y) = 0 y (0, 1) u (1, y) = b(y) y (0, 1) x 1 OABC u(x, y) OA OC BC 0 AB C u(x, 1) + u (x, 1) = 0 y B u(0, y) = 0 u(1, y) = b(y) x O u(x, 0) = a(x) A x x,y y P ij = (i, j) = (i x, j y) 2 u i+1,j 2u i,j + u i 1,j ( x) 2 + u i,j+1 2u i,j + u i,j 1 ( y) 2 = f(i x, j y) 65
66 f x y x = y = h u i+1,j + u i,j+1 4u i,j + u i 1,j + u i,j 1 h 2 = f(ih, jh) (i, j = 1, 2,, N 1) u i,0 = a(ih) (i = 0, 1, 2,, N 1) u 0,j = 0 (j = 0, 1, 2,, N 1) u N,j u N 1,j = b(jh) (j = 1, 2,, N 1) h u i,n + u i,n u i,n 1 h = 0 (i = 1, 2,, N 1) 4 1 u i,j 4 (i + 1, j), (i, j + 1), (i 1, j), (i, j 1) 1 Poisson 66
67 x y x 0, y 0 [ ] L ( 2 = 1 h x y 2 u(x + h, y) 2u(x, y) + u(x h, y) u(x, y + h) 2u(x, y) + u(x, y h) h u(x, y) = + h 2 h 2 = 4 ( ) u(x + h, y) + u(x, y + h) + u(x h, y) + u(x, y h) u(x, y) h 2 4 (x, y) h 4 u(x, y) ) h u = 4 u(x, y) u(x, y) h2 67
68 h (x, y) (x, y) u u = 0 Laplace h u(x, y) = 0 4 Laplace Laplace 4.2. Laplace Laplace 2 1 d 2 u/dx 2 ( ) 2 u(x + h) u(x h) u(x) h 2 2 d 2 u/dx 2 = 0 ( 2 u(x + h) u(x h) h 2 2 ) u(x) = u(x) u 3 u t = u [resp. ] u u < 0[> 0] [ ] [ ] [ ] Laplace 68
69 1 2 u t 2 = 2 u x 2 u(x, t) 3 Poisson u = f(x, y, z) u(x, y, z) f f > 0[resp. < 0] u < 0[> 0] u u < 0[> 0] [ ] u(x, y, z) [ ]
70 u t (x, t) = u d 2 (x, t) + a(1 u)u x [0, 1], t [0, T ], a > 0, d > 0 x2 u(0, t) = U(1, t) = 0 t [0, T ] u(x, 0) = φ(x) x [0, 1], φ(0) = φ(1) = 0 1 x u 2 2 u(x, t) u(x, t) t x 2 u(x, t + ) u(x, t) u(x + x, t) 2u(x, t) + u(x x, t) = d + a(1 u(x, t))u(x, t) ( x) 2 u k+1 j = λu k j+1 + (1 2λ)u k+1 j + λu k j 1 + a(1 u k j )u k j λ = d/( x) 2 0 φ(x) 1 λ : [ a + 2d ] 1 > ( x) 2 14 [ 2] 70
71 4.3. (1) u/ t 0 ( ) 0 a dπ 2 lim t u(x, t) = 0 x (0, 1) ( ) dπ 2 < a lim t u(x, t) = W (x) x (0, 1) W (x) u, t = 0 1 t x 2 0 = d w j+1 2w j + w j 1 ( x) 2 + a(1 w j )w j w 0 = w N = 0 ξ 1 = 4d sin ( 2 x π) ( x) 2 2 ( ) 0 ξ 1 W j = 0 ( ) ξ 1 < a 0 W j W j > ( ) 0 a ξ 1 u 0 j µ > 0 u0 j = µ sin(j xπ) lim k uk j = 0 ( j = 0, 1,, N) ( ) ξ 1 < a u 0 j δ (0 < δ 1 ξ 1/a) u 0 j = δ sin(j xπ) lim k uk j = W j ( j = 0, 1,, N) W j lim ξ 1 = dπ 2 x
72 2 2 1 a 3 2λ 0 0 u o j a < 2λ < 1 a x d T = 1/ T λ 1/2 u k+1 j u k j = d uk j+1 2u k j + u k j 1 + a(1 u k+1 ( x) 2 j )u k j 2 () u k j uk+1 j 72
73 4.6.2 x x u t (x, t) = u d 2 x (x, t) + (R 2 1 a 1 u b 1 v)u x [0, 1], t [0, T ], a > 0, d > 0 v t (x, t) = v d 2 x (x, t) + (R 2 2 a 2 v b 2 u)v x [0, 1], t [0, T ], a > 0, d > 0 u(x, 0) = φ(x), v(0, x) = ψ(x) x [0, 1], φ(0) = φ(1) = 0 u v u u (t, 0) = (t, 0) = (t, 1) = (t, 1) = 0 x x x x t 0 t x 2 u k+1 j v k+1 j u k j v k j = d uk j+1 2u k j + u k j 1 ( x) 2 = d vk j+1 2v k j + v k j 1 ( x) 2 + (R 1 a 1 u k j b 1 v k j )u k j + (R 2 a 2 v k j b 2 u k j )v k j 4.5. R 1 = R 2 = R, a 1 = a 2 = a, b 1 = b 2 = b x 1 [0, 1]
74 4.7 Burgers Burgers Navie-Stokes 1 Burgers Burgers Burgers u t + cu u x ν 2 u x 2 = 0 Navie-Stokes 1 Navie-Stokes 1 Burgers Burgers u t + c u x ν 2 u x 2 = 0 u/ t = ν 2 u/ x u/ t + c u/ x 2 c 0 1 Burgers ν u t + cu u x = 0 Burgers Hopf u t + ( ) 1 x 2 u2 = 0 Burgers 1 u t + c u x = 0 74
75 Burgers u t + cu u x ν 2 u = 0 x (0, 1) x2 u(x, 0) = f(x) u(0, t) = u(1, t) = Burgers Burgers u t + cu u x = 0 3 t x x 1 c > 0 3 cu u/ x u u u k j u(x, t + ) u(t, x) u(x, t) u(x x, t) + cu(x, t) x = 0 u k+1 j u k+1 j u k j = u k j + cu k j u k j u k j 1 x = 0 [ 1 c ] x (uk j u k j 1) u k j uk+1 j u k+1 j u k j u k + cu k+1 j u k j 1 j x = 0 u k+1 j = u k j 1 + c x (uk j uk j 1 ) 75
76 u k j (uk j + u k j 1)/2 u k+1 j u k j u k+1 j = u k j c 2 + c uk j + u k j 1 2 uk j u k j 1 x = 0 ( ) (u k j + u k x j 1) (u k j u k j 1) 3 16 Burgers sin 16 [ ] 76
77 sin x 17 sin x 17 [ ] 77
78 4.7.3 Burgers 1 1 cuu x { u u x u uk j uk j 1 x if u > 0 u uk j+1 uk j x if u < 0 u k+1 j u k j + uk j + u k j 2 uk j u k j 1 x Burgers + uk j u k j 2 ν uk j+1 2u k j + u k j 1 ( x) 2 uk j+1 u k j x u u Lax-Wendorf [ ] = 0 Burgers 1 78
79 2 1 79
80 4.8 Blow-up Solution 18 u t = 2 u x + 2 u1+a (t, x) (0, ) (0, 1) u u (t, 0) = (t, 1) = 0 t (0, infty) x x u(0, x) = f(x) x (0, 1) t x 2 u k+1 j u k+1 j u k j u k+1 j = [ ( x) 2 + (uk j ) a u k 1 u k 0 x = uk j+1 2u k j + u k J 1 ( x) 2 + (u k j ) 1+a = 0, ] u k j + ( x) 2 uk j+1 + u k j 1 ( x) 2 u k N uk N 1 x u k 1 = u k 0, u k N = uk N =
81 ( ) 1 k = t min 1, max j u k j a 81
82 / 2 u t + c u 1 x = uv v t + c v 2 x = uv t, x 19 u k+1 j v k+1 j u k j vj k + c 1 u k j+1 u k j x + c 2 v k j+1 v k j x = u k j v k j = u k j v k j u k+1 j, v k+1 j u k+1 j v k+1 j = u k j c 1 x (uk j+1 u k j ) u k j v k j = v k j c 2 x (vk j+1 v k j ) + u k j v k j 2 20 c 1, c 2 19 [ ]
83 4.10 KdV KdV J.Scott-Russell Korteweg de Vries KdV u t + cu u x + ν 3 u x 3 = 0 N.J.Zabusky M.D.Kruskal KdV sech KdV 3 Zabusky-Kruskal 21 u(x, t + ) u(x, t ) 2 u(x + x) + u(x, t) + u(x x, t) = c 3 u(x + 2 x, t) 2u(x + x, t) + 2u(x x, t) u(x 2 x, t) 2 x u(x + x) + u(x, t) + u(x x, t) ν 2( x) 2 21 [ ]p
84 t u(x, t + ) u(x, t) u(x + x) + u(x, t) + u(x x, t) = c 3 u(x + 2 x, t) 2u(x + x, t) + 2u(x x, t) u(x 2 x, t) 2 x u(x + x) + u(x, t) + u(x x, t) ν 2( x) 2 u k+1 j = u k j c 6 x (uk j+1 + u k j + u k j 1)(u k j+1 u k x j 1) ν 2( x) 3 (uk j+2 2u k j+1 + 2u k j 1 u k j 2) KdV Derichlet [0, 1] 0 f(x) sin Zabusky,Kruskal sech 2, x, c, ν 84
85 2 = 0.001, N = 100, c = 1, ν = s 1 = 1, s 2 = 15, s 3 = 70, t 1 = 16, t 2 = 55, A = 1, B = 0.4 s 3 A s 1, s 2 2 B t 1, t 2 c ν KdV 85
86 bababababababababababababababababababababababab 86
87 Laplace Poisson 9.6 Lax Kreiss 9.7 Bergers KdV bababababababababababababababababababababababab 87
88 [ ] [ ] [ ] [ ] [MKI] [ ] 1977 [ ] 1969 [ 1] [ 2] [ ] [ ] 1994 [ ] [ ] [ ] SGC
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1 2003 4 24 ( ) 1 1.1 q i (i 1,,N) N [ ] t t 0 q i (t 0 )q 0 i t 1 q i (t 1 )q 1 i t 0 t t 1 t t 0 q 0 i t 1 q 1 i S[q(t)] t1 t 0 L(q(t), q(t),t)dt (1) S[q(t)] L(q(t), q(t),t) q 1.,q N q 1,, q N t C :
all.dvi
29 4 Green-Lagrange,,.,,,,,,.,,,,,,,,,, E, σ, ε σ = Eε,,.. 4.1? l, l 1 (l 1 l) ε ε = l 1 l l (4.1) F l l 1 F 30 4 Green-Lagrange Δz Δδ γ = Δδ (4.2) Δz π/2 φ γ = π 2 φ (4.3) γ tan γ γ,sin γ γ ( π ) γ tan
t = h x z z = h z = t (x, z) (v x (x, z, t), v z (x, z, t)) ρ v x x + v z z = 0 (1) 2-2. (v x, v z ) φ(x, z, t) v x = φ x, v z
I 1 m 2 l k 2 x = 0 x 1 x 1 2 x 2 g x x 2 x 1 m k m 1-1. L x 1, x 2, ẋ 1, ẋ 2 ẋ 1 x = 0 1-2. 2 Q = x 1 + x 2 2 q = x 2 x 1 l L Q, q, Q, q M = 2m µ = m 2 1-3. Q q 1-4. 2 x 2 = h 1 x 1 t = 0 2 1 t x 1 (t)
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1 1.1 / Fik Γ= D n x / Newton Γ= µ vx y / Fouie Q = κ T x 1. fx, tdx t x x + dx f t = D f x 1 fx, t = 1 exp x 4πDt 4Dt lim fx, t =δx 3 t + dxfx, t = 1
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1 1 f (t) lim t a f (t) = 0 f (t) t a 1.1 (1) lim(t 1) 2 = 0 t 1 (t 1) 2 t 1 (2) lim(t 1) 3 = 0 t 1 (t 1) 3 t 1 2 f (t), g(t) t a lim t a f (t) g(t) g(t) f (t) = o(g(t)) (t a) = 0 f (t) (t 1) 3 1.2 lim
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II 1 II 212 II Gauss-Bonnet II 1 1 1.1......................................... 1 1.2............................................ 2 1.3.................................. 3 1.4.............................................
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(2 X Poisso P (λ ϕ X (t = E[e itx ] = k= itk λk e k! e λ = (e it λ k e λ = e eitλ e λ = e λ(eit 1. k! k= 6.7 X N(, 1 ϕ X (t = e 1 2 t2 : Cauchy ϕ X (t
![(2 X Poisso P (λ ϕ X (t = E[e itx ] = k= itk λk e k! e λ = (e it λ k e λ = e eitλ e λ = e λ(eit 1. k! k= 6.7 X N(, 1 ϕ X (t = e 1 2 t2 : Cauchy ϕ X (t](/thumbs/67/56612994.jpg "(2 X Poisso P (λ ϕ X (t = E[e itx ] = k= itk λk e k! e λ = (e it λ k e λ = e eitλ e λ = e λ(eit 1. k! k= 6.7 X N(, 1 ϕ X (t = e 1 2 t2 : Cauchy ϕ X (t")
6 6.1 6.1 (1 Z ( X = e Z, Y = Im Z ( Z = X + iy, i = 1 (2 Z E[ e Z ] < E[ Im Z ] < Z E[Z] = E[e Z] + ie[im Z] 6.2 Z E[Z] E[ Z ] : E[ Z ] < e Z Z, Im Z Z E[Z] α = E[Z], Z = Z Z 1 {Z } E[Z] = α = α [ α ]
() (, y) E(, y) () E(, y) (3) q ( ) () E(, y) = k q q (, y) () E(, y) = k r r (3).3 [.7 ] f y = f y () f(, y) = y () f(, y) = tan y y ( ) () f y = f y
![() (, y) E(, y) () E(, y) (3) q ( ) () E(, y) = k q q (, y) () E(, y) = k r r (3).3 [.7 ] f y = f y () f(, y) = y () f(, y) = tan y y ( ) () f y = f y](/thumbs/85/92979039.jpg "() (, y) E(, y) () E(, y) (3) q ( ) () E(, y) = k q q (, y) () E(, y) = k r r (3).3 [.7 ] f y = f y () f(, y) = y () f(, y) = tan y y ( ) () f y = f y")
5. [. ] z = f(, y) () z = 3 4 y + y + 3y () z = y (3) z = sin( y) (4) z = cos y (5) z = 4y (6) z = tan y (7) z = log( + y ) (8) z = tan y + + y ( ) () z = 3 8y + y z y = 4 + + 6y () z = y z y = (3) z =
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4. ϵ(ν, T ) = c 4 u(ν, T ) ϵ(ν, T ) T ν π4 Planck dx = 0 e x 1 15 U(T ) x 3 U(T ) = σt 4 Stefan-Boltzmann σ 2π5 k 4 15c 2 h 3 = W m 2 K 4 5.
A 1. Boltzmann Planck u(ν, T )dν = 8πh ν 3 c 3 kt 1 dν h 6.63 10 34 J s Planck k 1.38 10 23 J K 1 Boltzmann u(ν, T ) T ν e hν c = 3 10 8 m s 1 2. Planck λ = c/ν Rayleigh-Jeans u(ν, T )dν = 8πν2 kt dν c
00 3 9 ........................................................................................................................................... 4..3................................. 5.3.......................................
QMII_10.dvi
65 1 1.1 1.1.1 1.1 H H () = E (), (1.1) H ν () = E ν () ν (). (1.) () () = δ, (1.3) μ () ν () = δ(μ ν). (1.4) E E ν () E () H 1.1: H α(t) = c (t) () + dνc ν (t) ν (), (1.5) H () () + dν ν () ν () = 1 (1.6)