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- たみえ やまのかみしゃ
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1 16 17 : 1g99p038-8
2 Shortley-Weller
3
4 1 3
5 1.1 (finite difference method:fdm) 4
6 1. Shortley-Weller
7 6
8 .1 7
9 ...1 y(x) (initial condition) y = f(x; y) (.1) y(x 0 ) = y 0 (.) (initial value problem)...1 f(x; y) x x 0 A, y y 0 B f(x, y) M Lipschitz f(x, y) f(x, z) K y z (K ) y(x 0 ) = y 0 (.1) x x 0 r = min(a, B/M) Cauchy-Lipschitz ( ) (.1),(.) Y 0 = y 0 Y i+n = α 0 Y i + + α N 1 Y i + hφ(x i,, x i+n, Y i,, Y i+n ; h) (.3) h = (b a)/n, x j = a + jh, j = 0, 1,, n 8
10 α 0 + α α N 1 = 1 (α i ) Φ f (.3) N = 1 1 N 1 f Runge-Kutta Adams-Bashforth ( ) Moulton ( ) Runge-Kutta 1 ( ) ( ) 9
11 y = f(x, y, y ), a < x < b (.4) y(a) = α, y(b) = β (.5) (boundary value problem) (.5) (boundary condition) 1. y(a) = α, y(b) = β 1 Dirichlet. y (a) = α, y(b) = β Neumann 3. a 0 y(a) a 1 y (a) = α, b 0 y(b) + b 1 y b) = β 3 (Dirichlet ).3.. (Lees(1961)) f(x, y, z) D = {(x, y, z) a x b, < y <, < z < } f y, f z f y 0, f z M (M ) (.4),(.5) a x b C 10
12 3 11
13 3.1 1
14 f (x), f (x) f(x) C m x h > 0 f(x ± h) = f(x) ± hf (x) + h! f (x) ± h3 3! f (3) (x) + + (±h)m 1 (m 1)! f (m 1) (x) + R m (3.1) R m = (±h)m m! f (m) (x ± θh), 0 < θ < 1 f(x + h) f (3) f(x + h) f(x) h = f (x) + h f (ξ) (x < ξ < x + h) f(x h) f (3) f(x) f(x h) h = f (x) h f (ξ) (x h < ξ < x) O(h) f(x) C 3 (3.1) f(x + h) f(x h) f (4) (x) f(x + h) f(x h) h = f (x) + h 6 f (3) (ξ) (x h < ξ < x + h) O(h ) f (x) f(x + h) f(x) h : 1 f(x) f(x h) : 1 h f(x + h) f(x h) : 1 h f(x) C 4 (3.1) f(x + h) f(x h) f (5) (x) f(x + h) f(x) + f(x h) h = f (x) + h 1 f (4) (ξ) (x h < ξ < x + h) 13
15 f (x) O(h ) f(x + h) f(x) + f(x h) h : h h 3.. Ly y + f(x, y, y ), y(a) = α, y(b) = β a < x < b (3.) L h y, y (3.) ( y(x + h) y(x) + y(x h) L h y(x) = + f x, y(x), h ) y(x + h) y(x h) h L h y(x) C 4 L h y(x) Ly(x) = h 1 y(4) (ξ) + h 6 y(3) (η)f z (x, y, ζ) = O(h ) ( ) y(x + h) y(x h) ξ, η (x h, x + h), ζ = y (x) + θ y (x) (0 < θ < 1) [a, b] n + 1 h n L h(y j ) Y j+1 Y j +Y j 1 h h = b a n + 1, x j = a + jh (0 j n + 1) ( + f x j, Y j, Y j+1 Y j 1 h Y 0 = α, Y n+1 = β ) = 0 (j = 1,, n) (3.3) Y j h x j y j 14
16 3.1 (H.B.Keller) f(x, y, z). K, K 0 < K < f y < K, (x, y, z) D h M (3.3) Y 1,, Y n y(x) C 4 max y(x i) Y j 1 max 1 j n K L hy(x i ) 0 (h 0) 1 j n max y(x i) Y j O(h ) 1 j n (3.3) Y j τ j = L h y(x j ) = (L h L)y(x j ) x = x j L h L h τ = max τ j j ( ) 3..3 u = p(x)u + q(x)u + r(x), u(a) = α, u(b) = β a < x < b (3.4) h = b a n+1, x j = a + jh(j = 0, 1,, n), p j = p(x j ), q j = q(x j ), r j = r(x j ), u j = u(x j ) U j (3.4) U j+1 U j + U j 1 h + p i U j+1 U j 1 h + q j U j + r j = 0, (j = 1,, n) ( 1 hp j )U j 1 + (h q j + )U j + ( 1 + hp j )U j+1 = h r j (j = 1,,, n) 15
17 a j = 1 hp j b j = h q j + c j = 1 + hp j d j = h r j a j U j 1 + b j U j + c j U j+1 = d j (j = 1,,, n) U 0 = α, U n+1 = β (3.5) U j n (3.5) A, U, v b 1 c 1 O a b c A = a n 1 b n 1 c n 1 O a n b n U 1 d 1 αa 1 U d U =. U n 1 U n AU = v (3.6), v =. d n 1 d n βc n Newton (3.6) τ j x j O(h ) Keller ( 3.1) O(h ) 16
18 Shortley-Weller Lu(x) d dx ( ) p(x) du dx + q(x)u = f(x), a < x < b u(a) = α, u(b) = β (3.7) p C 4 [a, b], q, f C[a, b], p > 0, q 0 a = x 0 < x 1 < < x n < x n+1 = b h i = x i x i 1, h = max h i i x i+ 1 = 1 (x i + x i+1 ), x i 1 p i± 1 = 1 (x i 1 + x i ) = p(x i± 1 ), u i = u(x i ) x = x i d du (p(x) ) dx dx x=x i ( d p(x) du ) dx dx ( d p(x) du ) dx dx x=x i = p i+ u i+1 u i 1 h i+1 p i 1 h i+1 +h i ( h 3 +O(h i+1 h i ) + O i+1 + h 3 ) i h i+1 + h i x=x i p i+ u i+1 u i 1 h i+1 p i 1 h i+1 +h i u i u i 1 h i u i u i 1 h i (3.8) h i Shortley-Weller S-W (3.8) (3.7) L hu i = h i+1 + h i {( p ) ( i 1 p i+ 1 U i 1 + h i h i+1 + p i h i 1 = f i (i = 1,,, n) 17 + h ( i+1 + h i q i )U i + p } i+ 1 )U i+1 h i+1
19 a i = p i 1 h i b i = h i+1 + h i q i h i+1 +h i ( a i U i 1 + (a i + a i+1 + b i )U i a i+1 U i+1 ) = f i (i = 1,,, n) (3.9) U 0 = α, U n+1 = β (3.9) H(A + B)U = v (3.10) H, A, B, U, v ( H = diag,, h 1 + h A = h n + h n+1 ), B = diag(b 1,, b n ), a 1 + a a O a a + a 3 a U = a n 1 a n 1 + a n a n O a n a n + a n+1 U 1 f 1 + a 1α h 1 +h U f. U n 1 U n, v = x i L h τ i. f n 1 f n + a n+1β h n +h n+1, τ i = L hu(x i ) f(x i ) = L hu(x i ) Lu(x i ) { O(h) (hi h i+1 ) = O(h ) (h i = h i+1 ) u = [u 1,, u n ] t, τ = [τ 1,, τ n ] t H(A + B)u = v + τ (3.11) 18
20 (3.10), (3.11) H(A + B)(u U) = τ A A + B L A + B A O (A + B) 1 A 1 ([1]) u U = (A + B) 1 H 1 τ A 1 H 1 τ τ A 1 H 1 e (3.1) u U, τ u U, τ e 1 φ(x) d ( p(x) du ) = 1, u(a) = u(b) = 0 dx dx σ L hϕ(x j ) = L hϕ(x j ) Lϕ(x j ) = O(h j+1 h j ) 0 (h 0) ϕ = [ϕ 1,, ϕ n ] t, ρ = [ρ 1,, ρ n ] t HAϕ = e + ρ 1 e A 1 H 1 e ϕ < (3.1) { O(h) ((h1,, h n+1 ) (h,, h)) u U τ ϕ = O(h ) (h 1 = = h n+1 = h) (A + B) 1 H 1 τ max i u i U i = { O(h 3 ) O(h ) (3.13) Manteuffel-White,Jr. [9] [10], [11], [1] (3.1) f y 0 (3.), (3.3) (3.13) 1998 Ferreira-Grigorieff [7] S-W 3 1 L 1 (3.13) 19
21 4 0
22 4.1, 4. CPU.6GHz 56MB MATLAB 1
23 d ( p(x) du ) dx dx + q(x)u = f(x), 0 < x < 1 (4.1) u(0) = 0, u(1) = 0 p(x) = 1 q(x) = 100 f(x) = π cos πx 100 cos πx e10 1 e 0 1 [e10x + e 10 e 10x ] cos πx :
![[ 0, 1 ] 100 4. max u i U i =1.](/docs-images/88/116912327/images/24-0.jpg "330e-004 i 4.: 100 [ 0, 0.3 ], [ 0.")
24 [ 0, 1 ] max u i U i =1.330e-004 i 4.: 100 [ 0, 0.3 ], [ 0.7, 1 ] 3
![[ 0, 0.3 ], [ 0.7, 1 ] 30 [ 0, 0.3 ], [ 0.7, 1 ] 51 1.7 4.](/docs-images/88/116912327/images/25-0.jpg "3 4.1 max u i U i = 4.45e-005 i 4.3: [ 0, 0.3 ], [ 0.")
25 [ 0, 0.3 ], [ 0.7, 1 ] 30 [ 0, 0.3 ], [ 0.7, 1 ] max u i U i = 4.45e-005 i 4.3: [ 0, 0.3 ], [ 0.7, 1 ] 51 4
26 [ 0, 0.3 ], [ 0.7, 1 ] [ 0, 0.3 ], [ 0.7, 1 ] [ 0, 1 ] m h=0.3/m e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e : [ 0, 0.3 ], [ 0.7, 1 ] 5
27 4.3. d ( p(x) du ) dx dx + q(x)u = f(x), 0 < x < 1 (4.) u(0) = 0, u(1) = 0 p(x) = x + 1 q(x) = e x f(x) = 4π (x + 1)e sin πx (sin πx cos πx) π cos(πx)e sin πx + e sin πx e x u = e sin πx : 6
28 [ 0, 1 ] max u i U i =1.08e-003 i 4.5: 100 [ 0.1, 0.4 ] 7
![[ 0.1, 0.4 ] 30 [ 0.1, 0.4 ] [ 0.1, 0.4 ] 3 4.](/docs-images/88/116912327/images/29-0.jpg "6 4. max u i U i =9.6604e-004 i 4.6: 8")
29 [ 0.1, 0.4 ] 30 [ 0.1, 0.4 ] [ 0.1, 0.4 ] max u i U i =9.6604e-004 i 4.6: 8
30 [ 0.1, 0.4 ] [ 0.1, 0.4 ] [ 0, 1 ] m h=0.3/m e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e : [ 0.1, 0.4 ] 9
31 4. 4.5, 4.6 [ 0.1, 0.4 ] [ 0, 0.5 ] [ 0, 0.5 ] : max u i U i =.517e-004 i [ 0, 0.5 ] 30
32 [ 0, 0.5 ] [ 0, 0.5 ] [ 0, 1 ] m h=0.5/m e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e : [ 0, 0.5 ] 31
![, [ a, a+0. ] a = 0 0.8 0.1 4.8 4.4 10 3 10 4.8: [ a, a+0. ] a=0 0.8 [ a, a+0. ] [ 0, 1 ] a 0 8.470e-004 1.0074e-003 0.1 1.9450e-003 1.9450e-003 0..85e-003.85e-003 0.3.651e-004.](/docs-images/88/116912327/images/33-0.jpg "651e-004 0.4 1.3534e-003 1.3534e-003 0.5 4.314e-004 1.1793e-003 0.6 4.617e-005 1.3701e-003 0.7.945e-004 1.439e-003 0.8 9.667e-004 1.116e-003 4.4: [ a, a+0. ] a=0 0.8 4.8, 4.4 x = 0.")
33 , [ a, a+0. ] a = : [ a, a+0. ] a=0 0.8 [ a, a+0. ] [ 0, 1 ] a e e e e e e e e e e e e e e e e e e : [ a, a+0. ] a= , 4.4 x = 0.3, 0.5 3
34 5 33
35 a a a 5.1: 34
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38 [1] [ ], 003 [], 1971 [3], 1967 [4], 1973 [5] [ ], 00 [6] MATLAB/SCILAB 003 [7] J.A.Ferreira and R.D.Grigorieff, On the supraconvergence of elliptic finite difference schemes, Appl. Number. Math. 8 (1998), 75-9 [8] H.B.Keller, Numerical Methods for Two-Point Boundary Value Problems, Blaisdell, 1968 [9] T.A.Manteuffel and A.B.White,Jr., The numerical solution of second-order boundary value problems on nonuniform meshes, Math. of Comp. 47 (1986), [10] T.Yamamoto, Harmonic relations between Green s functions and Green s matrices for boundary value problems, RIMS, 000 [11] T.Yamamoto, II, RIMS, 00 [1] T.Yamamoto, III, RIMS,
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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
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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)
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2011 I 2 II III 17, 18, 19 7 7 1 2 2 2 1 2 1 1 1.1.............................. 2 1.2 : 1.................... 4 1.2.1 2............................... 5 1.3 : 2.................... 5 1.3.1 2.....................................
Trapezoidal Rule θ = 1/ x n x n 1 t = 1 [f(t n 1, x n 1 ) + f(t n, x n )] (6) 1. dx dt = f(t, x), x(t 0) = x 0 (7) t [t 0, t 1 ] f t [t 0, t 1 ], x x
![Trapezoidal Rule θ = 1/ x n x n 1 t = 1 [f(t n 1, x n 1 ) + f(t n, x n )] (6) 1. dx dt = f(t, x), x(t 0) = x 0 (7) t [t 0, t 1 ] f t [t 0, t 1 ], x x](/thumbs/92/107866474.jpg "Trapezoidal Rule θ = 1/ x n x n 1 t = 1 [f(t n 1, x n 1 ) + f(t n, x n )] (6) 1. dx dt = f(t, x), x(t 0) = x 0 (7) t [t 0, t 1 ] f t [t 0, t 1 ], x x")
University of Hyogo 8 8 1 d x(t) =f(t, x(t)), dt (1) x(t 0 ) =x 0 () t n = t 0 + n t x x n n x n x 0 x i i = 0,..., n 1 x n x(t) 1 1.1 1 1 1 0 θ 1 θ x n x n 1 t = θf(t n 1, x n 1 ) + (1 θ)f(t n, x n )
第5章 偏微分方程式の境界値問題
October 5, 2018 1 / 113 4 ( ) 2 / 113 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 Poisson 5.1.1 d {2, 3} Lipschitz (A.5 ) Γ D Γ N = \ Γ D Γ p Γ N Γ
dy = sin cos y cos () y () 1 y = sin 1 + c 1 e sin (3) y() () y() y( 0 ) = y 0 y 1 1. (1) d (1) y = f(, y) (4) i y y i+1 y i+1 = y( i + ) = y i
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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)
1.2 y + P (x)y + Q(x)y = 0 (1) y 1 (x), y 2 (x) y 1 (x), y 2 (x) (1) y(x) c 1, c 2 y(x) = c 1 y 1 (x) + c 2 y 2 (x) 3 y 1 (x) y 1 (x) e R P (x)dx y 2
1 1.1 R(x) = 0 y + P (x)y + Q(x)y = R(x)...(1) y + P (x)y + Q(x)y = 0...(2) 1 2 u(x) v(x) c 1 u(x)+ c 2 v(x) = 0 c 1 = c 2 = 0 c 1 = c 2 = 0 2 0 2 u(x) v(x) u(x) u (x) W (u, v)(x) = v(x) v (x) 0 1 1.2
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)
t θ, τ, α, β S(, 0 P sin(θ P θ S x cos(θ SP = θ P (cos(θ, sin(θ sin(θ P t tan(θ θ 0 cos(θ tan(θ = sin(θ cos(θ ( 0t tan(θ
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n=1 1 n 2 = π = π f(z) f(z) 2 f(z) = u(z) + iv(z) *1 f (z) u(x, y), v(x, y) f(z) f (z) = f/ x u x = v y, u y = v x
n= n 2 = π2 6 3 2 28 + 4 + 9 + = π2 6 2 f(z) f(z) 2 f(z) = u(z) + iv(z) * f (z) u(x, y), v(x, y) f(z) f (z) = f/ x u x = v y, u y = v x f x = i f y * u, v 3 3. 3 f(t) = u(t) + v(t) [, b] f(t)dt = u(t)dt
Sturm-Liouville Green KEN ZOU Hermite Legendre Laguerre L L [p(x) d2 dx 2 + q(x) d ] dx + r(x) u(x) = Lu(x) = 0 (1) L = p(x) d2 dx
![Sturm-Liouville Green KEN ZOU Hermite Legendre Laguerre L L [p(x) d2 dx 2 + q(x) d ] dx + r(x) u(x) = Lu(x) = 0 (1) L = p(x) d2 dx](/thumbs/88/116809375.jpg "Sturm-Liouville Green KEN ZOU Hermite Legendre Laguerre L L [p(x) d2 dx 2 + q(x) d ] dx + r(x) u(x) = Lu(x) = 0 (1) L = p(x) d2 dx")
Sturm-Liouville Green KEN ZOU 2006 4 23 1 Hermite Legendre Lguerre 1 1.1 2 L L p(x) d2 2 + q(x) d + r(x) u(x) = Lu(x) = 0 (1) L = p(x) d2 2 + q(x) d + r(x) (2) L = d2 2 p(x) d q(x) + r(x) (3) 2 (Self-Adjoint
5. [1 ] 1 [], u(x, t) t c u(x, t) x (5.3) ξ x + ct, η x ct (5.4),u(x, t) ξ, η u(ξ, η), ξ t,, ( u(ξ,η) ξ η u(x, t) t ) u(x, t) { ( u(ξ, η) c t ξ ξ { (
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