8 1 1., y y (, +1) (-1, ) (, ) (+1, ) y (, -1) 1.1: (,y ) y y ±1 = ± y ±1 = y ± y (, ), = (,y ) (,y ) +1, = ( +, y )=, + 1, = (, y )=, (1.) (1.3) ( )
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1 7 1 () Brgers 1.1 a + b y + c y + d + e + f + g =0. (1.1) y b 4ac > 0 t c =0 b 4ac =0 t = κ b 4ac < 0 + y =4πGρ
2 8 1 1., y y (, +1) (-1, ) (, ) (+1, ) y (, -1) 1.1: (,y ) y y ±1 = ± y ±1 = y ± y (, ), = (,y ) (,y ) +1, = ( +, y )=, + 1, = (, y )=, (1.) (1.3) ( ) +! ) ( +1, 1, = +! ( ) ( ) + 3 3! 3 3! ( 3 ) +... (1.) 3 ( 3 3 ) +... (1.3) ( ) + O( 3 ) (1.4) ( ) = +1, 1, + O( ) (1.5) (, ) ( / ) ( ) ( ) = +1, 1, (1.6)
3 ( ) ( ) (1.) (1.3) = +1,, =, 1, () (1.7) () (1.8) ( +1, + 1, =, + ) + O( 4 ) (1.9) ( ) = +1,, + 1, + O( ) (1.10) ( / ) ( ) = +1,, + 1, (1.11) ( ) y =,+1, +, 1 y. (1.1) t + c = 0 (1.13) c c>0 c (1.13) (, t) =( ct, 0) (1.14)
4 10 1 t = 0 t > 0 c 1 c 1 1.: 1 t>0 t =0 ct 1. 0 = 1 <0 = =0 t> FTCS 1 (1.13) t n t t n+1 = t n + t n = (,t n ) n + c n +1 n 1 = 0 (1.15) t FTCS (Forward n Tme and Centered Dfference n Space) = n 1 ν(n +1 n 1 ) (1.16) ν ν c t (1.17) 1.16 t n t n+1 = t n + t t n 1 (t n+1 ) FTCS 1.3 t n+1 t n 1 1. ( )
5 t bondary n+1 n : FTCS. (t =0) () 3. t t end (a) t ( ) FTCS (1.16) (b) ( ) (c) t : FTCS =1,..., 50 =1 =51,..., 100 =0 ν = c t/ = FTCS 1.4
6 FTCS Von Nemann FTCS n = cos(θ) (1.18) (1.16) θ k θ = k θ = π n θ = π/3 6 1 θ = π θ = π/3 = = : n θ = π/3 = cos(θ) : θ = π = cos(θ)+ νsnθsn(θ) (1.19) n+ = (1 ν sn θ)cos(θ)+νsnθsn(θ) (1.0) = Re [ (1 νsnθ) e θ] (1.1) Re n+k = Re [ (1 νsnθ) k e θ] (1.) ( ) (1.18)
7 n = gn e θ (1.3) g 1 g 1 Von Nemann n = gn ep(θ) FTCS g = 1 1 ν(eθ e θ ) (1.4) = 1 νsnθ (1.5) g =1+ν sn θ 1 (1.6) θ =0 FTCS FTCS t n+1 = t n + t ±1 = ± = n + t t + 1 t t +... (1.7) n +1 = n (1.8) n 1 = n (1.9) FTCS (1.7) (1.8) (1.9) n = 1 ν(n +1 n 1 ) (1.30) t t + 1 ( t t = ν ) (1.31) t + c = 1 t t c (1.3)
8 14 1 t = c t (1.33) = c (1.34) t + c = c t c (1.35) 1 FTCS La-Fredrch FTCS n (n +1 + n 1 )/ n = g n ep(θ) g = 1 (n +1 + n 1) ν (n +1 n 1) (1.36) g = 1 (eθ + e θ ) 1 (eθ e θ ) (1.37) = cosθ νsnθ (1.38) g = cos θ + ν sn θ (1.39) 1.6 g θ (g, θ) La-Fredrch ν = c t/ ν 1 Corant, Fredrch, Lewy (CFL ) (1.36) = 1 ν n ν n 1 (1.40) ν 1 t = t n+1 n 1 +1 n +1 t = t n 1
9 ν = 1 ν=0.5 g -1 θ 1 1.6: La-Fredrch t bondary n+1 n c t : La-Fredrch ν<1 ν >1 1.7 La-Fredrch t = t n+1 t = t n ν = c t/ <1 ν>1 c t < t n 1 n +1 t > / c La-Fredrch La-Fredrch
10 : La-Fredrch ν = t bondary c > 0 n+1 n c t : 1 1 c >0 n t + c n n 1 = 0 (1.41) = n c t (n n 1 ) (1.4) t n+1 t n t n+1 g = 1 ν(1 e θ ) (1.43)
11 = (1 ν + νcosθ) νsnθ (1.44) g = (1 ν + νcosθ) + ν sn θ (1.45) = 1 ν(1 ν)(1 cosθ) (1.46) 0 ν 1 θ g =(1 ν) n + νn 1. (1.47) 0 ν 1 t = t n+1 t = t n 1.9 ( c t, t n ) n 1 n : 1 ν = ν = ν 1,0.75,0.5 1 g θ ( g,θ) 1 (1.4) 1 t + c = 1 c (1 ν) 1 6 c( ) (ν 3ν +1) (1.48) 3
12 La-Wendroff La-Wendroff = n + t t + 1 t t + O( t3 ) (1.49) 3 / t = c / / t = c / = n c t + 1 c t + O( t3 ) (1.50) / / = n 1 c tn +1 n ( ) t c ( n +1 n + n 1) (1.51) La-Wendroff La-Wendroff La-Wendroff von Nemann g = 1 ν (eθ e θ )+ ν (eθ +e θ ) (1.5) = 1 νsnθ + ν cosθ ν (1.53) g = [1 ν (1 cosθ)] + ν sn θ (1.54) = 1 ν (1 ν )(1 cosθ) (1.55) ν 1 θ g 1 1 La-Wendroff La-Wendroff / +1/ = n +1 + n = n c t 1 c t (n +1 n ) (1.56) (n+1/ +1/ n+1/ 1/ ) (1.57) La-Wendroff 1 La-Wendroff Godnov 1 / t + c / =0
13 t bondary n+1 n+1/ n : La-Wendroff t n 1 +1 t n+1/ 1/ +1/ t n+1 = k a k n +k (1.58) t n (, t n ) t n+1 (, t n+1 ) 0 ν 1 n 1 n n 1 n n Godnov (1994) κ ũ n+1 = + κ t (n +1 n + n 1) (1.59) κ Q v κ +1/ = Q v n +1 n (1.60)
14 : La-Wendroff ν = ν = t + c = 0 (1.61) t + f = 0 (1.6) f = c (1.63) (1.6) 1.13 ( 1/ << +1/ ) (1.6) = 1/ = +1/ +1/ d + f( +1/ ) f( 1/ )=0. (1.64) t 1/ n = +1/ 1/ (, t n )d (1.65) ±1/ f ±1/ = n t (f n +1/ f n 1/) (1.66)
15 1.5. Brgers 1 f -1/ f +1/ : (1.66) f n ±1/ f n ±1/ FTCS La-Fredrch f n +1/ = 1 (f n +1 + f n ) (1.67) f n +1/ = 1 [ (1 1 ν )f +1 n +(1+ 1 ] ν )f n (1.68) Upwnd () f +1/ n = 1 [ (f n +1 + f n ) c (n +1 n )] (1.69) c >0 f n +1/ = f n c<0 f +1/ = f n +1 La-Wendroff f +1/ n = 1 [ ] (1 ν)f n +1 +(1+ν)f n (1.70) 1.5 Brgers c Brgers t + = 0 (1.71)
16 1 d/dt = / t + / d dt = 0 (1.7) = : Brgers : Brgers >0 <0 t/ =0.8 Brgers
17 1.5. Brgers 3 t + ( ) = 0 (1.73) f() f() = / c c>0 f n +1/ = f n +1/ ( (t)+ +1 (t))/ +1 (t)+ (t) > 0 f n +1/ = f n = (t) / +1 (t)+ (t) 0 f n +1/ = f n +1 = +1 (t) / 1.15 () =1+ɛsn(k) (ɛ =0.01, 0 k π) Brgers c = Brgers 1.16 () =1+0.1sn(k) Brgers / : () =1+0.1sn(k) Brgers t/ =0.8
18 Brgers Godnov La-Wendroff La-Wendroff f n +1/ = c[n + 1 (1 ν)(n +1 n )] (1.74) 1 c>0 f n +1/ = c La- Wendroff 1 La-Wendroff f n +1/ = c[ n + 1 (1 ν)b +1/( n +1 n )] (1.75) B +1/ (1.75) (1.66) n n 1 n = ν[1 1 (1 ν)b 1/]+ 1 ν(1 ν)b +1/ r (1.76) r n n 1 n +1 n n -1 (1.77) n+1 n 1.17: 1.17 n n 1 (1.76)
19 n+1 n n 1 n (1.76) 1 (1.78) ν B 1/ B +1/ r 1 ν (1.79) CFL 0 ν 1 B 1/ B +1/ r (1.80) 0 B +1/ (1.81) 0 B +1/ r (1.8) 1.18 r <0 B +1/ =0 B +1/ B = r 1 LW mnmod O 1 r 1.18: B +1/ (r) LW La-Wendroff mnmod mnmod La-Wendroff B +1/ =1( LW) r <1/
20 6 1 mnmod ( mnmod) 0 (r<0) mnmod(r) = r (0 r 1) 1 (r>1) (1.83) 1.7 TVD 1 U = d d. (1.84) d du/dt =0 Total Varaton (TV) TV( n ) n +1 n (1.85) Total Varaton TV( ) TV( n ) (1.86) Total Varaton Dmnshng (TVD) TVD 1.8 T t = κ T B t = η B
21 t =0 (, t) (, 0) t (> 0) (, t) t = κ (1.87) κ t = 0 t > : (eplct ) 1 (1.87) t n t t n+1 = t n + t (FTCS n = (,t n ) n t (1.88) = κ n +1 n + n 1 (1.88) = n + κ t (n +1 n + n 1 ) (1.89) t n t n+1 = t n + t t n 1 (t n+1 ) (
22 8 1 Von Nemann FTCS n = g n ep(θ) FTCS g n+1 e θ = g n e θ + κ t gn [e (+1)θ e θ + e ( 1)θ ]. (1.90) g =1 κ t (1 cosθ). (1.91) g κ t (1 cosθ) 1. (1.9) 0 κ t (1 cosθ) =κ t θ sn 1. (1.93) θ ( ) 0 κ t 1. (1.94) FTCS 1 t t 1/4 (mplct ) t n+1 (mplct) eplct Crank-Ncolson λ n t = κ [λ n (1 λ) n +1 n + ] n 1 (1.95) A = b( n ). (1.96) 1. A b
23 Von Nemann λ >1/ κ t/ > 0 t Crank-Ncolson (1) (1994) () (000) (3) Nmercal Comptaton of Internal and Eternal Flows, C. Hrsch, John Wley & Sons, 1990 (1)
Evoltion of onentration by Eler method (Dirihlet) Evoltion of onentration by Eler method (Nemann).2 t n =.4n.2 t n =.4n : t n
5 t = = (, y, z) t (, y, z, t) t = κ (68) κ [, ] (, ) = ( ) A ( /2)2 ep, A =., t =.. (69) 4πκt 4κt = /2 (, t) = for ( =, ) (Dirihlet ondition) (7) = for ( =, ) (Nemann ondition) (7) (68) (, t) = ( ) (
1 1 u m (t) u m () exp [ (cπm + (πm κ)t (5). u m (), U(x, ) f(x) m,, (4) U(x, t) Re u k () u m () [ u k () exp(πkx), u k () exp(πkx). f(x) exp[ πmxdx
1 1 1 1 1. U(x, t) U(x, t) + c t x c, κ. (1). κ U(x, t) x. (1) 1, f(x).. U(x, t) U(x, t) + c κ U(x, t), t x x : U(, t) U(1, t) ( x 1), () : U(x, ) f(x). (3) U(x, t). [ U(x, t) Re u k (t) exp(πkx). (4)
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
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64 3 g=9.85 m/s 2 g=9.791 m/s 2 36, km ( ) 1 () 2 () m/s : : a) b) kg/m kg/m k
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6 2 2 x y x y t P P = P t P = I P P P ( ) ( ) ,, ( ) ( ) cos θ sin θ cos θ sin θ, sin θ cos θ sin θ cos θ y x θ x θ P
6 x x 6.1 t P P = P t P = I P P P 1 0 1 0,, 0 1 0 1 cos θ sin θ cos θ sin θ, sin θ cos θ sin θ cos θ x θ x θ P x P x, P ) = t P x)p ) = t x t P P ) = t x = x, ) 6.1) x = Figure 6.1 Px = x, P=, θ = θ P
重力方向に基づくコントローラの向き決定方法
( ) 2/Sep 09 1 ( ) ( ) 3 2 X w, Y w, Z w +X w = +Y w = +Z w = 1 X c, Y c, Z c X c, Y c, Z c X w, Y w, Z w Y c Z c X c 1: X c, Y c, Z c Kentaro Yamaguchi@bandainamcogames.co.jp 1 M M v 0, v 1, v 2 v 0 v
(Compton Scattering) Beaming 1 exp [i (k x ωt)] k λ k = 2π/λ ω = 2πν k = ω/c k x ωt ( ω ) k α c, k k x ωt η αβ k α x β diag( + ++) x β = (ct, x) O O x
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Evolutes and involutes of fronts Masatomo Takahashi Muroran Institute of Technology
Evolutes and involutes of fronts Masatomo Takahashi Muroran Institute of Technology evolute involute, evolvent involute : involvo evolvent : evolvo 6 4 6 4 4 6 4 6 H 3 γ : I R C γ(t) 0, t(t) := γ(t) γ(t),
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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)
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m(ẍ + γẋ + ω 0 x) = ee (2.118) e iωt P(ω) = χ(ω)e = ex = e2 E(ω) m ω0 2 ω2 iωγ (2.119) Z N ϵ(ω) ϵ 0 = 1 + Ne2 m j f j ω 2 j ω2 iωγ j (2.120)
2.6 2.6.1 mẍ + γẋ + ω 0 x) = ee 2.118) e iωt Pω) = χω)e = ex = e2 Eω) m ω0 2 ω2 iωγ 2.119) Z N ϵω) ϵ 0 = 1 + Ne2 m j f j ω 2 j ω2 iωγ j 2.120) Z ω ω j γ j f j f j f j sum j f j = Z 2.120 ω ω j, γ ϵω) ϵ
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2 464-8602 1 2 2 2 N ΔμN ( N 2/3 ) N - (seed) (nucleation) 1.4 2 2.1 1 Makio Uwaha. E-mail:uwaha@nagoya-u.jp; http://slab.phys.nagoya-u.ac.jp/uwaha/ 2 [1] [2] [3](e) 3 2.1: [1] 2.1 ( ) 1 (cluster) ( N
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
201711grade1ouyou.pdf
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微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 初版 1 刷発行時のものです.
微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. ttp://www.morikita.co.jp/books/mid/00571 このサンプルページの内容は, 初版 1 刷発行時のものです. i ii 014 10 iii [note] 1 3 iv 4 5 3 6 4 x 0 sin x x 1 5 6 z = f(x, y) 1 y = f(x)
2.2 h h l L h L = l cot h (1) (1) L l L l l = L tan h (2) (2) L l 2 l 3 h 2.3 a h a h (a, h)
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EP7000取扱説明書
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数学演習：微分方程式
( ) 1 / 21 1 2 3 4 ( ) 2 / 21 x(t)? ẋ + 5x = 0 ( ) 3 / 21 x(t)? ẋ + 5x = 0 x(t) = t 2? ẋ = 2t, ẋ + 5x = 2t + 5t 2 0 ( ) 3 / 21 x(t)? ẋ + 5x = 0 x(t) = t 2? ẋ = 2t, ẋ + 5x = 2t + 5t 2 0 x(t) = sin 5t? ẋ
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
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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
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DT-870/5100 &DT-5042RFB ...3 1-1...3 1-1...6 1-3...16 2....17...21 3-1...21 3-2...21 3-2...22 3-3...23 3-4...24...25 4-1....25 4-2...27 4-3...28 4-4...33 4-5...36...37 5-1....39 5-2...40 5-3...43...49
II ( ) (7/31) II ( [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Re
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06 年 8 月 日 ( 月 )-6 日 ( 金 ) 千葉大学総合校舎 号館 4 階情報演習室 宇宙磁気流体 プラズマシミュレーションサマースクール 差分法の基礎 三好隆博 広島大学大学院理学研究科 時限目の目標 線形移流方程式 コンピュータ を計算機で解く! 内容 はじめに 差分法 移流方程式の差分法 高次精度風上差分法 はじめに はじめに 微分方程式 未知関数とその導関数を含む方程式 自然現象などを記述する基礎方程式
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arxiv:1311.0281v1(astro-ph.co) R µν 1 2 Rg µν + Λg µν = 8πG c 4 T µν Λ f(r) R f(r) Galileon φ(t) Massive Gravity etc... Action S = d 4 x g (L GG + L m ) L GG = K(φ,X) G 3 (φ,x)φ + G 4 (φ,x)r + G 4X (φ)
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.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
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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
chap9.dvi
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ρ /( ρ) + ( q, v ) : ( q, v ), L < q < q < q < L 0 0 ( t) ( q ( t), v ( t)) dq ( t) v ( t) lmr + 0 Φ( r) dt lmr + 0 Φ ( r) dv ( t) Φ ( q ( t) q ( t)) + Φ ( q+ ( t) q ( t)) dt ( ) < 0 ( q (0), v (0)) (
.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 =,
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c y /2 ddy = = 2π sin θ /2 dθd /2 [ ] 2π cos θ d = log 2 + a 2 d = log 2 + a 2 = log 2 + a a 2 d d + 2 = l
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c 28. 2, y 2, θ = cos θ y = sin θ 2 3, y, 3, θ, ϕ = sin θ cos ϕ 3 y = sin θ sin ϕ 4 = cos θ 5.2 2 e, e y 2 e, e θ e = cos θ e sin θ e θ 6 e y = sin θ e + cos θ e θ 7.3 sgn sgn = = { = + > 2 < 8.4 a b 2
第1章 微分方程式と近似解法
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[ ] 0.1 lim x 0 e 3x 1 x IC ( 11) ( s114901) 0.2 (1) y = e 2x (x 2 + 1) (2) y = x/(x 2 + 1) 0.3 dx (1) 1 4x 2 (2) e x sin 2xdx (3) sin 2 xdx ( 11) ( s
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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
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1 1. x 1 (1) x 2 + 2x + 5 dx d dx (x2 + 2x + 5) = 2(x + 1) x 1 x 2 + 2x + 5 = x + 1 x 2 + 2x x 2 + 2x + 5 y = x 2 + 2x + 5 dy = 2(x + 1)dx x + 1
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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