第5章 偏微分方程式の境界値問題
|
|
|
- ことこ もちやま
- 5 years ago
- Views:
Transcription
1 October 5, / 113
2 4 ( ) 2 / 113
3 Poisson 5.1 Poisson ( A.7.1) Poisson Poisson 1 (A.6 ) Γ p p N u D Γ D b 5.1.1: = Γ D Γ N 3 / 113
4 Poisson d {2, 3} Lipschitz (A.5 ) Γ D Γ N = \ Γ D Γ p Γ N Γ p Γ N \ Γ p 9 = Laplace ν ( A.5.4) ν = ν Poisson 4 / 113
5 Poisson (Poisson ) b : R, p N : Γ N R, u D : R u = b in, ν u = p N on Γ N, u = u D on Γ D (5.1.1) (5.1.2) (5.1.3) u : R 5 / 113
6 Poisson (5.1.1) Poisson b = 0 Laplace Poisson Laplace (5.1.3) u u D Γ D u u D (5.1.2) ν u Γ N u ( ) 6 / 113
7 Poisson u D H 1 (; R) (5.1.3) u U (u D ) = { v H 1 (; R) v = ud on Γ D } 4.6 U (u D ) Hilbert U = { v H 1 (; R) v = 0 on Γ D } (5.1.4) U Hilbert Poisson (5.1.1) v U Gauss-Green ( A.8.2) uv dx = u v dx ν uv dγ = bv dx (5.1.5) Γ N 7 / 113
8 Poisson Γ D v = 0 (5.1.2) v U Γ N ν uv dγ = Γ N p N v dγ Γ N (5.1.6) (5.1.6) (5.1.5) u v dx = bv dx + p N v dγ (5.1.7) Γ N (5.1.7) Poisson (5.1.7) u v (5.1.7) v 4.6 a (u, v) = u v dx, (5.1.8) l (v) = bv dx + p N v dγ (5.1.9) Γ N 8 / 113
9 Poisson / 113
10 Poisson (Poisson ) U (5.1.4) b L 2 (; R), p N L 2 (Γ N ; R) u D H 1 (; R) a (, ) l ( ) (5.1.8) (5.1.9) v U a (u, v) = l (v) (5.1.10) ũ = u u D U ( u) (5.1.1) u 2 Γ N ν u u 2 (5.1.8) u v Poisson Poisson u 10 / 113
11 Poisson / 113
12 Poisson (5.1.3) Dirichlet 1 Dirichlet Dirichlet Dirichlet (5.1.2) Neumann 2 Neumann Neumann Neumann Neumann ( 5.2.6) Dirichlet Neumann Dirichlet Neumann u D = 0 p N = 0 u D 0 p N 0 12 / 113
13 Poisson Poisson Poisson Poisson Poisson ( Poisson ) b : R, c : R, p R : R, c : R u + c u = b in, (5.1.11) ν u + c u = p R on, (5.1.12) u : R 13 / 113
14 Poisson Poisson (5.1.12) Robin Robin U = H 1 (; R) (5.1.13) (5.1.11) v U Gauss-Green ( A.8.2) ( u + c u) v dx = ( u v + c uv) dx ν uv dγ = bv dx (5.1.14) (5.1.12) v U ν uv dγ = (p R c u) v dγ (5.1.15) 14 / 113
15 Poisson Poisson (5.1.15) (5.1.14) ( u v + c uv) dx + c uv dγ = bv dx + p R v dγ (5.1.16) (5.1.16) / 113
16 Poisson Poisson (5.1.16) u v v a : U U R l : U R a (u, v) = ( u v + c uv) dx + c uv dγ, (5.1.17) l (v) = bv dx + p R v dγ (5.1.18) ( Poisson ) U (5.1.13) b L 2 (; R), c L (; R), p R L 2 ( ; R), c L ( ; R) a (, ) l ( ) (5.1.17) (5.1.18) v U a (u, v) = l (v) (5.1.19) u U 16 / 113
17 Poisson Poisson (5.1.10) (5.1.19) 2 ( A.7.1) 1 a 1 l U Hilbert U 1 (4.4 ) 17 / 113
18 5.2.1 ( Hilbert 1 ) a : U U R U 1 v U α > 0 a (v, v) α v 2 U a 18 / 113
19 U R d 1 x, y R d a (x, y) = x (Ay) A R d d A = A T a A ( Hilbert 1 ) a : U U R U 1 u, v U β > 0 a (u, v) β u U v U a U = R d 1 a (x, y) = x (Ay) A ( (4.4.3) ) 19 / 113
20 5.2.3 ( ) a : U U R U 1 l = l ( ) = l, U v U a (u, v) = l (v) u U U = R d 1 a (x, y) = x (Ay) A R d d b R d y R d (Ax) y = b y (5.2.1) x R d 20 / 113
21 Lax-Milgram Lax-Milgram Lax-Milgram 1 a Hilbert Riesz ( ) ( [2] p.29 Theorem1.3, [3] p. 297 Theorem 1, [9] p ) (Lax-Milgram ) a l U u U α u U 1 α l U 21 / 113
22 Lax-Milgram U = R d (5.2.1) x A A x = A 1 b (5.2.2) x R d 1 α b R d α A A A ( A T + A ) /2 x R d x (Ax) α x 2 R d x {( A T + A ) x } 2α x 2 R d Poisson Lax-Milgram 22 / 113
23 Lax-Milgram (Poisson ) Γ D (= Γ D dγ) ũ = u u D U Lax-Milgram U = { u H 1 (; R) } u = 0 on ΓD Hilbert ˆl (v) = l (v) a (ud, v) (5.2.3) v U a (ũ, v) = ˆl (v) ũ = u u D U Lax-Milgram 23 / 113
24 Lax-Milgram 1 a Poincaré ( A.9.4) a (v, v) = v v dx = v 2 L 2 (;R d ) 1 c 2 v 2 H 1 (;R) 1/c 2 α a 2 a Hölder ( A.9.1) a (u, v) = u v dx u L 2 (;R d ) v L 2 (;R d ) u H 1 (;R) v H 1 (;R) β = 1 24 / 113
25 Lax-Milgram 3 ˆl U Lipschitz ( 4.4.2) γ L(H 1 (;R);H 1/2 ( ;R)) = sup v H 1/2 ( ;R) v H 1 (;R)\{0 H 1 (;R) } v H 1 (;R) (5.2.4) c 1 > 0 Hölder ˆl (v) bv dx + p Nv dγ + u D v dx Γ N b L 2 (;R) v L 2 (;R) + pn L 2 (Γ N ;R) v L 2 (Γ N ;R) + u D L2 (;R d ) v L 2 (;R d ) ( ) b L 2 (;R) + c1 pn L 2 (Γ N ;R) + ud H 1 (;R) v H 1 (;R) b L 2 (; R), p N L 2 (Γ N; R) u D H 1 (; R) ( ) l U ũ = u u D U 25 / 113
26 Lax-Milgram (Neumann ) Γ D = u U a Γ D > 0 Poincaré ( A.9.4) Neumann Γ D = 0 Poincaré a Lax Milgram u D = 1 u dx (5.2.5) Poincaré ( A.9.3) a (v, v) = v v dx = v 2 L 2 (;R d ) 1 c v 2 ud 2 L 2 (;R d ) 26 / 113
27 Lax-Milgram a Neumann (5.2.5) u D ũ = u u D U 27 / 113
28 Lax-Milgram Poisson ( 5.1.3) ( Poisson ) c L (; R) 2 c L ( ; R) u U Lax-Milgram U = H 1 (; R) Hilbert 28 / 113
29 Lax-Milgram 1 a ess inf x c (x) ess inf x c (x) c 1 > 0 c 2 > 0 γ : H 1 (; R) L 2 ( ; R) γ 1 L(L 2 ( ;R);H 1 (;R)) = sup v H 1 (;R) v L 2 ( ;R)\{0 L 2 ( ;R) } v L 2 ( ;R) (5.2.6) c 3 > 0 a (v, v) v 2 L 2 (;R d ) + c1 v 2 L 2 (;R) + c2 v 2 L 2 ( ;R) ( ) min {1, c 2} + c2 v 2 c 2 H 1 (;R) 3 ( ) α a 29 / 113
30 Lax-Milgram 2 a (5.2.4) γ L(H 1 (;R);H 1/2 ( ;R)) c4 c L (; R) c L ( ; R) a (u, v) u L2 (;R d ) v L 2 (;R d ) + c L (;R) u L 2 (;R) v L 2 (;R) + c L ( ;R) u L 2 ( ;R) v L 2 ( ;R) ( ) 1 + c L (;R) + c2 4 c L ( ;R) u H1 (;R d ) v H 1 (;R d ) ( ) β a 30 / 113
31 Lax-Milgram 3 l U (5.2.4) γ L(H 1 (;R);H 1/2 ( ;R)) c4 l (v) bv dx + p Rv dγ b L 2 (;R) v L 2 (;R) + pr L 2 ( ;R) v L 2 ( ;R) ( ) b L 2 (;R) + c4 pr L 2 ( ;R) v H 1 (;R) b L 2 (; R) p R L 2 ( ; R) ( ) l U Lax-Milgram u U 31 / 113
32 ( ) a : U U R U 1 l = l ( ) = l, U f : U R {f (u) = 12 } a (u, u) l (u) min u U u U U = R d {f (x) = 12 } x (Ax) b x min x R d (5.2.7) x R d 32 / 113
33 5.2.8 ( [2] p.24 Theorem1.1, [6] p , [9] p ) ( ) a l U u U / 113
34 U = R d A (5.2.7) x R d (5.2.2) Poisson ( 5.1.2) a (Poisson ) a ˆl (5.1.8) (5.2.3) { f (ũ) = 1 } 2 a (ũ, ũ) ˆl (ũ). min ũ U ũ = u u D U 34 / 113
35 5.3 Poisson ( 5.1.1) b, p, u D (5.2.3) ˆl U Poisson u u D U = { u H 1 (; R) u = 0 on Γ D } Poisson 8 9 H 1 ( ) C 1 35 / 113
36 5.3.1 Poisson ( 5.1.1) b, p, u D u = b in, ν u = p N on Γ N, u = u D on Γ D b L 2 (; R), p N H 1 (; R), u D H 2 (; R) Dirichlet Neumann Γ N Γ D B u H 2 ( \ B; R ) b L 2 (; R) Poisson u H 2 ( \ B; R ) 36 / 113
37 ν C (Γ N ; R) p N H 1 (; R) p N H 1/2 (Γ N ; R) ν u = ν u H 1/2 ( Γ N \ B; R ) u H 2 ( \ B; R ) Sobolev ( ) d {2, 3} α (0, 1/2) H 2 (; R) C 0,α ( ; R ) u u u 37 / 113
38 5.3.2 x 0 Θ α θ Γ1 Γ2 B(x 0,r 0 ) 5.3.1: 2 38 / 113
39 x 0 u ( ) r (0, r 0 ] u u (r, θ) = i {1,2, } k i u i (r) τ i (θ) + u R (5.3.1) ( [7], [8] p.273 8, [10], [5] p.ix Preface p.182 Chapter 4) u R (5.3.1) 1 k i u i (r) r τ i (θ) θ (0, α) Γ 1 Γ 2 Dirichlet (u = 0) Neumann ( ν u = 0) i {1, 2, } τ i (θ) = sin iπ α θ, τ i (θ) = cos iπ α θ (5.3.2) (5.3.3) 39 / 113
40 (5.3.2) τ i (0) = τ i (α) = 0 (5.3.3) (dτ i /dθ) (0) = (dτ i /dθ) (α) = 0 Γ 1 Γ 2 Dirichlet Neumann i {1, 2, } τ i (θ) = sin iπ 2α θ (5.3.4) 40 / 113
41 Laplace ( (r ω 2 sin ωθ) = r r r r 2 θ 2 ) (r ω sin ωθ) = 0 (5.3.5) ω ω > 1/4 1 ω > 1/4 Γ 1 Γ 2 (α 2π) ω 1/4 ω = 1 (5.3.5) z = x 1 + ix 2 = re iθ C (i ) x = (x 1, x 2 ) R 2 f (z) = z ω u i = Im [z ω ] = r m sin ωθ Cauchy-Riemann (5.3.5) r ω sin ωθ Laplace ( Poisson ) τ i (θ) sin ωθ u i (r) = r ω Laplace x 0 r 0 B (x 0, r 0 ) 41 / 113
42 1 Γ 1 Γ 2 Dirichlet (u = 0) u (r, θ) = kr π/α sin π α θ + u R (5.3.6) 2 Γ 1 Γ 2 Neumann ( ν u = 0) u (r, θ) = kr π/α cos π α θ + u R (5.3.7) 3 Γ 1 Dirichlet Γ 2 Neumann u (r, θ) = kr π/(2α) sin π 2α θ + u R (5.3.8) k α 42 / 113
43 5.3.1 ( ) 2 x 0 α (0, 2π) u x 0 B (x 0, r 0 ) u = r ω τ (θ) τ (θ) C ((0, α), R) k {0, 1, 2, } p (1, ) ω > k 2 p (5.3.9) u W k,p (B (x 0, r 0 ) ; R) 43 / 113
44 u = r ω τ (θ) k r ω k τ (θ) τ (θ) C ((0, α), R) u k B (x 0, r 0 ) p Lebesgue r0 α 0 0 r p(ω k) r τ (θ) dθdr < p (ω k) + 1 > 1 (5.3.9) 44 / 113
45 Poisson u r ω ( ) 2 x 0 Θ α (0, 2π) Poisson ( 5.1.1) u x 0 u H s (B (x 0, r 0 ) ; R) 1 x 0 Γ 1 Γ 2 α [π, 2π) s (3/2, 2] 2 Γ 1 Γ 2 α [π/2, π) s (3/2, 2] α [π, 2π) s (5/4, 3/2] 45 / 113
46 Γ 1 Γ 2 (5.3.6) (5.3.7) ω = π/α α [π, 2π) ω (1/2, 1] (5.3.9) s 1 2 p = = 1 2 < ω s 2 2 p = = 1 ω (1/2, 1] s (1) Γ 1 Γ 2 (5.3.8) ω = π/ (2α) α [π/2, π) ω (1/2, 1] (5.3.9) s (2) α [π, 2π) ω (1/4, 1/2] (5.3.9) s 1 2 p = = 1 4 < ω s 2 2 p = = 1 2 ω (1/4, 1/2] s (2) 46 / 113
47 B(x 0,r 0 ) Γ N B(x 0,r 0 ) Γ D (a) α > π (b) α > π/ : 2 47 / 113
48 5.3.2 (α = 2π) x 0 ϵ > 0 u H 3/2 ϵ (B (x 0, r 0 ) ; R) (5.3.10) x 0 x 0 (α = π) (5.3.10) u W 1, ( ) 2 x 0 Θ α (0, 2π) x 0 1 Γ 1 Γ 2 α < π 2 Γ 1 Γ 2 α < π/2 Poisson ( 5.1.1) u W 1, (B (x 0, r 0 ) ; R2) 48 / 113
49 Γ 1 Γ 2 (5.3.6) (5.3.7) ω = π/α α < π ω > 1 (5.3.9) (1) Γ 1 Γ 2 (5.3.8) ω = π/ (2α) α < π/2 ω > 1 (5.3.9) (2) (2) Γ 1 Γ Poisson c : R Dirichlet Neumann 49 / 113
50 5.4 Stokes p N Γ p u D Γ D b 5.4.1: 50 / 113
51 R d d {2, 3} Lipschitz Γ D (Dirichlet ) Γ N = \ Γ D (Neumann ) Γ p Γ N Γ p Γ N \ Γ p Γ D > 0 b : R d p N : Γ N R d u D : R d u : R d 51 / 113
52 5.4.1 u u x dx 2 2 u u x dx 2 2 dx 2 u u 2 u dx dx 1 1 x 1 u 1 u u x dx : 2 u ( u T) T 52 / 113
53 1 1 d {2, 3} 1 u (0, l) du/dx d {2, 3} u d ( u T) T = ( ui / x j ) ij R d d 2 ( ) u ( u T) T ( u T ) T = E (u) + R (u) (5.4.1) E (u) = E T (u) = (ε ij (u)) ij = 1 2 { u T + ( u T) T } (5.4.2) R (u) = R T (u) = (r ij (u)) ij = 1 2 { ( u T ) T u T } (5.4.3) 53 / 113
54 E (u) (a) (c) d R (u) (d) d ε 22 ε 12 r 12 1 ε (a) ε 11 (b) ε 22 (c) ε 12 = ε 21 (d) r : 2 E (u) R (u) 54 / 113
55 (5.4.2) (5.4.3) u 0 R d ( ) u u 2 Almansi Green 55 / 113
56 Cauchy Cauchy d = (b) 3 d = ν (d = 2 d = 3 ) p R d p i, j {1,, d} σ ij x i x j S = (σ ij ) R d d Cauchy 56 / 113
57 Cauchy x2 u D Γ D dγ p N Γ p x1 b (a) dγ σ11 σ12 dγ1 dγ σ21 dγ2 σ22 ν p (b) Cauchy df =pdγ =S(u)νdγ 5.4.4: 2 Cauchy S p 57 / 113
58 Cauchy x2 ν df=pdγ=sνdγ σ11 dγ σ33 p x3 σ22 x : 3 Cauchy S p 58 / 113
59 Cauchy Cauchy S p (Cauchy ) p S Cauchy S T ν = Sν = p (5.4.4) d = 2 i {1, 2} x i σ 1i dγ 1 + σ 2i dγ 2 = p i dγ ( (b)) ν 1 = dγ 1 /dγ ν 2 = dγ 2 /dγ σ 1i ν 1 + σ 2i ν 2 = p i (5.4.5) 59 / 113
60 Cauchy (5.4.5) (5.4.4) σ 21 = σ 12 ( 5.4.6) d = 3 σ22 σ21 σ11 σ12 (0,ε) 2 σ12 σ11 σ21 σ : 2 (ϵ 1) 60 / 113
61 d S (u) = S T (u) = (σ ij (u)) ij = CE (u) = c ijkl ε kl (u) (k,l) {1,,d} 2 ij (5.4.6) C = (c ijkl ) ijkl : R d d d d 4 S (u) E (u) c ijkl = c jikl, c ijkl = c ijlk (5.4.7) 61 / 113
62 C L A = (a ij ) ij R d d B = (b ij ) ij R d d A (CA) α A 2, A (CB) β A B (5.4.8) (5.4.9) α β A B = i,j{1,,d} a ijb ij (5.4.8) C (5.4.9) C C u ( ) (5.4.6) Hooke C u 62 / 113
63 C d = 3 1 C 3 4 = 81 2 (5.4.7) 36 3 w w = 1 w E (u) (CE (u)), S (u) = 2 E (u) 2 c ijkl = c klij (5.4.10) / 113
64 λ L µ L S (u) = 2µ L E (u) + λ L tr (E (u)) I λ L µ L Lamé tr (E (u)) = i {1,,d} e ii (u) µ L e Y ν P E (u) = 1 + ν P S (u) ν P tr (S (u)) I e Y e Y e Y ν P (Young ) Poisson k b k b = λ L + 2µ L 3, e Y = 2µ L (1 + ν P ), λ L = 2µ Lν P 1 2ν P 64 / 113
65 5.4.4 Cauchy (5.4.6) Hooke σ11 σ σ ε σ σ 21 x ε x 2 b σ 12 2 σ 12 + ε x 1 b 1 σ σ σ12 (0,ε) 2 ε σ21 σ22 x : (ϵ 1) 65 / 113
66 x 1 x 2 σ 11 + σ 21 + b 1 = 0, x 1 x 2 σ 12 + σ 22 + b 2 = 0 x 1 x 2 d {2, 3} T S (u) = b T [ (5.4.11) T S (u) = C (5.4.11) { ( 1 2 u T + ( u T) )}] T u 2 C (5.4.11) 66 / 113
67 (5.4.11) ( ) b : R d, p N : Γ N R d u D : R d T S (u) = b T in, S (u) ν = p N on Γ N, u = u D on Γ D (5.4.12) (5.4.13) (5.4.14) u : R d 67 / 113
68 u U = { v H 1 ( ; R d) v = 0R d on Γ D } (5.4.15) (5.4.12) v U Gauss-Green (A.8.2) ( T S (u) ) v dx = (S (u) ν) v dγ + S (u) E (v) dx Γ N = b v dx (5.4.16) (5.4.13) v U Γ N 68 / 113
69 (S (u) ν) v dγ = Γ N p N v dγ Γ N (5.4.17) (5.4.16) 2 1 (5.4.17) S (u) E (v) dx = b v dx + p N v dγ Γ N v U a(u, v) = S (u) E (v) dx, (5.4.18) l (v) = b v dx + p N v dγ (5.4.19) Γ N 69 / 113
70 5.4.3 ( ) U (5.4.15) b L 2 ( ; R d), p N L 2 ( Γ N ; R d), u D H 1 ( ; R d) C L ( ; R d d d d) a (, ) l ( ) (5.4.18) (5.4.19) v U a (u, v) = l (v) ũ = u u D U 70 / 113
71 5.4.6 v l (v) a (u, v) ( ) Γ D > 0 ũ = u u D U 71 / 113
72 Lax-Milgram U Hilbert v U ˆl (v) = l (v) a (ud, v) a (ũ, v) = ˆl (v) ũ = u u D U Lax-Milgram 72 / 113
73 1 a Γ D > 0 Korn 2 ( A.9.6) c v 2 H 1 (;R d ) c E (v) 2 L 2 (;R d d ) (5.4.8) C v U a (v, v) = E (v) (CE (v)) dx c 1 E (v) 2 L 2 (;R d d ) c1 c v 2 H 1 (;R d ) c 1 (5.4.8) α c 1/c α a 2 a (5.4.9) β β a 73 / 113
74 3 ˆl U Lipschitz ( 4.4.2) γ L(H 1 (;R d );H 1/2 ( ;R d )) c2 > 0 Hölder ˆl (v) b v dx + p N v dγ + Γ N β E (u D) E (v) dx b L2 (;R d ) v L 2 (;R d ) + pn L 2 (Γ N ;R d ) v L 2 (Γ N ;R d ) + β E (u D) L2 (;R d d ) E (v) L 2 (;R d d ) ( b L2 (;R d ) + c2 pn L 2 (Γ N ;R d ) ) + β E (u D) L2 (;R d d ) v H1 (;R d ) Lax-Milgram ũ = u u D U 74 / 113
75 Stokes 5.5 Stokes Stokes Stokes u D b 5.5.1: Stokes 75 / 113
76 Stokes R d d {2, 3} Lipschitz b : R d Dirichlet u D : R d u D = 0 in (5.5.1) µ Stokes Stokes u : R d p : R (ν ) u = ( u T) T ν ν u 76 / 113
77 Stokes (Stokes ) b : R d, u D : R d µ R T ( µ u T) + T p = b T in, (5.5.2) u = 0 in, u = u D on, p dx = 0 (u, p) : R d+1 (5.5.3) (5.5.4) (5.5.5) (5.5.2) Stokes (5.5.3) Newton (5.5.2) T ( µ u T pi ) = b T in (5.5.6) 77 / 113
78 Stokes I d (5.4.2) E (u) Cauchy S (u, p) = pi + 2µE (u) (5.5.7) (5.5.2) T S (u, p) = b T in, (5.5.8) (5.5.3) 5.6 (5.5.2) 78 / 113
79 Stokes u U = { u H 1 ( ; R d) u = 0R d on } (5.5.9) (5.5.2) v U Gauss-Green (A.8.2) { T ( µ u T) T p + b T} v dx = (µ ν u pν) v dγ { ( + µ u T ) ( v T) + p v + b v } dx { ( = µ u T ) ( v T) + p v + b v } dx = 0 79 / 113
80 Stokes v U Stokes p { } P = q L 2 (; R) q dx = 0 (5.5.10) (5.5.3) q P q u dx = 0 q P Stokes a (u, v) = µ ( u T) ( v T) dx, (5.5.11) 80 / 113
81 Stokes b (v, q) = q v dx, l (v) = b v dx (5.5.12) (5.5.13) Stokes (Stokes ) U P (5.5.9) (5.5.10) u D H 1 ( ; R d) (5.5.1) µ a (, ), b (, ) l ( ) (5.5.11), (5.5.12) (5.5.13) (v, q) U P a (u, v) + b (v, p) = l (v), b (u, q) = 0 (5.5.14) (5.5.15) (ũ, p) = (u u D, p) U P 81 / 113
82 5.6 Stokes u Stokes u p Stokes 82 / 113
83 U P Hilbert a : U U R b : U P R U U U P (4.4.4 ) a = a L(U,U;R) = b = b L(U,P ;R) = a (u, v) sup, u,v U\{0 U } u U v U sup u U\{0 U }, q P \{0 P } b (u, q) u U q P 83 / 113
84 Hilbert ( Hilbert U div ) b : U P R 1 U div = {v U b (v, q) = 0 for all q P } U Hilbert 84 / 113
85 5.6.2 ( ) a : U U R b : U P R l U r P (v, q) U P a (u, v) + b (v, p) = l, v, b (u, q) = r, q (u, p) U P 85 / 113
86 ( [4] p.61 Corollary 4.1, [1] p.42 Theorem 1.1, [6] p , [9] p ) 86 / 113
87 5.6.3 ( ) a : U U R U div ( v U div α > 0 a (v, v) α v 2 U ) b : U P R β > 0 inf sup q P \{0 P } v U\{0 U } b (v, q) v U q P β (5.6.1) 87 / 113
88 5.6.3 ( ) (u, p) U P α, β, a b c > 0 u U + p P c ( l U + r P ) (5.6.1) Ladysenskaja-Babuška-Brezzi Babuška-Brezzi-Kikuchi 88 / 113
89 5.6.3 Stokes (Stokes ) ũ = u u D U div ũ = 0 R d (5.5.14) (5.5.15) (ũ, p) U P U P Hilbert (v, q) U P a (ũ, v) + b (v, p) = ˆl (v), b (ũ, q) = ˆr (q), (ũ, p) U P ˆl (v) = l (v) a (ud, v), ˆr (q) = b (u D, q) a U div a U b 89 / 113
90 U div U div div U div τ τ ( divv / v U < ) (v 1, v 2 U div τv 1 = τv 2 v 1 = v 2) v U div τv = divv = 0 v U div v U div U div = {0 U } τ U div P ( [4] ) inf sup q P \{0 P } v U\{0 U } inf q P \{0 P } v U\{0 U } b (v, q) v U q P = inf sup 1 > 0 τ 1 L(P ;U div) sup q P \{0 P } v U\{0 U } ( τv, q) P τ 1 ( q) U q P inf q P \{0 P } ( divv, q) L 2 (;R) v U q P (q, q) P τ 1 ( q) U q P ˆl U (5.5.1) ˆr (q) = 0 P (u u D, p) U P 90 / 113
91 5.6.2 ( 5.6.2) a : U U R ( ) a : U U R b : U P R (l, r) U P L (v, q) = 1 a (v, v) + b (v, q) l, v r, q 2 (v, q) U P L (u, q) L (u, p) L (v, p) (u, p) U P 91 / 113
92 5.6.5 ( [4] p.62 Theorem 4.2, [9] p ) ( ) a (a (u, v) = a (v, u)) ( v U a (v, v) 0 ) ( 5.6.5) q P Lagrange f (v) = 1 a (v, v) l, v 2 min {f (v) b (v, q) r, q = 0} (5.6.2) (v,q) U P 92 / 113
93 (v, q) L (v, q) Lagrange q P Lagrange (5.6.2) ( 2.9.2) 93 / 113
94 (Poisson ) Lax-Milgram (5.1, 5.2 ) 2 (5.3 ) 3 Lax-Milgram (5.4 ) 4 Stokes Stokes (5.5 ) 94 / 113
95 b : R, u D : R u + u = b in, u = u D on u : R u b u D x A x B 95 / 113
96 (cnt.) x A x B Γ D u D=(0, 0) T Γ p p N=(0, 1) T b=(0, 0) T 5.8.1: 96 / 113
97 (cnt.) 5.3 b : (0, t T ) R d, p N : Γ N (0, t T ) R d, u D : (0, t T ) R d, u D0 : R d, u DT : R d ρ > 0 ρü T S (u) = b T in (0, t T ), S (u) ν = p N on Γ N (0, t T ), u = u D on Γ D (0, t T ), u = u D0 in {0}, u = u DT in {t T } u : (0, t T ) R d t (0, t T ) u = u/ t 97 / 113
98 (cnt.) b = 0 R d, p N = 0 R d u D = 0 R d (x, t) (0, t T ) ( ) u (x, t) = ϕ (x) e λt ϕ : R d λ R 5.5 b : (0, t T ) R d, u D : (0, t T ) R d, µ > 0 ρ > 0 ρ u + ρ (u ) u µ u + p = b in (0, t T ), u = 0 in (0, t T ), u = u D on { (0, t T )} { {0}} (u, p) : (0, t T ) R d R Navier-Stokes 1 Navier-Stokes 2 98 / 113
99 (cnt.) 5.6 e Y µ L Poisson ν P e Y = 2µ L (1 + ν P ) 99 / 113
100 5 5.1 Dirichlet U = H0 1 (; R) v U ( u + u) v dx = ( u + u) v dx = v u ν dγ + ( u v + uv) dx = ( u v + uv) dx = bv dx v U a (u, v) = l (v) 100 / 113
101 5 (cnt.) ũ = u u D U a (u, v) = ( u v + uv) dx, l (v) = bv dx Lax-Milgram U = H 1 0 (; R) Hilbert v H 1 0 (; R) a (v, v) = v 2 H 1 (;R) a ˆl U ˆl ˆl (v) bv dx + ( u D v + u D v ) dx 101 / 113
102 5 (cnt.) b L 2 (;R) v L 2 (;R) + u D L 2 (;R d ) v L 2 (;R d ) + u D L 2 (;R) v L 2 (;R) ( ) b L2 (;R) + u D H1 (;R) v H1 (;R) b L 2 (; R) u D H 1 (; R) 5.2 x A Dirichlet Neumann α = π/ (2) x A B A u H 2 ( B A ; R 2) x A x B Neumann Neumann α π/ (1) p N x B B B (0, 0) T (0, 1) T Γ p p N L ( B B ; R 2) u C 0,1 ( B B ; R 2) H 2 ( B B ; R 2) 102 / 113
103 5 (cnt.) 5.3 U = { u H 1 ( (0, t T ) ; H 1 ( ; R d)) u = 0R d on Γ D (0, t T ), u = 0 R d on (0, t T ) } u D0, u DT H 1 ( ; R d) u D H 1 ( (0, t T ) ; H 1 ( ; R d)) b L 2 ( (0, t T ) ; L 2 ( ; R d)), p N L 2 ( (0, t T ) ; L 2 ( Γ N ; R d)) 1 v U (0, t T ) v U tt 0 (b ( u, v) a (u, v) + l (v)) dt = / 113
104 5 (cnt.) ũ = u u D U b (u, v) = ρu v dx, a (u, v) = S(u) E(v) dx, l (v) = b v dx + p N v dγ Γ N 104 / 113
105 5 (cnt.) 5.4 ϕ U = { ϕ H 1 ( ; R d) ϕ = 0R d on Γ D } ϕ U u (x, t) = ϕ (x) e λt ρü T T S (u) = 0 T R v U d u = u D on Γ D (0, t T ) v U λ 2 b (ϕ, v) + a (ϕ, v) = 0 (ϕ, λ) U R ( ) U ( ) a (, ) (0 ) b (, ) ( ) U 105 / 113
106 5 (cnt.) (ϕ i, λ i ) i N λ 2 i 0 λ i = ±iω i (i ) ϕ i (x) ( e iωit + e iωit) = ϕ i cos ω i t ω i ϕ i 5.5 u p U = { u H 1 ( (0, t T ) ; H 1 ( ; R d)) u = 0 R d on (0, t T ) {0} }, V = { u H 1 ( (0, t T ) ; H 1 ( ; R d)) u = 0 R d on (0, t T ) {t T } }, P = {p L 2 ( (0, t T ) ; L 2 (; R) ) } p dx = / 113
107 5 (cnt.) v V Navier-Stokes (0, t T ) u = u D on (0, t T ) {0} Navier-Stokes q P (0, t T ) (v, q) U Q tt (b( u, v) + c(u)(u, v) + a(u, v) + d(v, p)) dt = 0 0 tt 0 d(u, q) dt = 0 (u u D, p) U Q a (u, v) = µ ( u T) ( v T) dx, tt l (v) dt, 107 / 113
108 5 (cnt.) b (u, v) = ρu v dx, c(u)(w, v) = ρ ((u ) w) v dx, d (v, q) = q v dx, l (v) = b v dx + p N v dγ Γ N 108 / 113
109 5 (cnt.) 5.6 P.5.1 (a) ε 11 = ε 22 = 1 + ν P e Y σ 0 (1) P.5.1 (b) π/4 ε 11 = γ 12/ 2 2 = γ 12 2 = ε 12 = σ 0 2µ L (2) (1) (2) e Y = 2µ L (1 + ν P ) 109 / 113
110 5 (cnt.) σ0 σ0 σ0 σ0 x 2 x 1 σ0 σ0 σ0 γ 12 /2 σ0 (a) 2 (b) π/4 P.5.1: 110 / 113
111 [1] F. Brezzi and M. Fortin. Mixed and Hybrid Finite Element Methods. Springer, New York; Tokyo, [2] P. G. Ciarlet. Finite Element Methods. Handbook of Numerical Analysis, P.G. Ciarlet, J.L. Lions, general editors. Elsevier, Amsterdam; Tokyo: North-Holl, [3] L. C. Evans and R. F. Gariepy. Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton, [4] V. Girault and P. A. Raviart. Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms. Springer, Berlin; Tokyo, / 113
112 (cnt.) [5] P. Grisvard. Elliptic Problems in Nonsmooth Domains. Pitman Advanced Pub. Program, Boston, [6]. :.,, [7] R. S. Lehman. Developments at an analytic corner of solutions of elliptic partial differential equations. Journal of Applied Mathematics and Mechanics, Vol. 8, pp , [8] G. Strang, G. J. Fix,,..,, / 113
113 (cnt.) [9]..,, [10],,.. bit, Vol. 5, pp , / 113
all.dvi
72 9 Hooke,,,. Hooke. 9.1 Hooke 1 Hooke. 1, 1 Hooke. σ, ε, Young. σ ε (9.1), Young. τ γ G τ Gγ (9.2) X 1, X 2. Poisson, Poisson ν. ν ε 22 (9.) ε 11 F F X 2 X 1 9.1: Poisson 9.1. Hooke 7 Young Poisson G
all.dvi
5,, Euclid.,..,... Euclid,.,.,, e i (i =,, ). 6 x a x e e e x.:,,. a,,. a a = a e + a e + a e = {e, e, e } a (.) = a i e i = a i e i (.) i= {a,a,a } T ( T ),.,,,,. (.),.,...,,. a 0 0 a = a 0 + a + a 0
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
第1章 微分方程式と近似解法
April 12, 2018 1 / 52 1.1 ( ) 2 / 52 1.2 1.1 1.1: 3 / 52 1.3 Poisson Poisson Poisson 1 d {2, 3} 4 / 52 1 1.3.1 1 u,b b(t,x) u(t,x) x=0 1.1: 1 a x=l 1.1 1 (0, t T ) (0, l) 1 a b : (0, t T ) (0, l) R, u
II ( ) (7/31) II ( [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Re
![II ( ) (7/31) II ( [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Re](/thumbs/94/118770263.jpg "II ( ) (7/31) II ( [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Re")
II 29 7 29-7-27 ( ) (7/31) II (http://www.damp.tottori-u.ac.jp/~ooshida/edu/fluid/) [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Reynolds [ (4.6), (45.8)] [ p.186] Navier Stokes I Euler Navier
all.dvi
38 5 Cauchy.,,,,., σ.,, 3,,. 5.1 Cauchy (a) (b) (a) (b) 5.1: 5.1. Cauchy 39 F Q Newton F F F Q F Q 5.2: n n ds df n ( 5.1). df n n df(n) df n, t n. t n = df n (5.1) ds 40 5 Cauchy t l n mds df n 5.3: t
~nabe/lecture/index.html 2
2001 12 13 1 http://www.sml.k.u-tokyo.ac.jp/ ~nabe/lecture/index.html nabe@sml.k.u-tokyo.ac.jp 2 1. 10/ 4 2. 10/11 3. 10/18 1 4. 10/25 2 5. 11/ 1 6. 11/ 8 7. 11/15 8. 11/22 9. 11/29 10. 12/ 6 1 11. 12/13
Untitled
II 14 14-7-8 8/4 II (http://www.damp.tottori-u.ac.jp/~ooshida/edu/fluid/) [ (3.4)] Navier Stokes [ 6/ ] Navier Stokes 3 [ ] Reynolds [ (4.6), (45.8)] [ p.186] Navier Stokes I 1 balance law t (ρv i )+ j
1 4 1 ( ) ( ) ( ) ( ) () 1 4 2
7 1995, 2017 7 21 1 2 2 3 3 4 4 6 (1).................................... 6 (2)..................................... 6 (3) t................. 9 5 11 (1)......................................... 11 (2)
OHP.dvi
7 2010 11 22 1 7 http://www.sml.k.u-tokyo.ac.jp/members/nabe/lecture2010 nabe@sml.k.u-tokyo.ac.jp 2 1. 10/ 4 2. 10/18 3. 10/25 2, 3 4. 11/ 1 5. 11/ 8 6. 11/15 7. 11/22 8. 11/29 9. 12/ 6 skyline 10. 12/13
(1.2) T D = 0 T = D = 30 kn 1.2 (1.4) 2F W = 0 F = W/2 = 300 kn/2 = 150 kn 1.3 (1.9) R = W 1 + W 2 = = 1100 N. (1.9) W 2 b W 1 a = 0
1 1 1.1 1.) T D = T = D = kn 1. 1.4) F W = F = W/ = kn/ = 15 kn 1. 1.9) R = W 1 + W = 6 + 5 = 11 N. 1.9) W b W 1 a = a = W /W 1 )b = 5/6) = 5 cm 1.4 AB AC P 1, P x, y x, y y x 1.4.) P sin 6 + P 1 sin 45
2002 11 21 1 http://www.sml.k.u-tokyo.ac.jp/members/nabe/lecture2002 http://www.sml.k.u-tokyo.ac.jp/members/nabe/lecture nabe@sml.k.u-tokyo.ac.jp 2 1. 10/10 2. 10/17 3. 10/24 4. 10/31 5. 11/ 7 6. 11/14
2 (March 13, 2010) N Λ a = i,j=1 x i ( d (a) i,j x j ), Λ h = N i,j=1 x i ( d (h) i,j x j ) B a B h B a = N i,j=1 ν i d (a) i,j, B h = x j N i,j=1 ν i
1. A. M. Turing [18] 60 Turing A. Gierer H. Meinhardt [1] : (GM) ) a t = D a a xx µa + ρ (c a2 h + ρ 0 (0 < x < l, t > 0) h t = D h h xx νh + c ρ a 2 (0 < x < l, t > 0) a x = h x = 0 (x = 0, l) a = a(x,
Report98.dvi
1 4 1.1.......................... 4 1.1.1.......................... 7 1.1..................... 14 1.1.................. 1 1.1.4........................... 8 1.1.5........................... 6 1.1.6 n...........................
n (1.6) i j=1 1 n a ij x j = b i (1.7) (1.7) (1.4) (1.5) (1.4) (1.7) u, v, w ε x, ε y, ε x, γ yz, γ zx, γ xy (1.8) ε x = u x ε y = v y ε z = w z γ yz
1 2 (a 1, a 2, a n ) (b 1, b 2, b n ) A (1.1) A = a 1 b 1 + a 2 b 2 + + a n b n (1.1) n A = a i b i (1.2) i=1 n i 1 n i=1 a i b i n i=1 A = a i b i (1.3) (1.3) (1.3) (1.1) (ummation convention) a 11 x
.2 ρ dv dt = ρk grad p + 3 η grad (divv) + η 2 v.3 divh = 0, rote + c H t = 0 dive = ρ, H = 0, E = ρ, roth c E t = c ρv E + H c t = 0 H c E t = c ρv T
NHK 204 2 0 203 2 24 ( ) 7 00 7 50 203 2 25 ( ) 7 00 7 50 203 2 26 ( ) 7 00 7 50 203 2 27 ( ) 7 00 7 50 I. ( ν R n 2 ) m 2 n m, R = e 2 8πε 0 hca B =.09737 0 7 m ( ν = ) λ a B = 4πε 0ħ 2 m e e 2 = 5.2977
note1.dvi
(1) 1996 11 7 1 (1) 1. 1 dx dy d x τ xx x x, stress x + dx x τ xx x+dx dyd x x τ xx x dyd y τ xx x τ xx x+dx d dx y x dy 1. dx dy d x τ xy x τ x ρdxdyd x dx dy d ρdxdyd u x t = τ xx x+dx dyd τ xx x dyd
1. R n Ω ε G ε 0 Ω ε B n 2 Ωε = with Bu = 0 on Ω ε i=1 x 2 i ε +0 B Bu = u (Dirichlet, D Ω ε ), Bu = u ν (Neumann, N Ω ε ), Ω ε G ( ) / 25
.. IV 2012 10 4 ( ) 2012 10 4 1 / 25 1. R n Ω ε G ε 0 Ω ε B n 2 Ωε = with Bu = 0 on Ω ε i=1 x 2 i ε +0 B Bu = u (Dirichlet, D Ω ε ), Bu = u ν (Neumann, N Ω ε ), Ω ε G ( ) 2012 10 4 2 / 25 1. Ω ε B ε t
1 1.1 H = µc i c i + c i t ijc j + 1 c i c j V ijklc k c l (1) V ijkl = V jikl = V ijlk = V jilk () t ij = t ji, V ijkl = V lkji (3) (1) V 0 H mf = µc
013 6 30 BCS 1 1.1........................ 1................................ 3 1.3............................ 3 1.4............................... 5 1.5.................................... 5 6 3 7 4 8
構造と連続体の力学基礎
12 12.1? finite deformation infinitesimal deformation large deformation 1 [129] B Bernoulli-Euler [26] 1975 Northwestern Nemat-Nasser Continuum Mechanics 1980 [73] 2 1 2 What is the physical meaning? 583
I , : ~/math/functional-analysis/functional-analysis-1.tex
I 1 2004 8 16, 2017 4 30 1 : ~/math/functional-analysis/functional-analysis-1.tex 1 3 1.1................................... 3 1.2................................... 3 1.3.....................................
u = u(t, x 1,..., x d ) : R R d C λ i = 1 := x 2 1 x 2 d d Euclid Laplace Schrödinger N := {1, 2, 3,... } Z := {..., 3, 2, 1,, 1, 2, 3
2 2 1 5 5 Schrödinger i u t + u = λ u 2 u. u = u(t, x 1,..., x d ) : R R d C λ i = 1 := 2 + + 2 x 2 1 x 2 d d Euclid Laplace Schrödinger 3 1 1.1 N := {1, 2, 3,... } Z := {..., 3, 2, 1,, 1, 2, 3,... } Q
18 I ( ) (1) I-1,I-2,I-3 (2) (3) I-1 ( ) (100 ) θ ϕ θ ϕ m m l l θ ϕ θ ϕ 2 g (1) (2) 0 (3) θ ϕ (4) (3) θ(t) = A 1 cos(ω 1 t + α 1 ) + A 2 cos(ω 2 t + α
18 I ( ) (1) I-1,I-2,I-3 (2) (3) I-1 ( ) (100 ) θ ϕ θ ϕ m m l l θ ϕ θ ϕ 2 g (1) (2) 0 (3) θ ϕ (4) (3) θ(t) = A 1 cos(ω 1 t + α 1 ) + A 2 cos(ω 2 t + α 2 ), ϕ(t) = B 1 cos(ω 1 t + α 1 ) + B 2 cos(ω 2 t
20 4 20 i 1 1 1.1............................ 1 1.2............................ 4 2 11 2.1................... 11 2.2......................... 11 2.3....................... 19 3 25 3.1.............................
takei.dvi
0 Newton Leibniz ( ) α1 ( ) αn (1) a α1,...,α n (x) u(x) = f(x) x 1 x n α 1 + +α n m 1957 Hans Lewy Lewy 1970 1 1.1 Example 1.1. (2) d 2 u dx 2 Q(x)u = f(x), u(0) = a, 1 du (0) = b. dx Q(x), f(x) x = 0
: 1g99p038-8
16 17 : 1g99p038-8 1 3 1.1....................................... 4 1................................... 5 1.3.................................. 5 6.1..................................... 7....................................
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
Design of highly accurate formulas for numerical integration in weighted Hardy spaces with the aid of potential theory 1 Ken ichiro Tanaka 1 Ω R m F I = F (t) dt (1.1) Ω m m 1 m = 1 1 Newton-Cotes Gauss
量子力学 問題
3 : 203 : 0. H = 0 0 2 6 0 () = 6, 2 = 2, 3 = 3 3 H 6 2 3 ϵ,2,3 (2) ψ = (, 2, 3 ) ψ Hψ H (3) P i = i i P P 2 = P 2 P 3 = P 3 P = O, P 2 i = P i (4) P + P 2 + P 3 = E 3 (5) i ϵ ip i H 0 0 (6) R = 0 0 [H,
III 1 (X, d) d U d X (X, d). 1. (X, d).. (i) d(x, y) d(z, y) d(x, z) (ii) d(x, y) d(z, w) d(x, z) + d(y, w) 2. (X, d). F X.. (1), X F, (2) F 1, F 2 F
III 1 (X, d) d U d X (X, d). 1. (X, d).. (i) d(x, y) d(z, y) d(x, z) (ii) d(x, y) d(z, w) d(x, z) + d(y, w) 2. (X, d). F X.. (1), X F, (2) F 1, F 2 F F 1 F 2 F, (3) F λ F λ F λ F. 3., A λ λ A λ. B λ λ
B 1 B.1.......................... 1 B.1.1................. 1 B.1.2................. 2 B.2........................... 5 B.2.1.......................... 5 B.2.2.................. 6 B.2.3..................
OHP.dvi
t 0, X X t x t 0 t u u = x X (1) t t 0 u X x O 1 1 t 0 =0 X X +dx t x(x,t) x(x +dx,t). dx dx = x(x +dx,t) x(x,t) (2) dx, dx = F dx (3). F (deformation gradient tensor) t F t 0 dx dx X x O 2 2 F. (det F
(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 ]
II Karel Švadlenka * [1] 1.1* 5 23 m d2 x dt 2 = cdx kx + mg dt. c, g, k, m 1.2* u = au + bv v = cu + dv v u a, b, c, d R
![II Karel Švadlenka * [1] 1.1* 5 23 m d2 x dt 2 = cdx kx + mg dt. c, g, k, m 1.2* u = au + bv v = cu + dv v u a, b, c, d R](/thumbs/95/125540821.jpg "II Karel Švadlenka * [1] 1.1* 5 23 m d2 x dt 2 = cdx kx + mg dt. c, g, k, m 1.2* u = au + bv v = cu + dv v u a, b, c, d R")
II Karel Švadlenka 2018 5 26 * [1] 1.1* 5 23 m d2 x dt 2 = cdx kx + mg dt. c, g, k, m 1.2* 5 23 1 u = au + bv v = cu + dv v u a, b, c, d R 1.3 14 14 60% 1.4 5 23 a, b R a 2 4b < 0 λ 2 + aλ + b = 0 λ =
II A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka )
II 214-1 : October 2, 214 Version : 1.1 Kawahira, Tomoki TA (Kondo, Hirotaka ) http://www.math.nagoya-u.ac.jp/~kawahira/courses/14w-biseki.html pdf 1 2 1 9 1 16 1 23 1 3 11 6 11 13 11 2 11 27 12 4 12 11
120 9 I I 1 I 2 I 1 I 2 ( a) ( b) ( c ) I I 2 I 1 I ( d) ( e) ( f ) 9.1: Ampère (c) (d) (e) S I 1 I 2 B ds = µ 0 ( I 1 I 2 ) I 1 I 2 B ds =0. I 1 I 2
9 E B 9.1 9.1.1 Ampère Ampère Ampère s law B S µ 0 B ds = µ 0 j ds (9.1) S rot B = µ 0 j (9.2) S Ampère Biot-Savart oulomb Gauss Ampère rot B 0 Ampère µ 0 9.1 (a) (b) I B ds = µ 0 I. I 1 I 2 B ds = µ 0
December 28, 2018
e-mail : kigami@i.kyoto-u.ac.jp December 28, 28 Contents 2............................. 3.2......................... 7.3..................... 9.4................ 4.5............. 2.6.... 22 2 36 2..........................
tomocci ,. :,,,, Lie,,,, Einstein, Newton. 1 M n C. s, M p. M f, p d ds f = dxµ p ds µ f p, X p = X µ µ p = dxµ ds µ p. µ, X µ.,. p,. T M p.
tomocci 18 7 5...,. :,,,, Lie,,,, Einstein, Newton. 1 M n C. s, M p. M f, p d ds f = dxµ p ds µ f p, X p = X µ µ p = dxµ ds µ p. µ, X µ.,. p,. T M p. M F (M), X(F (M)).. T M p e i = e µ i µ. a a = a i
) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4
![) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4](/thumbs/92/107866707.jpg ") ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4")
1. k λ ν ω T v p v g k = π λ ω = πν = π T v p = λν = ω k v g = dω dk 1) ) 3) 4). p = hk = h λ 5) E = hν = hω 6) h = h π 7) h =6.6618 1 34 J sec) hc=197.3 MeV fm = 197.3 kev pm= 197.3 ev nm = 1.97 1 3 ev
N cos s s cos ψ e e e e 3 3 e e 3 e 3 e
3 3 5 5 5 3 3 7 5 33 5 33 9 5 8 > e > f U f U u u > u ue u e u ue u ue u e u e u u e u u e u N cos s s cos ψ e e e e 3 3 e e 3 e 3 e 3 > A A > A E A f A A f A [ ] f A A e > > A e[ ] > f A E A < < f ; >
9. 05 L x P(x) P(0) P(x) u(x) u(x) (0 < = x < = L) P(x) E(x) A(x) P(L) f ( d EA du ) = 0 (9.) dx dx u(0) = 0 (9.2) E(L)A(L) du (L) = f (9.3) dx (9.) P
9 (Finite Element Method; FEM) 9. 9. P(0) P(x) u(x) (a) P(L) f P(0) P(x) (b) 9. P(L) 9. 05 L x P(x) P(0) P(x) u(x) u(x) (0 < = x < = L) P(x) E(x) A(x) P(L) f ( d EA du ) = 0 (9.) dx dx u(0) = 0 (9.2) E(L)A(L)
TOP URL 1
TOP URL http://amonphys.web.fc.com/ 1 19 3 19.1................... 3 19.............................. 4 19.3............................... 6 19.4.............................. 8 19.5.............................
Introduction to Numerical Analysis of Differential Equations Naoya Enomoto (Kyoto.univ.Dept.Science(math))
Introduction to Numerical Analysis of Differential Equations Naoya Enomoto (Kyoto.univ.Dept.Science(math)) 2001 1 e-mail:s00x0427@ip.media.kyoto-u.ac.jp 1 1 Van der Pol 1 1 2 2 Bergers 2 KdV 2 1 5 1.1........................................
n ξ n,i, i = 1,, n S n ξ n,i n 0 R 1,.. σ 1 σ i .10.14.15 0 1 0 1 1 3.14 3.18 3.19 3.14 3.14,. ii 1 1 1.1..................................... 1 1............................... 3 1.3.........................
numb.dvi
11 Poisson kanenko@mbkniftycom alexeikanenko@docomonejp http://wwwkanenkocom/ , u = f, ( u = u+f u t, u = f t ) 1 D R 2 L 2 (D) := {f(x,y) f(x,y) 2 dxdy < )} D D f,g L 2 (D) (f,g) := f(x,y)g(x,y)dxdy (L
43433 8 3 . Stochastic exponentials...................................... 3. Girsanov s theorem......................................... 4 On the martingale property of stochastic exponentials 5. Gronwall
IA
IA 31 4 11 1 1 4 1.1 Planck.............................. 4 1. Bohr.................................... 5 1.3..................................... 6 8.1................................... 8....................................
SO(3) 7 = = 1 ( r ) + 1 r r r r ( l ) (5.17) l = 1 ( sin θ ) + sin θ θ θ ϕ (5.18) χ(r)ψ(θ, ϕ) l ψ = αψ (5.19) l 1 = i(sin ϕ θ l = i( cos ϕ θ l 3 = i ϕ
SO(3) 71 5.7 5.7.1 1 ħ L k l k l k = iϵ kij x i j (5.117) l k SO(3) l z l ± = l 1 ± il = i(y z z y ) ± (z x x z ) = ( x iy) z ± z( x ± i y ) = X ± z ± z (5.118) l z = i(x y y x ) = 1 [(x + iy)( x i y )
4 2016 3 8 2.,. 2. Arakawa Jacobin., 2 Adams-Bashforth. Re = 80, 90, 100.. h l, h/l, Kármán, h/l 0.28,, h/l.., (2010), 46.2., t = 100 t = 2000 46.2 < Re 46.5. 1 1 4 2 6 2.1............................
2005 2006.2.22-1 - 1 Fig. 1 2005 2006.2.22-2 - Element-Free Galerkin Method (EFGM) Meshless Local Petrov-Galerkin Method (MLPGM) 2005 2006.2.22-3 - 2 MLS u h (x) 1 p T (x) = [1, x, y]. (1) φ(x) 0.5 φ(x)
II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2
![II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2](/thumbs/101/149159646.jpg "II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2")
II No.1 [n/] [1]H n x) H n x) = 1) r n! r!n r)! x)n r r= []H n x) n,, H n x) = 1) n H n x) [3] H n x) = 1) n dn x e dx n e x [4] H n+1 x) = xh n x) nh n 1 x) ) d dx x H n x) = H n+1 x) d dx H nx) = nh
1 (Berry,1975) 2-6 p (S πr 2 )p πr 2 p 2πRγ p p = 2γ R (2.5).1-1 : : : : ( ).2 α, β α, β () X S = X X α X β (.1) 1 2
2005 9/8-11 2 2.2 ( 2-5) γ ( ) γ cos θ 2πr πρhr 2 g h = 2γ cos θ ρgr (2.1) γ = ρgrh (2.2) 2 cos θ θ cos θ = 1 (2.2) γ = 1 ρgrh (2.) 2 2. p p ρgh p ( ) p p = p ρgh (2.) h p p = 2γ r 1 1 (Berry,1975) 2-6
TOP URL 1
TOP URL http://amonphys.web.fc.com/ 3.............................. 3.............................. 4.3 4................... 5.4........................ 6.5........................ 8.6...........................7
xia2.dvi
Journal of Differential Equations 96 (992), 70-84 Melnikov method and transversal homoclinic points in the restricted three-body problem Zhihong Xia Department of Mathematics, Harvard University Cambridge,
( ) ) ) ) 5) 1 J = σe 2 6) ) 9) 1955 Statistical-Mechanical Theory of Irreversible Processes )
( 3 7 4 ) 2 2 ) 8 2 954 2) 955 3) 5) J = σe 2 6) 955 7) 9) 955 Statistical-Mechanical Theory of Irreversible Processes 957 ) 3 4 2 A B H (t) = Ae iωt B(t) = B(ω)e iωt B(ω) = [ Φ R (ω) Φ R () ] iω Φ R (t)
RIMS Kôkyûroku Bessatsu B3 (22), Numerical verification methods for differential equations: Computer-assisted proofs based on infinite dimension
微分方程式の精度保証付き数値計算 : 逐次反復に基づく Title計算機援用証明 (Progress in Mathematics of Systems) Author(s) 渡部, 善隆 Citation 数理解析研究所講究録別冊 = RIMS Kokyuroku Bessa (22), B3: 45-55 Issue Date 22-4 URL http://hdl.handle.net/2433/9624
meiji_resume_1.PDF
β β β (q 1,q,..., q n ; p 1, p,..., p n ) H(q 1,q,..., q n ; p 1, p,..., p n ) Hψ = εψ ε k = k +1/ ε k = k(k 1) (x, y, z; p x, p y, p z ) (r; p r ), (θ; p θ ), (ϕ; p ϕ ) ε k = 1/ k p i dq i E total = E
Kullback-Leibler
Kullback-Leibler 206 6 6 http://www.math.tohoku.ac.jp/~kuroki/latex/206066kullbackleibler.pdf 0 2 Kullback-Leibler 3. q i.......................... 3.2........... 3.3 Kullback-Leibler.............. 4.4
i
009 I 1 8 5 i 0 1 0.1..................................... 1 0.................................................. 1 0.3................................. 0.4........................................... 3
No δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i x j δx j (5) δs 2
No.2 1 2 2 δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i δx j (5) δs 2 = δx i δx i + 2 u i δx i δx j = δs 2 + 2s ij δx i δx j
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)
( ) (, ) ( )
( ) (, ) ( ) 1 2 2 2 2.1......................... 2 2.2.............................. 3 2.3............................... 4 2.4.............................. 5 2.5.............................. 6 2.6..........................
TOP URL 1
TOP URL http://amonphys.web.fc2.com/ 1 30 3 30.1.............. 3 30.2........................... 4 30.3...................... 5 30.4........................ 6 30.5.................................. 8 30.6...............................
Gmech08.dvi
145 13 13.1 13.1.1 0 m mg S 13.1 F 13.1 F /m S F F 13.1 F mg S F F mg 13.1: m d2 r 2 = F + F = 0 (13.1) 146 13 F = F (13.2) S S S S S P r S P r r = r 0 + r (13.3) r 0 S S m d2 r 2 = F (13.4) (13.3) d 2
() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi)
0. A A = 4 IC () det A () A () x + y + z = x y z X Y Z = A x y z ( 5) ( s5590) 0. a + b + c b c () a a + b + c c a b a + b + c 0 a b c () a 0 c b b c 0 a c b a 0 0. A A = 7 5 4 5 0 ( 5) ( s5590) () A ()
b3e2003.dvi
15 II 5 5.1 (1) p, q p = (x + 2y, xy, 1), q = (x 2 + 3y 2, xyz, ) (i) p rotq (ii) p gradq D (2) a, b rot(a b) div [11, p.75] (3) (i) f f grad f = 1 2 grad( f 2) (ii) f f gradf 1 2 grad ( f 2) rotf 5.2
Z: Q: R: C: sin 6 5 ζ a, b
Z: Q: R: C: 3 3 7 4 sin 6 5 ζ 9 6 6............................... 6............................... 6.3......................... 4 7 6 8 8 9 3 33 a, b a bc c b a a b 5 3 5 3 5 5 3 a a a a p > p p p, 3,
k m m d2 x i dt 2 = f i = kx i (i = 1, 2, 3 or x, y, z) f i σ ij x i e ij = 2.1 Hooke s law and elastic constants (a) x i (2.1) k m σ A σ σ σ σ f i x
k m m d2 x i dt 2 = f i = kx i (i = 1, 2, 3 or x, y, z) f i ij x i e ij = 2.1 Hooke s law and elastic constants (a) x i (2.1) k m A f i x i B e e e e 0 e* e e (2.1) e (b) A e = 0 B = 0 (c) (2.1) (d) e
) a + b = i + 6 b c = 6i j ) a = 0 b = c = 0 ) â = i + j 0 ˆb = 4) a b = b c = j + ) cos α = cos β = 6) a ˆb = b ĉ = 0 7) a b = 6i j b c = i + 6j + 8)
4 4 ) a + b = i + 6 b c = 6i j ) a = 0 b = c = 0 ) â = i + j 0 ˆb = 4) a b = b c = j + ) cos α = cos β = 6) a ˆb = b ĉ = 0 7) a b = 6i j b c = i + 6j + 8) a b a b = 6i j 4 b c b c 9) a b = 4 a b) c = 7
2009 2 26 1 3 1.1.................................................. 3 1.2..................................................... 3 1.3...................................................... 3 1.4.....................................................
A = A x x + A y y + A, B = B x x + B y y + B, C = C x x + C y y + C..6 x y A B C = A x x + A y y + A B x B y B C x C y C { B = A x x + A y y + A y B B
9 7 A = A x x + A y y + A, B = B x x + B y y + B, C = C x x + C y y + C..6 x y A B C = A x x + A y y + A B x B y B C x C y C { B = A x x + A y y + A y B B x x B } B C y C y + x B y C x C C x C y B = A
1. z dr er r sinθ dϕ eϕ r dθ eθ dr θ dr dθ r x 0 ϕ r sinθ dϕ r sinθ dϕ y dr dr er r dθ eθ r sinθ dϕ eϕ 2. (r, θ, φ) 2 dr 1 h r dr 1 e r h θ dθ 1 e θ h
IB IIA 1 1 r, θ, φ 1 (r, θ, φ)., r, θ, φ 0 r
2016 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 1 16 2 1 () X O 3 (O1) X O, O (O2) O O (O3) O O O X (X, O) O X X (O1), (O2), (O3) (O2) (O3) n (O2) U 1,..., U n O U k O k=1 (O3) U λ O( λ Λ) λ Λ U λ O 0 X 0 (O2) n =
50 2 I SI MKSA r q r q F F = 1 qq 4πε 0 r r 2 r r r r (2.2 ε 0 = 1 c 2 µ 0 c = m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq
49 2 I II 2.1 3 e e = 1.602 10 19 A s (2.1 50 2 I SI MKSA 2.1.1 r q r q F F = 1 qq 4πε 0 r r 2 r r r r (2.2 ε 0 = 1 c 2 µ 0 c = 3 10 8 m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq F = k r
微分方程式の解を見る
( ) norikazu[at]ms.u-tokyo.ac.jp http://www.infsup.jp/saito/materials/829gairon.pdf 28 2 9 NS ( ) 28 2 9 / 5 Newton 2 3 Navier-Stokes 4 5 6 NS ( ) 28 2 9 2 / 5 2 x 2 + bx + c = x = b ± b 2 4c 2 3 x 3 +
21 2 26 i 1 1 1.1............................ 1 1.2............................ 3 2 9 2.1................... 9 2.2.......... 9 2.3................... 11 2.4....................... 12 3 15 3.1..........
IA hara@math.kyushu-u.ac.jp Last updated: January,......................................................................................................................................................................................
5 H Boltzmann Einstein Brown 5.1 Onsager [ ] Tr Tr Tr = dγ (5.1) A(p, q) Â 0 = Tr Âe βĥ0 Tr e βĥ0 = dγ e βh 0(p,q) A(p, q) dγ e βh 0(p,q) (5.2) e βĥ0
![5 H Boltzmann Einstein Brown 5.1 Onsager [ ] Tr Tr Tr = dγ (5.1) A(p, q) Â 0 = Tr Âe βĥ0 Tr e βĥ0 = dγ e βh 0(p,q) A(p, q) dγ e βh 0(p,q) (5.2) e βĥ0](/thumbs/99/141438380.jpg "5 H Boltzmann Einstein Brown 5.1 Onsager [ ] Tr Tr Tr = dγ (5.1) A(p, q) Â 0 = Tr Âe βĥ0 Tr e βĥ0 = dγ e βh 0(p,q) A(p, q) dγ e βh 0(p,q) (5.2) e βĥ0")
5 H Boltzmann Einstein Brown 5.1 Onsager [ ] Tr Tr Tr = dγ (5.1) A(p, q) Â = Tr Âe βĥ Tr e βĥ = dγ e βh (p,q) A(p, q) dγ e βh (p,q) (5.2) e βĥ A(p, q) p q Â(t) = Tr Â(t)e βĥ Tr e βĥ = dγ() e βĥ(p(),q())
Part () () Γ Part ,
Contents a 6 6 6 6 6 6 6 7 7. 8.. 8.. 8.3. 8 Part. 9. 9.. 9.. 3. 3.. 3.. 3 4. 5 4.. 5 4.. 9 4.3. 3 Part. 6 5. () 6 5.. () 7 5.. 9 5.3. Γ 3 6. 3 6.. 3 6.. 3 6.3. 33 Part 3. 34 7. 34 7.. 34 7.. 34 8. 35
1 filename=mathformula tex 1 ax 2 + bx + c = 0, x = b ± b 2 4ac, (1.1) 2a x 1 + x 2 = b a, x 1x 2 = c a, (1.2) ax 2 + 2b x + c = 0, x = b ± b 2
filename=mathformula58.tex ax + bx + c =, x = b ± b 4ac, (.) a x + x = b a, x x = c a, (.) ax + b x + c =, x = b ± b ac. a (.3). sin(a ± B) = sin A cos B ± cos A sin B, (.) cos(a ± B) = cos A cos B sin
Hanbury-Brown Twiss (ver. 2.0) van Cittert - Zernike mutual coherence
Hanbury-Brown Twiss (ver. 2.) 25 4 4 1 2 2 2 2.1 van Cittert - Zernike..................................... 2 2.2 mutual coherence................................. 4 3 Hanbury-Brown Twiss ( ) 5 3.1............................................
ii 3.,. 4. F. (), ,,. 8.,. 1. (75%) (25%) =7 20, =7 21 (. ). 1.,, (). 3.,. 1. ().,.,.,.,.,. () (12 )., (), 0. 2., 1., 0,.
24(2012) (1 C106) 4 11 (2 C206) 4 12 http://www.math.is.tohoku.ac.jp/~obata,.,,,.. 1. 2. 3. 4. 5. 6. 7.,,. 1., 2007 (). 2. P. G. Hoel, 1995. 3... 1... 2.,,. ii 3.,. 4. F. (),.. 5... 6.. 7.,,. 8.,. 1. (75%)
2019 1 5 0 3 1 4 1.1.................... 4 1.1.1......................... 4 1.1.2........................ 5 1.1.3................... 5 1.1.4........................ 6 1.1.5......................... 6 1.2..........................
( ) sin 1 x, cos 1 x, tan 1 x sin x, cos x, tan x, arcsin x, arccos x, arctan x. π 2 sin 1 x π 2, 0 cos 1 x π, π 2 < tan 1 x < π 2 1 (1) (
6 20 ( ) sin, cos, tan sin, cos, tan, arcsin, arccos, arctan. π 2 sin π 2, 0 cos π, π 2 < tan < π 2 () ( 2 2 lim 2 ( 2 ) ) 2 = 3 sin (2) lim 5 0 = 2 2 0 0 2 2 3 3 4 5 5 2 5 6 3 5 7 4 5 8 4 9 3 4 a 3 b
( 12 ( ( ( ( Levi-Civita grad div rot ( ( = 4 : 6 3 1 1.1 f(x n f (n (x, d n f(x (1.1 dxn f (2 (x f (x 1.1 f(x = e x f (n (x = e x d dx (fg = f g + fg (1.2 d dx d 2 dx (fg = f g + 2f g + fg 2... d n n
http://www.ns.kogakuin.ac.jp/~ft13389/lecture/physics1a2b/ pdf I 1 1 1.1 ( ) 1. 30 m µm 2. 20 cm km 3. 10 m 2 cm 2 4. 5 cm 3 km 3 5. 1 6. 1 7. 1 1.2 ( ) 1. 1 m + 10 cm 2. 1 hr + 6400 sec 3. 3.0 10 5 kg
JFE.dvi
,, Department of Civil Engineering, Chuo University Kasuga 1-13-27, Bunkyo-ku, Tokyo 112 8551, JAPAN E-mail : atsu1005@kc.chuo-u.ac.jp E-mail : kawa@civil.chuo-u.ac.jp SATO KOGYO CO., LTD. 12-20, Nihonbashi-Honcho
7 π L int = gψ(x)ψ(x)φ(x) + (7.4) [ ] p ψ N = n (7.5) π (π +,π 0,π ) ψ (σ, σ, σ )ψ ( A) σ τ ( L int = gψψφ g N τ ) N π * ) (7.6) π π = (π, π, π ) π ±
![7 π L int = gψ(x)ψ(x)φ(x) + (7.4) [ ] p ψ N = n (7.5) π (π +,π 0,π ) ψ (σ, σ, σ )ψ ( A) σ τ ( L int = gψψφ g N τ ) N π * ) (7.6) π π = (π, π, π ) π ±](/thumbs/91/106462075.jpg "7 π L int = gψ(x)ψ(x)φ(x) + (7.4) [ ] p ψ N = n (7.5) π (π +,π 0,π ) ψ (σ, σ, σ )ψ ( A) σ τ ( L int = gψψφ g N τ ) N π * ) (7.6) π π = (π, π, π ) π ±")
7 7. ( ) SU() SU() 9 ( MeV) p 98.8 π + π 0 n 99.57 9.57 97.4 497.70 δm m 0.4%.% 0.% 0.8% π 9.57 4.96 Σ + Σ 0 Σ 89.6 9.46 K + K 0 49.67 (7.) p p = αp + βn, n n = γp + δn (7.a) [ ] p ψ ψ = Uψ, U = n [ α
Lebesgue Fubini L p Banach, Hilbert Höld
II (Analysis II) Lebesgue (Applications of Lebesgue Integral Theory) 1 (Seiji HIABA) 1 ( ),,, ( ) 1 1 1.1 1 Lebesgue........................ 1 1.2 2 Fubini...................... 2 2 L p 5 2.1 Banach, Hilbert..............................
1 1.1 ( ). z = a + bi, a, b R 0 a, b 0 a 2 + b 2 0 z = a + bi = ( ) a 2 + b 2 a a 2 + b + b 2 a 2 + b i 2 r = a 2 + b 2 θ cos θ = a a 2 + b 2, sin θ =
1 1.1 ( ). z = + bi,, b R 0, b 0 2 + b 2 0 z = + bi = ( ) 2 + b 2 2 + b + b 2 2 + b i 2 r = 2 + b 2 θ cos θ = 2 + b 2, sin θ = b 2 + b 2 2π z = r(cos θ + i sin θ) 1.2 (, ). 1. < 2. > 3. ±,, 1.3 ( ). A
第8章 位相最適化問題
8 February 25, 2009 1 (topology optimizaiton problem) ( ) H 1 2 2.1 ( ) V S χ S : V {0, 1} S (characteristic function) (indicator function) 1 (x S ) χ S (x) = 0 (x S ) 2.1 ( ) Lipschitz D R d χ Ω (Ω D)
08-Note2-web
r(t) t r(t) O v(t) = dr(t) dt a(t) = dv(t) dt = d2 r(t) dt 2 r(t), v(t), a(t) t dr(t) dt r(t) =(x(t),y(t),z(t)) = d 2 r(t) dt 2 = ( dx(t) dt ( d 2 x(t) dt 2, dy(t), dz(t) dt dt ), d2 y(t) dt 2, d2 z(t)
W u = u(x, t) u tt = a 2 u xx, a > 0 (1) D := {(x, t) : 0 x l, t 0} u (0, t) = 0, u (l, t) = 0, t 0 (2)
3 215 4 27 1 1 u u(x, t) u tt a 2 u xx, a > (1) D : {(x, t) : x, t } u (, t), u (, t), t (2) u(x, ) f(x), u(x, ) t 2, x (3) u(x, t) X(x)T (t) u (1) 1 T (t) a 2 T (t) X (x) X(x) α (2) T (t) αa 2 T (t) (4)
A
A04-164 2008 2 13 1 4 1.1.......................................... 4 1.2..................................... 4 1.3..................................... 4 1.4..................................... 5 2
.5 z = a + b + c n.6 = a sin t y = b cos t dy d a e e b e + e c e e e + e 3 s36 3 a + y = a, b > b 3 s363.7 y = + 3 y = + 3 s364.8 cos a 3 s365.9 y =,
[ ] IC. r, θ r, θ π, y y = 3 3 = r cos θ r sin θ D D = {, y ; y }, y D r, θ ep y yddy D D 9 s96. d y dt + 3dy + y = cos t dt t = y = e π + e π +. t = π y =.9 s6.3 d y d + dy d + y = y =, dy d = 3 a, b
数値計算：有限要素法
( ) 1 / 61 1 2 3 4 ( ) 2 / 61 ( ) 3 / 61 P(0) P(x) u(x) P(L) f P(0) P(x) P(L) ( ) 4 / 61 L P(x) E(x) A(x) x P(x) P(x) u(x) P(x) u(x) (0 x L) ( ) 5 / 61 u(x) 0 L x ( ) 6 / 61 P(0) P(L) f d dx ( EA du dx
構造と連続体の力学基礎
II 37 Wabash Avenue Bridge, Illinois 州 Winnipeg にある歩道橋 Esplanade Riel 橋6 6 斜張橋である必要は多分無いと思われる すぐ横に道路用桁橋有り しかも塔基部のレストランは 8 年には営業していなかった 9 9. 9.. () 97 [3] [5] k 9. m w(t) f (t) = f (t) + mg k w(t) Newton
r d 2r d l d (a) (b) (c) 1: I(x,t) I(x+ x,t) I(0,t) I(l,t) V in V(x,t) V(x+ x,t) V(0,t) l V(l,t) 2: 0 x x+ x 3: V in 3 V in x V (x, t) I(x, t
1 1 2 2 2r d 2r d l d (a) (b) (c) 1: I(x,t) I(x+ x,t) I(0,t) I(l,t) V in V(x,t) V(x+ x,t) V(0,t) l V(l,t) 2: 0 x x+ x 3: V in 3 V in x V (x, t) I(x, t) V (x, t) I(x, t) V in x t 3 4 1 L R 2 C G L 0 R 0
A 2 3. m S m = {x R m+1 x = 1} U + k = {x S m x k > 0}, U k = {x S m x k < 0}, ϕ ± k (x) = (x 0,..., ˆx k,... x m ) 1. {(U ± k, ϕ± k ) 0 k m} S m 1.2.
A A 1 A 5 A 6 1 2 3 4 5 6 7 1 1.1 1.1 (). Hausdorff M R m M M {U α } U α R m E α ϕ α : U α E α U α U β = ϕ α (ϕ β ϕβ (U α U β )) 1 : ϕ β (U α U β ) ϕ α (U α U β ) C M a m dim M a U α ϕ α {x i, 1 i m} {U,