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1 213 2 katurada AT meiji.ac.jp ,

2 R d Alembert , d Alembert R n Duhamel Fourier R 2, R Huygens (W-IBP) (W-IBP) : Fourier (1)

3 : : (HE), (DBC) : (IC) {c n } Fourier (2) : Fourier S f : Fourier t = : f C 1 Green Fourier Fourier Neumann Neumann Neumann Neumann Dirichlet Neumann (H-IBP) (N-H-IBP) (H-IBP) Neumann Dirichlet, Neumann : Fourier Laplace Poisson

4 r = x Laplace Dirichlet Fourier Fourier : Euler Poisson Laplace, Gauss Green Green third identity E Gauss : Poisson Fourier Dirichlet, Poincaré-Perron , Potential A 183 A A A.2.1 Newton A A A.2.4 Laplace A.2.5 Poisson A

5 A.4 : B 19 B B B B B B B B B.2.7 Fourier B B B C 21 C C.2 compact C.3 Weierstrass M C C C D Fourier 27 D.1 Fourier Fourier D.2 Fourier D D D.5 L D D D D D.7.3 Fourier S E Fourier 215 E.1 Fourier ( ) E F Green 222 F.1 1 Poisson F

6 F F.2 2 Green G misc 229 G.1 P S G.2 Kirchhoff G G G.5 gnuplot G.6 Mathematica G.7 ( )BASIC G.7.1 ( )BASIC G G H Fourier 245 H H I 248 I.1 von Neumann I I.3 :

7 .1 Fourier 1, (1) 1, d Alembert () 2 (2), Duhamel ( ), Foueri ( ) 3 (3) 2 3 Huygens ( ) 4 (4) 1, 5 (1) Fourier,, Dirichlet ( ) Neumann ( ), 6 (2),, 7 (3) Fourier 6

8 8 (4) Fourier 9 (5) Fourier 1 (6), 11 Poisson ( ) Laplace ( ) (1), Poisson Dirichlet 12 Poisson Laplace (2) 2 Laplace Dirichlet 13 Poisson Laplace (3) Green,,, Gauss 14 Poisson Laplace (4) Dirichlet, 15 3 WWW, I,II,,, (26),, 7

9 : 14 ( B ) : :.2 ( ) ( ) ( ) ( 15 ) ( ) () WWW ( ) 8

10 .3 2 (partial differential equation, PDE) PDE PDE ( (ordinary differential equation, ODE)) PDE ODE PDE Laplace (Laplace, Laplace operator, Laplacian) := x 2 1 x 2 2 x 2 n (i) (wave equation) u tt = u. u = u(x, t) (ii) (, heat equation) (iii) u t = u. u = u(x, t) Poisson (Poisson equation) u = f. u = u(x) f = f(x) 3 PDE ( ) 3 (,, ) 1 ( ) 1 Poisson 9

11 Fourier 3 Fourier ( B ) : PDE 1. Fourier (Fourier Fourier ) 2. ( 2 ) 3. 3 Lebesgue () PDE () (Fourier 3 )PDE 4 : () ( ).4 ( ) ( ) ( 2, 3 ) [57] ( 2 (functional) 1

12 )Laplace Maxwell Navier-Stokes [23] ( ) [64] () () Poincaré-Perron [22] ( ) (?) () [57] [33], [67] [21] 3 [56] 3 () [33] [56] [32] [72] [18] (i) ( ) ( ) (ii) ( ) 4 (iii) ( )

13 [45] Schrödinger [71] () [4] [3] [48] ( ) [73] [69] 5 ( 4 ) 1 [2], () katurada@meiji.ac.jp WWW ( 5 12

14 [1],, (28). [2] V I,, (1999). [3],, (1996). [4], 2,, (1997). [5],, (1963). [6],, (1966). [7],, (1979). [8],, (1989). [9],, (22). [1],,, (25) [11] [12], 1, 2,, (1997).,, (1982). [13], 1, [14], 2, tahensuu2/ [15], Laplacian, polar-laplace.pdf [16], y + py + qy = f(x) Green, jp/~mk/lecture/ode/green/green.pdf ( ). [17],, ODE.pdf (1991, ). [18],, (1998). [19] (George Gamow) ( ),, (1977). [2], 2, (1992). [21],, (198). 13

15 [22],, (1978, 21 ). [23],,, (1989). [24],,,, (1991). James Gleick, Chaos making a new science, Viking Penguin (1987). [25] T.W.,,, (1996). [26],,, (1998). [27], Fourier,, (1978). [28] A. N.,, 19 III, (29). [29], 3, (1991). [3],, (1995). [31] [32] (1978).,, (1999). [33],, (26). [34], I, (198). [35], II, (1985). [36], 2,, 2 14, (1998). [37],,, (199). [38],,, (25). [39],, (23). [4], ( 3 ), (1983). [41],, (199). [42],, (1981). [43] ( ),, (1998). [44],, (1959). [45],, (24). 14

16 [46],, (1983) [47] ( ),, (196). [48], (1999).,,,, [49], ( ), G5, (196). [5],, 1,, (1971). [51], ε-δ, (21). [52],, (1974, 24). [53] ( ),, (1977). Philosophiae Naturalis Principia Mathematica, 1687 [54],,, (1957). John von Neumann, Mathematische Grundlagen der Quantenmechanik (1932). [55],,, III,, (1969). [56] ( ), (1996). [57],,,, I, II ( ), (1977, 1979). [58],, (1991). [59],,, (1991)., Fourier, Lebesgue [6],, (27). (1995) [61], (2). [62],,,, (25). Par M. Fourier (Jean Baptiste Joseph Fourier), Théorie analytique de la chaleur ( Paris 1822 ) [63],,,, (1988). [64] I. G. Petrovskiĭ ( ),,, (1958). [65] E. T.,,,,, (1976). 15

17 [66],, (23). [67],,,, (1996). 24 [68],, 4 1 (1999). [69],, (1965). [7],, (1997). [71],,, (26). 1 (1997) [72],, (1994). [73],, (22). [74],, I, (1961). [75] D.,,, (1998). Detlef Laugwitz, Bernhard Riemann ; Wendepunkte in der Auffassung der Mathematik (Vita Methematica, Bd. 1) published by Birkhäuser Verlag AG (1996). [76] Sheldon Axler, Paul Bourdon, Ramey Wade, Harmonic Funtion Theory, second edition, Springer (21). [77] Gerald B. Folland, Introduction to Partial Differential Equations, second edition, Princeton University Press (1995). [78] M. H. Protter and H.F.Weinberger, Maximum principles in differential equations, Springer- Verlag (1984). [79] Oliver Dimon Kellog, Foundations of potential theory, Springer (1929). [8] O. Perron, Eine neue Behandlung der ersten Randwertaufgabe für u =, Math. Zeitschrift, 18 (1923), pp

18 1 ( [57] [22] ) [64] [4] [69] 1.1 x, t u = u(x, t) 1 2 u (x, t) = u(x, t) c(x) 2 t2 ( x Laplacian) (wave equation) c = c(x) x [, L] 1 t x u(x, t) u u c(x) 2 t (x, t) = 2 u (x, t) 2 x2 ((x, t) (, L) (, )) ( [57] ) c(x) T (x) c(x) = ρ(x) T (x) ρ(x) T (x) ρ(x) c(x) c(x) 1 2 u t (x, t) = 2 u (x, t) 2 x2 2 () t, (x, y) u(x, y, t) 1 1 c 2 u tt = x 1 2 u c 2 t (x, y, t) = 2 u 2 x (x, y, t) + 2 u (x, y, t) 2 y2 ( ) u x 1 + (u x ) 2 17

19 T ρ c T c = ρ u c 2 t (x, y, z, t) = 2 u 2 x (x, y, z, t) + 2 u 2 y (x, y, z, t) + 2 u (x, y, z, t). 2 z2 [18] Maxwell ( ) E, B ( ) (Maxwell (1873), ) E, B, ρ, j Maxwell (Maxwell s equations) (1.1.1) E = ρ, E = B ε t, B =, c2 B = j + E ε t 2 (c, ε 3 ) (ρ, j ) E =, E = B t, B =, c2 B = E t rot(rot f) = grad(div f) f 1 2 E c 2 t B c 2 t 2 = B ( B) = = ( E) t t = E ( E) = E = E, = 1 c 2 t ( E) = 1 c E 2 t = 1 c ( c 2 B ) = ( B) 2 = B ( B) = B = B. E, B c Maxwell (James Clerk Maxwell, ) (1864 ) 1887 Hertz (Heinrich Rudolph Hertz, ) () [9], [1] 2 Maxwell Heaviside 3 SI c = m/s ( ), ε = 17 4πc F/m. 18

20 u = u(x, y, z, t) ρ 2 u = µ u + (λ + µ) grad (div u) t2 ( ρ, µ, λ ) p := div u, s := rot u ρ 2 p = (λ + 2µ) p t2 (P ), ρ 2 s = µ s (S ) t2 P (primary wave) S (secondary wave) ( p.229) 2 u t 2 (x, y, t) = 2 u(x, y, t) 4 (Kirchhoff-Love ) ( 2 2 = x y 2 ) 2 = 4 x x 2 y y R d Alembert (1.2.1) 1 2 u c 2 t (x, t) = 2 u (x, t) ((x, t) R R) 2 x2 u = u(x, t) (c ) f : R R, g : R R C 2 (1.2.2) u(x, t) := f(x ct) + g(x + ct) u (1.2.1) ( ) () (1) C 2 f : R R, g : R R u (1.2.2) u (1.2.1) (2) C 2 u: R R R (1.2.1) (1.2.2) C 2 f : R R, g : R R 19

21 4 (1) u t (x, t) = f (x ct) ( c) + g (x + ct) c, u tt (x, t) = f (x ct) ( c) 2 + g (x + ct) c 2 = c 2 (f (x ct) + g (x + ct)), u x (x, t) = f (x ct) + g (x + ct), (2) (x, t) (ξ, η) (1.2.3) ξ = x ct, η = x + ct v(ξ, η) = u(x, t) u t = c v ξ + c v η, 2 u t = u 2 t t = u x = v ξ + v η, 2 u x = u 2 x x = t = ξ t ( c ξ + c η ξ + η t u xx (x, t) = f (x ct) + g (x + ct) η = c ξ + c η ) ( c v ) ( ) ξ + c v = c 2 2 v η ξ 2 2 v 2 ξ η + 2 v. η 2 x = ξ x ξ + η x η = ξ + η ( ξ + ) ( v η ξ + v ) = 2 v η ξ v 2 ξ η + 2 v η u c 2 t 2 u 2 x = 4 2 v 2 ξ η u 1 c 2 u tt = u xx 2 v ξ η =. v η ξ η 5 : G C 1 (R; R) s.t. v η = G(η). 4 f a b (f(ax + b)) = af (ax + b) u 1 5 f = f(x, y) C 1 (R 2 ; R) f x f(x, y) = f(, y) ( y R) f(x, y) = f(, y) + 2 x f x (t, y) dt

22 G g (η) = G(η) g 1 C 2 (R; R) v (v(ξ, η) g(η)) = G(η) =. η η v(ξ, η) g(η) η ξ : f C 2 (R; R) s.t. v(ξ, η) g(η) = f(ξ). u(x, t) = v(ξ, η) = f(ξ) + g(η) = f(x ct) + g(x + ct). (1.2.2) (1.2.1) ( ) 6 d Alembert f(x ct) x c g(x + ct) x c (1.2.2) (1.2.3) ( ) 2 ( ) 2 ( 1 1 = c 2 t x c t + x ) ( 1 c t x 2 η 2 ξ ) 1.2.2, d Alembert (1.2.1) u = u(x, t) ( t = ) (1.2.4) u(x, ) = ϕ(x), u (x, ) = ψ(x) (x R) t (1.2.1) (1.2.4) u = u(x, t) (initial value problem) Cauchy (Cauchy problem) (1.2.4) (initial condition) t 2 ϕ ψ (initial values) (1.2.1) u (1.2.2) (1.2.4) (1.2.2) f, g 6 ( ) 1 21

23 (1.2.2) f(x) + g(x) = 1 c f(x) + g(x) = ϕ(x) (x R), c( f (x) + g (x)) = ψ(x) (x R) x 1 f(x) = 1 ( ϕ(x) 1 2 c g(x) = 1 ( ϕ(x) c ψ(y) dy f() + g() (x R) x x ) ψ(y) dy + f() g(), ) ψ(y) dy f() + g() (1.2.5) u(x, t) = 1 2 (ϕ(x ct) + ϕ(x + ct)) + 1 2c x+ct x ct. ψ(y) dy. d Alembert Stokes (Euler ) (1 ) ϕ C 2 (R), ψ C 1 (R) (1.2.1), (1.2.4) C 2 d Alembert (1.2.5) d Alembert ( 8 ) 9 (1) ϕ(x) = sin x, ψ(x). { 1 (x ( 1, 1)) (2) ϕ(x) = ( ), ψ(x). (3) ϕ(x) =, ψ(x) = sin x. 7 Euler d Alembert Stokes (21 ) WWW ( jp/~ee881/) 9 (2) d Alembert u 22

24 1.2.3 d Alembert u(x, t) = 1 2 (ϕ(x ct) + ϕ(x + ct)) + 1 2c x+ct x ct ψ(y) dy. (x, t ) (t > ) u u(x, t ) ϕ x ± ct ψ I := [x ct, x + ct ] I I (x, t ) (domain of dependence) 1 t c(t t ) (x x ) = (x, t ) c(t t ) + (x x ) = (x ct, ) I (x + ct, ) x 1.1: (x, t ) x (x, ) ϕ, ψ u(x, t) (x, t) Σ := {(x, t) R R; c t x x } Σ (x, ) (domain of influence) 11 ( Σ (x, ) (x, t) x (x, t) x [x c t, x + c t ] x c t x x + c t c t x x c t x x c t. ) 1 ( ) {(x, t); x x c(t t )} ((x, t ) ) 11 23

25 t Σ ct + (x x ) = ct (x x ) = (x, ) x 1.2: (x, ) 1 2 u c 2 t (x, t) = 2 u (x, t) 2 x2 (propagation speed, speed of propagation) c 1.3 R n x 1 n x R n Fourier Fourier ( ) Duhamel ( () F : R n [, ) R, ϕ: R n R, ψ : R n R ) (1.3.1) 2 u t (x, t) = 2 c2 u(x, t) + F (x, t) ((x, t) R n (, )), u u(x, ) = ϕ(x), t (x, ) = ψ(x) (x Rn ) 3 24

26 1 F, ϕ 2 F, ψ 3 ϕ, ψ 2 v t (x, t) = 2 c2 v(x, t) ((x, t) R n (, )), v v(x, ) =, t (x, ) = ψ(x) (x Rn ). 2 w t 2 (x, t) = c2 w(x, t) w(x, ) = ϕ(x), w t (x, ) = (x Rn ). ((x, t) R n (, )), 2 U t (x, t) = 2 c2 U(x, t) + F (x, t) ((x, t) R n (, )), U U(x, ) =, t (x, ) = (x Rn ). v, w, U u u := v + w + U (, the superposition principle, principle of superposition) v 1 w, U 2, 3 u Duhamel (Duhamel s principle) v ψ w := v ϕ t v w 2 v tt (x, t) = c 2 v(x, t) ((x, t) R n (, )), v(x, ) = (x R n ), v t (x, ) = ϕ(x) (x R n ) w := v t w tt (x, t) = c 2 w(x, t) ((x, t) R n (, )), w(x, ) = ϕ(x) (x R n ), w t (x, ) = (x R n ). 12 Jean Marie Constant Duhamel ( , St Malo Paris ). 25

27 ( ) w t = v tt = c 2 v, w tt = t w t = t ( c 2 v ) = t w(x, ) = v t (x, ) = ϕ(x), ( n c 2 2 x 2 j=1 j w t (x, ) = v tt (x, ) = c 2 v(x, ) = c 2 =. v ) = c 2 n 2 x 2 j=1 j ( ) t v = c2 t v = c 2 w, f : R 2 (x, y) f(x, y) R f(x, ) = (x R) f x (x, ) = f y (x, ) = v(x, ) = (x R n ) v xi (x, ) =, v xi x i (x, ) = (x R n ) v(x, ) = (x R n ) ( ) 2 t 1 u tt = c 2 u () (1.3.2) u tt (x, t) = c 2 u(x, t) (R n R), u(x, ) = ϕ(x) (R n ), u t (x, ) = ψ(x) (R n ) ϕ v ψ u := t v ϕ + v ψ (1.3.2) 1 3 ( ( [64], 2 12) ) (1.3.3) U(x, t) := t v F (,s) (x, t s)ds v F (,s) s 2 v t (x, t) = 2 c2 v(x, t) ((x, t) R n (, )), v v(x, ) =, t (x, ) = F (x, s) (x Rn ) 26

28 v (F (, s) s 1 R n y F (y, s) R ) U t (x, t) = t U(x, ) = ds =. t t v (x, t s)ds + v (x, t s) s=t = F (,s) F (,s) t v (x, t s)ds F (,s) U (x, ) = t ds =. 2 U t 2 (x, t) = t2 t v (x, t s)ds + 2 F (,s) t v (x, t s) F (,s) = U(x, t) = t t 2 t v (x, t s)ds + F (x, t), 2 F (,s) v F (,s) (x, t s)ds s=t 2 U t 2 (x, t) c2 U(x, t) = = t t U 3 ( ) 2 t v (x, t s) 2 F (,s) c2 v F (,s) (x, t s) ds + F (x, t) ds + F (x, t) = F (x, t) (1 ) 1 d Alembert v ψ (x) = 1 2c x+ct x ct ψ(y) dy, w(x, t) = 1 (ϕ(x ct) + ϕ(x + ct)) 2 ( v ϕ t = w ) v F (,s) (x, t) = 1 2c x+ct x ct F (y, s) dy. 13 F (y, s) F (y, s) (y, s) F 27

29 x+ct ( y z 1 F (z, s) dz 2c x ct s F (, s) ) t ( t ) 1 x+c(t s) (1.3.4) U(x, t) = v F (,s) (x, t s) ds = F (y, s) dy ds 2c = 1 2c t x+c(t s) x c(t s) F (y, s) dyds. x c(t s) ( ) ( ) dx dt G(t) = Ax + f(t), x() = x(t) = dx dt x(t) := dx dt = Ax, x() = 1 t G(t s)f(s) ds = Ax + f(t), x() = t e A(t s) f(s) ds n n { x (n) (t) + a 1 x (n 1) (t) + + a n 1 x (t) + a n x(t) =, x() = x () = = x (n 2) () =, x (n 1) () = 1 G(t) { x (n) (t) + a 1 x (n 1) (t) + + a n 1 x (t) + a n x(t) = F (t), (1.3.5) x() = x () = = x (n 2) () = x (n 1) () = (1.3.6) x(t) = t G(t s)f (s) ds ( [16]) G(t) { x (n) (t) + a 1 x (n 1) (t) + + a n 1 x (t) + a n x(t) =, x() = x () = = x (n 2) () =, x (n 1) () = F (s) 28

30 G( )F (s): t G(t)F (s) v F (s) { x (n) (t) + a 1 x (n 1) (t) + + a n 1 x (t) + a n x(t) = F (t), x() = x () = = x (n 2) () = x (n 1) () = x(t) = t v F (s) (t s) ds 1 3 (1.3.6) (1.3.5) ( ) ω >, a, b F : [, ) R x (t) = ω 2 x(t) + F (t), x() = a, x () = b ( F ) Fourier (Fourier ) ( (1.3.1) 1 ) 1 1 ix ξ sin(c ξ t) (1.3.7) v(x, t) = e ψ(ξ) dξ. (2π) n/2 R c ξ n i x ξ = : ψ(ξ) = n x j ξ j (R n ), ψ ψ Fourier j=1 1 e ix ξ ψ(x) dx (ξ R n ). (2π) n/2 R n (1.3.7) () 14 [22] R n 14 29

31 Fourier 15 2 v (ξ) = x 2 j v x j (ξ) = iξ j v(ξ) x j v x j (ξ) = iξ j v x j (ξ) = (iξ j ) 2 v(ξ) = ξ 2 j v(ξ). j = 1, 2,, n Laplacian Fourier v(ξ) = ξ 2 v(ξ) v tt = c 2 v ( ) 2 d v(ξ, t) = c 2 ξ 2 v(ξ, t) dt ( ξ ) 2 16 v(ξ, t) = A(ξ) cos(c ξ t) + B(ξ) sin(c ξ t) A(ξ), B(ξ) v(ξ, ) =, d v (ξ, ) = ψ(ξ) dt A(ξ), B(ξ) = ψ(ξ) c ξ v(ξ, t) = sin(c ξ t) c ξ Fourier (1.3.7) 17 u () [ 1 u(x, t) = e ix ξ cos(c ξ t) ϕ(ξ) + sin(c ξ t) ] (1.3.8) ψ(ξ) dξ (2π) n/2 R c ξ n t [ ] ix ξ sin(c ξ (t s)) + e F (ξ, s)dξ ds R c ξ n 15 ix ξ v e (x) dx = e ix ξ v(x) dx = iξ j e ix ξ v(x) dx R x n j R x n j R n 16 x (t) = ω 2 x(t) (ω = c ξ ) x(t) = A cos ωt+b sin ωt (A, B ) 17 1 Fourier f(x) = e ix ξ f(ξ) dξ (2π) n/2 R n 3 ψ(ξ).

32 F (ξ, s) s x x F (x, s) Fourier 1 F (ξ, s) := e ix ξ F (x, s) dx. (2π) n/2 R n R 2, R Fourier (1.3.8) 18 n 2 3 (Kirchhoff 19 (Kirchhoff s formula), Poisson 2 ) (1.3.9) (1.3.1) 2 u t 2 (x, t) = c2 u(x, t) u(x, ) = ϕ(x), ((x, t) R n (, )), u t (x, ) = ψ(x) (x Rn ) c ϕ, ψ c >, ϕ C 3 (R n ), ψ C 2 (R n ) (1.3.9), (1.3.1) (1) n = 3 ( (Kirchhoff) ) u(x, t) = 1 [ ( ) 1 ϕ(y) dσ 4πc 2 y + 1 ] ψ(y) dσ y. t t y x =ct t y x =ct dσ y (2) n = 2 ( (Poisson) ) [ ( u(x, t) = 1 2πc t y x ct ) ϕ(y) c2 t 2 y x dy + 2 y x ct ] ψ(y) c2 t 2 y x dy. 2 ( ψ ( 2 ) ψ ϕ t ϕ ( 1 ) ) Gustav Robert Kirchhoff ( , Königsberg Berlin ) Kirchhoff Kirchhoff 2 Siméon Denis Poisson ( , Pithiviers, Sceaux ) 31

33 1.3.5 (R 3 Laplacian ) R 3 Laplacian u xx +u yy +u zz (1.3.11) u = 2 u r + 2 u 2 r r + 1 r 2 S u = 1 2 r r (r u) r 2 S u. S Laplace-Beltrami (Laplace- Beltrami operator) S u := 1 ( sin θ u ) u (1.3.12) sin θ θ θ sin 2 θ ϕ 2 = 2 u θ + 1 u 2 tan θ θ u sin 2 θ ϕ. 2 ( ) [13], [15] (1) () x R 3 r >, t y x = r u(y, t) ũ(r, t) : ũ(r, t) := 1 4πr 2 y x =r u(y, t) dσ y. 1 1 c 2 ũtt = ũ rr + 2 r ũr x y = x + rω, ω = sin θ cos ϕ sin θ sin ϕ cos θ ((θ, ϕ) Σ := [, π] [, 2π]) dσ y = r 2 sin θ dθ dϕ ũ(r, t) = 1 4π ( ) 2 1 ũ(r, t) = 1 c 2 t 4π u 1 c 2 u tt = u (θ,ϕ) Σ (θ,ϕ) Σ u(x + rω, t) sin θ dθ dϕ. ( ) 2 1 u(x + rω, t) sin θ dθ dϕ. c 2 t ( ) 2 1 ũ(r, t) = 1 [ ( ) ] c 2 t 4π (θ,ϕ) Σ r r r + 1 r 2 S u(x + rω, t) sin θ dθ dϕ. (1.3.12) π d (sin θf (θ)) dθ =, dθ 2π d (G(ϕ)) dϕ = dϕ 32 (G 2π )

34 S ( ) 2 1 ũ(r, t) = 1 c 2 t 4π (θ,ϕ) Σ [ ( ) ] u(x + rω, t) sin θ dθ dϕ. r r r ( ) [ 2 ( ) ] 2 1 ũ(r, t) = u(x + rω, t) sin θ dθ dϕ c 2 t r r r 4π (θ,ϕ) Σ [ ( ) ] 2 = + 2 ũ(r, t). r r r v(r, t) := rũ(r, t) ( 2 r 2 (ru(r)) = ru rr + 2u r ) 1 c v tt(r, t) = v 2 rr (r, t) ((r, t) (, ) (, )), v(r, ) = r ϕ(r), v t (r, ) = r ψ(r) (r (, )) ϕ(r) := 1 1 ϕ(y) dσ 4πr 2 y, ψ(r) := ψ(y) dσ y x =r 4πr 2 y. y x =r ũ(, t), ϕ, ψ R 1 c v tt(r, t) = v 2 rr (r, t) ((r, t) R (, )), v(r, ) = r ϕ(r), v t (r, ) = r ψ(r) (r R). d Alembert ( ) v(r, t) = 1 ( ) (r + ct) ϕ(r + ct) + (r ct) ϕ(r ct) c r+ct r ct s ψ(s) ds. r r ( ) ( ) 1 1 lim v(r, t) = lim ũ(r, t) = lim u(y, t) dσ r r r r 4πr 2 y = u(x, t). y x =r ( ) 1 ϕ 1 [ ] (r + ct) ϕ(r + ct) + (r ct) ϕ(r ct) = (ct + r) ϕ(ct + r) (ct r) ϕ(ct r) 2r 2r s ϕ(s) [ ] (r + ct) ϕ(r + ct) + (r ct) ϕ(r ct) = d ds (s ϕ(s)) lim r 1 2r s=ct = ( ) t ϕ(ct). t 33

35 ( ) 2 F (ξ) := ξ s ψ(s) ds (ψ ) F 1 2cr r+ct r ct s ψ(s) ds = 1 c 1 (F (r + ct) F (r ct)) 2r = 1 F (ct + r) F (ct r) c 2r u(x, t) = t ψ(ct) + t (t ϕ(ct)) 1 = t ψ(y) dσ 4π(ct) 2 y + y x =ct t = 1 ( ψ(y) dσ 4πc 2 y + t t y x =ct 1 c F (ct) = 1 c ct ψ(ct) = t ψ(ct). ( t y x =ct ) 1 ϕ(y) dσ 4π(ct 2 y ) y x =ct ϕ(y) t dσ y ). ( 3 (, ) ) u sin 2 θ θ ϕ2 [, π] 3 ϕ(r) := r ϕ(r), ψ(r) := r ψ(r) d Alembert ϕ C 2, ψ C 1 ( G.2 (p. 23) ) (2) (Poisson ) Hadamard () u 2 ũ(x, y, z, t) := u(x, y, t), ϕ(x, y, z) := ϕ(x, y), ψ(x, y, z) := ψ(x, y) ũ 3 1 c ũtt(x, y, z, t) = ũ(x, y, z, t) ((x, y, z, t) R 3 R), 2 ũ(x, y, z, ) = ϕ(x, y) ((x, y, z) R 3 ), ũ t (x, y, z, ) = ψ(x, y) ((x, y, z) R 3 ) Kirchhoff [ ( (1.3.13) ũ(x, t) = 1 1 4πc 2 t t y x =ct ϕ(y) dσy ) + 1 t y x =ct ψ(y) dσy ]. 34

36 ( 21 ) (a 1, a 2, a 3 ) R 3, r > S := { (x 1, x 2, x 3 ); (x 1 a 1 ) 2 + (x 2 a 2 ) 2 + (x 3 a 3 ) 2 = r 2}, D := { (x 1, x 2 ); (x 1 a 1 ) 2 + (x 2 a 2 ) 2 r 2} D f : D R f(x 1, x 2 ) f dσ = 2r r2 (x 1 a 1 ) 2 (x 2 a 2 ) dx 1dx 2. 2 S D f(x 1, x 2, x 3 ) := f(x 1, x 2 ) ((x 1, x 2, x 3 ) S) a = (a 1, a 2, a 3 ) R 3, r > a := (a 1, a 2 ), S := { y R 3 ; y a = r }, D := { z R 2 ; z a r } D f f dσ = 2r (1.3.13) [ ( ũ(x, t) = 1 1 4πc 2 t t 2ct z x ct + 1 t 2ct S z x ct D f(z) r2 z a 2 dz. ) ϕ(z) c2 t 2 z x dz 2 ] ψ(z) c2 t 2 z x dz 2 [ ( ) = 1 ϕ(z) 2πc t z x ct c2 t 2 z x dz + 2 z x ct ] ψ(z) c2 t 2 z x dz. 2 1 (d Alembert ) Huygens (x, t ) R n (, ) 21 S z = c± R 2 (x a) 2 (y b) 2 z = F (x, y) dσ = 1 + (F x ) 2 + (F y ) 2 dx dy Christiaan Huygens ( , Hague Hague (1657) ( ) (1678) ) 35

37 n = 1 [x ct, x + ct ] ({y R 1 ; y x ct } ) n = 2 {y R 2 ; y x ct } n = 3 {y R 3 ; y x = ct } n = 1 n 2 (i) n x ct : {y R n ; y x ct } (ii) n x ct : {y R n ; y x = ct } c {y R n ; y x ct } ( ) n (i) (ii) {y; y x = ct } (1km 3 (1 m 34 m/s 3 ) 1 ( ) 4 ()) 3 (SF ) 2 4 ( ) u(x, t ) (x R n, t > ) t = {y R n ; y x = ct } Huygens (Huygens principle) ( ) n = 3 Huygens n = 1 n = ( ) [, L] 36

38 (boundary condition) {, L} ρ = ρ(x) (> ), ϕ = ϕ(x), ψ = ψ(x) T (1.4.1) ρ(x) 2 u t (x, t) = T 2 u (x, t) 2 x2 ((x, t) (, L) (, )), (1.4.2) u(, t) = u(l, t) = (t (, )), (1.4.3) u(x, ) = ϕ(x), u (x, ) = ψ(x) t (x [, L]). u = u(x, t) ( (W-IBP) ) (initial boundary value problem) (mixed problem) (1.4.2) (1.4.4) u u (, t) = (L, t) = x x (t (, )) (1.4.2) u Dirichlet (Dirichlet boundary condition), (1.4.4) u Neumann (Neumann boundary condition) (W-IBP) u (W-IBP) (1), (2), (3) (1) u C 2 ((, L) (, )) C([, L] [, )). (2) u t, u x [, L] [, ) (3) (1.4.1), (1.4.2), (1.4.3) ((1) (2) u C 2 ((, L) (, )) C 1 ([, L] [, )) (1) k C k (2) ) (W-IBP) Fourier ρ(x) ρ ( ) ( 37

39 [64] ) c := T/ρ 1 2 u c 2 t (x, t) = 2 u (x, t) 2 x2 u(, t) = u(l, t) = u u(x, ) = ϕ(x), (x, ) = ψ(x) t ((x, t) (, L) (, )), (t (, )), (x [, L]) ( 2 Fourier ) u(x, t) = a n = 2 L L ( a n cos nπct L + b n sin nπct ) sin nπx L L, ϕ(x) sin nπx L dx, b n = 2 nπc L ψ(x) sin nπx L dx (n N). ϕ C 4 ψ C 3 ϕ() = ϕ(1) = ϕ () = ϕ (1) = ψ() = ψ(1) = u (W-IBP) ( ) 1 : a n, b n = O (n ) n ( ) ( a n cos nπct L + b n sin nπct ) sin nπx L L ( (x ) 24 ) n = 1 25 (fundamental tone) n > 1 n (harmonic overtone) 26 nπc (angular frequency) L 2π nc (, frequency) 2L c 1 ( u(x, t) 2L t 2L ( c : u x, t + 2L ) = u(x, t)) 2, 3 c 24 ( ) ( ) : 44Hz 44Hz 88Hz, 132Hz, 176Hz, ( 1 ) 1 ( 88 Hz) ( ) 2 38

40 () 1 ( ) Mathematica c=1; L=1; u[n_, x_, t_] := Sin[n Pi x/l] Cos[c n Pi t/l] Animate[Table[ Plot[{u[n, x, ], u[n, x, t]}, {x,, L}, PlotRange -> {-1, 1}], {n, 1, 3}], {t,, 2L/c,.1}] ( ) Dirichlet Neumann u x u (, t) = (L, t) = x Dirichlet (reflection from a hard boundary) Neumann (reflection from a free (soft) boundary) (G.7.3 ) (W-IBP) E k = E k (t), E p = E p (t), E = E(t) : E k (t) := 1 2 E p (t) := 1 2 L L ρ(x)u t (x, t) 2 dx, T u x (x, t) 2 dx, E(t) := E k (t) + E p (t) = 1 2 L [ ρ(x)ut (x, t) 2 + T u x (x, t) 2] dx : 39

41 1.4.1 ( ) u (W-IBP) ϕ C 1 [, L], ψ C[, L] E : E(t) = E() = 1 2 L (ρ(x)ψ(x) 2 + T ϕ (x) 2 )dx (t > ). u C 2 d dt E(t) = d dt = 1 2 = = = T L L 1 2 L L L t (u ) u(, t) = u(l, t) = L [ ρ(x)ut (x, t) 2 + T u x (x, t) 2] dx [ ρ(x)ut (x, t) 2 + T u x (x, t) 2] dx [ρ(x)u tt (x, t)u t (x, t) + T u xt (x, t)u x (x, t)] dx [T u xx (x, t)u t (x, t) + T u xt (x, t)u x (x, t)] dx [u xx (x, t)u t (x, t) + u xt (x, t)u x (x, t)] dx u t (, t) = u t (L, t) = (t (, )) (t (, )) L u xx (x, t)u t (x, t) dx = [u x (x, t)u t (x, t)] x=l x= u x (x, t)u tx (x, t) dx = L d dt E(t) = T E(t) t E(t) = E() = 1 2 L u x (x, t)u xt (x, t) dx. L dt =. ( ρ(x)ut (x, ) 2 + T u x (x, ) 2) dx u t (x, ) = ψ(x) u(x, ) = ϕ(x) u x (x, ) = ϕ (x) ( ) ε, t E ε (t) := 1 2 L ε ε [ ρ(x) ( ) 2 u + T t 4 ( ) ] 2 u dx x

42 ε > [ d u dt E u ε(t) = T t x ] x=l ε x=ε = T lim E ε (t) = E (t). ε ( u t (L ε, t) u x (L ε, t) u t ) (ε, t) u(ε, t). x u u [, L] [, ) x =, L t x ε t [, ) d dt E ε(t). E ε (t) ε = t d dt E (t) = d E(t) =. dt T > ε t [, T ] d dt E ε(t) (: compact ) ( (W-IBP) ) ϕ C 1 [, L], ψ C[, L] (W-IBP) u 1, u 2 v := u 1 u 2 ρ(x) 2 v t = T 2 v 2 x 2 v(, t) = v(l, t) = ((x, t) (, L) (, )), t (, ), v(x, ) =, v t (x, ) = (x [, L]) E(t) = 1 [ L ( ) 2 v ρ(x) (x, t) + T 2 t ( ) ] 2 v (x, t) dx 1 x 2 L [ρ(x) + T ] dx =. ( ) v v (x, t) =, t (x, t) = x v [, L] [, ) u 1 u 2 v(x, t) v(x, ) =. 41 ((x, t) [, L] (, )).

43 1.5 ( 1 ) (i) 1 n (ii) (Cauchy ) ( ) (iii) ( ) (iv) 1 (v) (vi) (vii) () 1, 2, 3 (ii) u = u(x, t) x Ω u(x, ) = ϕ(x), u t (x, ) = ψ(x) (x Ω) u u(x, t) = (x Ω, t R) u (iii) n 1, 2, 3 Fourier n Fourier Fourier Duhamel 1 c 2 u tt(x, t) = u(x, t) + F (x, t), u(x, ) = ϕ(x), u t (x, ) = ψ(x) F, ϕ, ψ F, ϕ, ψ 42

44 (iv) (v) ( ) (vi) (vii) Fourier 2 (iii) 1.6 : 1. ( ) f : I = (a, b) R c I, f (a, c) (c, b) A R s.t. A = lim x c f (x) x c f c f (c) = A f (a, c) (c, b) C 1 f I C 1 2. ( ) (c), (d), (e) ((a), (b) ) (a) (b) (c) (d) (e) d dx d dx d dx d dx 3. x a ϕ(x) ψ(x) x a ϕ(x) ψ(x) d dx b a f(y) dy = f(x). f(x, y) dy = b a f (x, y) dy. x f(y) dy = f (ϕ(x)) ϕ (x) f (ψ(x)) ψ (x). g(x, y) dy = x a g (x, y)dy + g(x, x). x g(x, y) dy = g(x, ϕ(x))ϕ (x) g(x, ψ(x))ψ (x) + ϕ(x) ψ(x) f : R n R x R n 1 lim f(y) dy = f(x) r r n ω n y x r ω n R n : ω n := dz, B 1 := {z R n ; z 1}. B 1 43 g (x, y) dy. x

45 : f : R R f f( x) = f(x) (x R) f f( x) = f(x) (x R) f(x) = x k (k ) k =, =, = 4. f : R R, g : R R (1) a R a a f(x) dx =, (2) f g a a g(x) dx = 2 a g(x) dx (3) k =, 1, 2,... f (2k) () =, g (2k+1) () = ( Taylor f Taylor g Taylor ) 5. f : [, ) R C 2 { f(x) (x ), F (x) := f( x) (x < ) ( f F ) (1) F f() = (2) f() = F C 1 (3) f() = F C 2 f () = ( f() = F F f ) 6. 1 < h < min{c a, b c} h θ (, 1) s.t. f(c + h) f(c) = f (c + θh). h lim x c f (x) = A, h (c + θh) c = θ h h x c f(c + h) f(c) lim h h f c f (c) = A 44 = A.

46 2 (a) () ( ) (b) ( ) (c) (f ϕ ψ ) F = f F ( F (x) := d dx ϕ(x) ψ(x) (d) (g g x ) d dx x a g(x, y) dy = d dx x f(y) dy = d [F (ϕ(x)) F (ψ(x))] dx a f(y) dy) = F (ϕ(x)) ϕ (x) F (ψ(x)) ψ (x) = f (ϕ(x)) ϕ (x) f (ψ(x)) ψ (x). G(u, v) := v G v g G (u, v) = (u, y) dy, u a x G(x, x) = G u (e) (g g x ) d dx x a g(x, y) dy = d dx G(u, v) := a g(u, y) dy (u, v) = g(u, v). v G x (x, x) + (x, x) = v a v G v g G (u, v) = (u, y)dy, u a x G(x, x) = G u a g(u, y)dy (u, v) = g(u, v). v G x (x, x) + (x, x) = v a g (x, y) dy + g(x, x). x g (x, y)dy + g(x, x). x 1.7 (, ) 45

47 1. f : R R, g : R R C 2, c u(x, t) := f(x ct) + g(x + ct) u: R 2 R u 1 1 c 2 u tt(x, t) = u xx (x, t) 2. u(x, t) = f(x ct) (c, f C 2 f ) 1 u tt (x, t) = u xx (x, t) c = 1 u(x, ), u t (x, ) ( : ) 3. (, plane wave) ν ν = 1 R n c U : R R C 2 u(x, t) = U(ν x ct) (x R n, t R) u: R n R R u tt (x, t) = c 2 u(x, t) ν x ν x 4. c C 2 u: R R (x, t) u(x, t) R ξ = x ct, η = x + ct (ξ, η) v(ξ, η) = u(x, t) 1 c 2 u tt(x, t) u xx (x, t) = 4v ξη (ξ, η) 5. f : R 2 (x, y) f(x, y) R C 1 (1), (2) (1) f x (x, y) g : R R s.t. f(x, y) = g(y) ((x, y) R 2 ) (2) f x (x, y) F (x), F : R R g : R R s.t. f(x, y) = F (x)+g(y) ((x, y) R 2 ) 6. Ω R 2 f C 1 (Ω), f x (Ω ) f(x, y) = g(y) ((x, y) Ω) g ( ) 46

48 7. ϕ: R R, ψ : R R 1 1 c 2 u tt(x, t) = u xx (x, t) ((x, t) R 2 ) u(x, ) = ϕ(x), u t (x, ) = ψ(x) (x R) d Alembert u(x, t) = 1 2 (ϕ(x ct) + ϕ(x + ct)) + 1 2c x+ct x ct ψ(y)dy ( ) 8. ϕ C 2 (R; R), ψ C 1 (R; R) u u(x, t) := 1 2 (ϕ(x ct) + ϕ(x + ct)) + 1 2c x+ct x ct ψ(y) dy ((x, t) R R) 1 c 2 u tt(x, t) = u xx (x, t) in R R, u(x, ) = ϕ(x) (x R), u t (x, ) = ψ(x) (x R) 9. 1 Duhamel 1 2 d Alembert (1) 1 c 2 v tt(x, t) = v xx (x, t) (in R R), v(x, ) = (x R), v t (x, ) = ϕ(x) (x R) v w := v t w ϕ (2) w 1 c 2 w tt(x, t) = w xx (x, t) (in R R), w(x, ) = ϕ(x) (x R), w t (x, ) = (x R) 1. c F : R [, ) R 2 U t (x, t) = 2 U 2 c2 (x, t) + F (x, t) x2 (x R, t > ), U(x, ) = (x R), U (x, ) = t (x R) U = U(x, t) (1) d Alembert (1.2.5) Duhamel (1.3.3) (p.26) (1.3.4) U U (x, t), t 2 U U (x, t), t2 U 47 (x, t), x 2 U (x, t) x2

49 (2) ( U ξ = x + ct, η = x ct, V (ξ, η) := ξ + η U 2, ξ η ) 2c (a) V 2 V ξ η (ξ, η) = 1 ( ξ + η 4 F 2, ξ η ) 2c ((ξ, η) Ω), V (ξ, ξ) = (ξ R), V V (ξ, ξ) = (ξ, ξ) ξ η (ξ R) Ω := {(ξ, η) R 2 ; ξ > η}. (b) (ξ, η ) Ω V (ξ, η ) = V (η, η ) + V ξ (ξ, η ) = V (ξ, ξ) + ξ η ξ η η ξ V ξ (ξ, η ) dξ, 2 V (ξ, η) dη η ξ ξ ( ξ ) ( ) V (ξ, η ) = g(ξ, η) dη dξ, g(ξ, η) := 1 ( ξ + η η 4c F 2 2, ξ η ) 2c (c) ( ) U ( F ) (1) ( (1.3.4)) 11. (WE) (NBC) (IC) I = [, ) 1 c u tt(x, t) = u 2 xx (x, t) u x (, t) = (x (, ), t (, )), (t (, )), u(x, ) = ϕ(x), u t (x, ) = ψ(x) (x I) c ϕ C 2 (I; R) ψ C 1 (I; R) ϕ () = ψ () = { { ϕ(x) (x ) ψ(x) (x ) (1) Φ(x) :=, Ψ(x) := Φ, Ψ Φ ϕ( x) (x < ) ψ( x) (x < ) C 2 (R; R), Ψ C 1 (R; R) (2) (WE), (NBC), (IC) ( ) (3) (NBC) Dirichlet (DBC) u(, t) = (t > ) ϕ, ψ 48

50 c u tt(x, y, z, t) = u 2 xx (x, y, z, t) + u yy (x, y, z, t) + u zz (x, y, z, t) u = u(x, y, z, t) w u(x, y, z, t) = w(r, t), r = x 2 + y 2 + z 2 (spherical wave) u(x, y, z, t) = h 1(r ct) + h 2(r + ct) r r (h 1, h 2 1 ) n 13. (Fourier ) [, 1] f() = f(1) = f c R (1.7.1) 1 c u tt(x, t) = u 2 xx (x, t) ((x, t) (, 1) (, )), (1.7.2) u(, t) = u(1, t) = (t (, )), (1.7.3) u(x, ) = ϕ(x), u t (x, ) = ψ(x) (x [, 1]) Fourier () (1.7.2) Neumann u x (, t) = u x (1, t) = (t (, )) 14. R n Ω u tt (x, t) = u(x, t) ((x, t) Ω R) u(x, t) = ((x, t) Ω R) u(x, ) = ϕ(x), u t (x, ) = ψ(x) (x Ω) Ω R C 2 ( Ω Ω ) E(t) := 1 [ ] n u t (x, t) 2 u + (x, t) 2 dx (t R) 2 x j t Ω j=1 15. (a 1, a 2, a 3 ) R 3, r > S := { (x 1, x 2, x 3 ); (x 1 a 1 ) 2 + (x 2 a 2 ) 2 + (x 3 a 3 ) 2 = r 2}, D := { (x 1, x 2 ); (x 1 a 1 ) 2 + (x 2 a 2 ) 2 r 2} D f : D R f(x 1, x 2 ) f dσ = 2r r2 (x 1 a 1 ) 2 (x 2 a 2 ) dx 1dx 2. 2 S D f(x 1, x 2, x 3 ) := f(x 1, x 2 ) ((x 1, x 2, x 3 ) S) (Poisson ) 49

51 Laplacian 2 x y = 2 2 r r r + 1 r 2 2 θ 2, 2 x y z 2 = 2 r r 2 r + 1 r ( 1 sin θ ( sin θ ) + 1 θ θ sin 2 θ ) 2. ϕ 2 (2 3 ) 17. Kirchhoff, Poisson ( ) & 1. u xx 1 c 2 u tt f (x ct) + g (x + ct) 2. u tt = c 2 f (x ct), u xx = f (x ct) u tt = u xx (c 2 1)f (x ct) =. f 1 c 2 = 1. c = 1. u t (x, t) = cf (x ct) u(x, ) = f(x), u t (x, ) = cf (x). 3. u tt = c 2 U (ν x ct), u xi x i = ν i 2 U (ν x ct) (i = 1, 2,, n) ( mind.meiji.ac.jp/~mk/tahensuu1/) 5. 1 f C R s.t. f(x) C ( ) f : R R C 1 f(b) f(a) = b a f (t)dt 6. Ω = R 2 \ {(, y); y } f : Ω R (y < ) f(x, y) := y 2 (y x < ) y 3 (y x > ) 7. ( 1.2.2) 5

52 8. ψ Ψ u(x, t) = 1 2 (ϕ(x ct) + ϕ(x + ct)) + 1 (Ψ(x + ct) Ψ(x ct)) 2c 9. (1) v(x, t) = 1 2c x+ct x ct ϕ(y) dy w(x, t) = 1 (ϕ(x ct) + ϕ(x + ct)). 2 (2) d Alembert ψ = u(x, t) = 1 (ϕ(x ct) + ϕ(x + ct)) 2 (1) w 1. (1) (2) [33] 11. ( ) f : I = (a, b) R c I, f (a, c) (c, b) A R s.t. A = lim x c f (x) x c f c f (c) = A. f (a, c), (c, b) C 1 f I C 1 (1) Φ x C 2 ϕ: [, ) R Φ(+) = lim x x> Φ(x) = lim x ϕ(x) = ϕ(), x> Φ( ) = lim x x< Φ(x) = lim x ϕ( x) = ϕ(), x< lim Φ(x) = ϕ() = Φ() Φ: R R x { { Φ (x) = ϕ (x) (x > ) ϕ ( x) (x < ), Φ (x) = ϕ (x) (x > ) ϕ ( x) (x < ). ϕ: [, ) R C 2 ϕ () = Φ (+) = lim x x> Φ ( ) = lim x x< Φ (x) = lim x ϕ (x) = ϕ () =, x> Φ (x) = lim x ( ϕ ( x)) = ϕ () = x< Φ (Φ () = ) Φ Φ (+) = lim x x> Φ ( ) = lim x x< Φ (x) = lim x ϕ (x) = ϕ (), x> Φ (x) = lim x (ϕ ( x)) = ϕ () x< 51

53 1.7.1 Φ (Φ () = ϕ ()) Φ Φ R C 2 Ψ R C 1 ( ) (2) Φ C 2 (R; R), Ψ C 1 (R; R) 1 c 2 U tt(x, t) = U xx (x, t) (x R, t > ), U(x, ) = Φ(x) (x R), U t (x, ) = Ψ(x) (x R) C 2 U(x, t) = 1 2 (Φ(x ct) + Φ(x + ct)) + 1 2c x+ct x ct Ψ(y) dy U x (x, t) = 1 2 (Φ (x ct) + Φ (x + ct)) + 1 [Ψ(x + ct) Ψ(x ct)] 2c U x (, t) = 1 2 (Φ ( ct) + Φ (ct)) + 1 [Ψ(ct) Ψ( ct)]. 2c Φ, Ψ Φ Φ ( ct) = Φ (ct), Ψ( ct) = Ψ(ct) U x (, t) = 1 2 ( Φ (ct) + Φ (ct)) + 1 (Ψ(ct) Ψ(ct)) =. 2c u(x, t) := U(x, t) u (WE), (NBC), (IC) (x [, ), t [, )) (Φ Ψ ϕ ψ ) (i) x ct Ψ ψ u(x, t) = 1 2 (Φ(x ct) + Φ(x + ct)) + 1 2c = 1 2 (ϕ(x ct) + ϕ(x + ct)) + 1 2c x+ct x ct x+ct x ct Ψ(y) dy ψ(y) dy. (ii) x ct < y < Ψ(y) ψ( y) x+ct u(x, t) = 1 2 (Φ(x ct) + Φ(x + ct)) + 1 Ψ(y) dy 2c x ct = 1 2 (ϕ ( (x ct)) + ϕ(x + ct)) + 1 ( (ψ( y)) dy + 2c x ct z = y x ct ψ( y) dy = ct x u(x, t) = 1 2 (ϕ(ct x) + ϕ(x + ct)) + 1 2c 52 ψ(z) dz = ( x+ct ct x ψ(y) dy + ψ(z) dz x+ct ct x ) ψ(y) dy. ) ψ(y) dy.

54 1 2 (ϕ(x + ct) + ϕ(x ct)) + 1 2c u(x, t) = 1 2 (ϕ(x + ct) + ϕ(ct x)) + 1 2c x+ct x ct ( x+ct ψ(y) dy (x ct ) ψ(y) dy + ct x ) ψ(y) dy (3) ϕ() = ϕ () = ψ() = 1 2 (ϕ(x ct) + ϕ(x + ct)) + 1 x+ct ψ(y) dy (x ct ) 2c x ct u(x, t) = 1 2 (ϕ(x + ct) ϕ(ct x)) + 1 x+ct ψ(y) dy. (x ct < ). 2c ct x (x ct < ). 12. (Laplace Laplacian ) v = rw u = 2 w r + 2 w 2 r r + 1 r 2 S = 1 2 r r (rw) r 2 S 1 c 2 v tt = v rr C 2 h 1, h 2 29 v(r, t) = h 1 (r ct) + h 2 (r + ct). w(r, t) = v(r, t) r = h 1(r ct) r + h 2(r + ct). r 13. u(x, t) = X(x)T (t) λ X (x) = λx(x) (x (, 1)), X() = X(1) =, X(x), T (t) = cλt (t) (t > ) ( 2 ) λ = λ n := n 2 π 2, X(x) = X n (x) := C n sin nπx, T (t) = A n cos nπct + B n sin nπct (C n, A n, B n ; n N). u(x, t) = (a n cos nπct + b n sin nπct) sin nπx 29 r r

55 ϕ(x) = a n sin nπx, ψ(x) = nπcb n sin nπx. a n = 2 1 ϕ(x) sin nπx dx, b n = 2 nπc 1 ψ(x) sin nπx dx. 14. E(t) = 1 2 Ω ( u 2 t + u u ) dx ( u = grad u) ( ) 3 E (t) = 1 ( u 2 2 Ω t t + u u ) ( dx = u t u tt + u ) Ω t u dx = (u t u + u u t ) dx. Ω Green u = (on Ω) u t = (on Ω) u u u t dx = n u t dσ u u t dx = u t u dx E Ω E (t) = Ω Ω (u t u u t u) dx = Ω Ω Ω dx =. c, I = [, ) (1) x f(x) = g(x) = f, g C 2 (R; R) ( ) u(x, t) := f(x ct) + g(x + ct) g( (x ct)) (x R, t ) u (i) 1 c 2 u tt(x, t) = u xx (x, t) (x R, t > ) (ii) u(, t) = (t ) (iii) x u(x, ) u t (x, ) 3 1 (f 2 ) = 2f f t (u t) 2 = 2u t u tt, t ( u u) = 2 u t u. 54

56 (2) ϕ C 2 (I; R) ψ C 1 (I; R) ϕ() = ϕ () = ϕ () = ψ() = ψ () = (1) f g (WE) (DBC) (IC) 1 c u tt(x, t) = u 2 xx (x, t) ((x, t) (, ) (, )) u(, t) = (t (, )) u(x, ) = ϕ(x), u t (x, ) = ψ(x) (t (, )) (3) x = 55

57 2 [57] () x I = [a, b] x t u(x, t) x (a, b) x V = (α, β) a < α < x < β < b t V J(t) = β α c(x) u(x, t) dx c = c(x) t t + t t J J ( β ) ( β J = J(t + t) J(t) J (t) t = α (c(x) u(x, t)) dx t t = α ) c(x) u t (x, t) dx t V x = α, x = β V x = α V k(α) u x (α, t) t Fourier (Fourier s law of heat conduction) ( ) 1 k = k(x) (thermal conductivity) x = β V k(β) u x (β, t) t 1 56

58 J t t + t x = α, x = β V ( β ) c(x) u t (x, t) dx t J = k(β)u x (β, t) t k(α)u x (α, t) t. β α α c(x) u t (x, t) dx = k(β)u x (β, t) k(α)u x (α, t) = β β α α x, β x c u t (x, t) = x (k(x) u x(x, t)). x=x (2.1.1) c u t (x, t) = x (k(x) u x(x, t)). c, k κ := k/c u t (x, t) = κ u xx (x, t). α x (k(x) u x(x, t))dx. : u = u(x, t) u t (x, t) = κ u xx (x, t). κ = 1 ( p ) Fourier 2 C 1 g 8 C 5 g x C = x (1 + 5). x = C. (1) ( ) (2) 57

59 2.1.1 ( ) ( ) SI W/K m.24 W/K m ( C),.1 W/K m ( C),.6 W/K m ( C), 16 W/K m, 79.5? W/K m (2 C), 237 W/K m (2 C), 39 W/K m k u k = k(u) u k k(u) ( ) 3 R n (2.2.1) c u t = div(k grad u) (c k ) (2.1.1) c k κ := k/c (2.2.2) u t = κ u 2 κ = 1 (2.2.1) Ω x x V V Ω V V ( ) c = c(x) V u J(t) := c(x) u(x, t) dx V J t t + t J ( ) J = J(t + t) J(t) J (t) t = c u t dx t. V V 2 : div grad = 58

60 Fourier : t, x ( ) k = k(x) () = k(x) grad u(x, t) ( k(x) 3 ) t t + t V S V ( ) ( k grad u) n dσ t S 4 n S ( : S x S S V ( grad u(x, t)) n S t S S ) V c(x)u t (x, t) dx = k(x) grad u(x, t) n dσ. V Gauss (p.154) c(x)u t (x, t) dx = div(k(x) grad u(x, t)) dx. V x V ( Ω) V S cu t (x, t) = div(k grad u(x, t)) x=x. u t (x, t) = 1 div(k(x) grad u(x, t)) (x Ω, t > ). c(x) f : R n R V f(x) dx = V f() = (: ) f : R n R lim ε + 1 B(x ; ε) B(x ;ε) f(x) dx = f(x ) B(x ; ε) := {x R n ; x x < ε} B(x ; ε) B(x ; ε) (n = 2 n = 3 ) 3 4 v(x) v n dσ S 59 S

61 2.2.1 ( ) ( ) u t = u u tt = u ( t ) ( ) A (2.2.3) dx dt = Ax () (2.2.4) d 2 x = Ax ( ) dt2 (2.2.3) x(t) = Ce At (C ) (2.2.4) x(t) = C sin ωt + D cos ωt, ω := A (C, D ) ( t ) Fourier t ( ) (H-IBP) (HE), (DBC), (IC) u = u(x, t) : (HE) (DBC) (IC) u t (x, t) = u xx (x, t) (x (, 1), t > ), u(, t) = u(1, t) = (t > ), u(x, ) = f(x) (x [, 1]). f : [, 1] R 6

62 (DBC) 5 (boundary condition) (DBC) Dirichlet (Dirichlet boundary condition) Dirichlet (homogeneous Dirichlet boundary condition) 6 (IC) (initial condition) f (initial value) (initial condition) (H-IBP) (initial boundary value problem) (mixed problem) Dirichlet ( Neumann Neumann ) (Jacques Salomon Hadamard, , Versailles Paris ) (well-posedness) (well-posed) ( ) 7 8 ( ) () ( ) 5 (, 1) {, 1} 6 Dirichlet = = homogeneous 7 () 8 : ( 2 ) ( u f ) ( ) ( ) [24] 61

63 2.3.3 (H-IBP) (H-IBP) u (H-IBP) 3 (1) u [, 1] [, ) (2) u t, u x, u xx (, 1) (, ) (3) (HE), (DBC), (IC) ( ) ( (weak solution) (generalized solution) ) 9 () (H-IBP) Fourier ( ) ( ) 9 62

64 : 1 A Ax = b x = A 1 b. Ax = b A A 1 x = A 1 b. x = A 1 b Ax = A(A 1 b) = (A A 1 )b = Ib = b. ( ) A 1 b ( ) () 2.4 T Q = Q T := [, 1] [, T ], Γ = Γ T := {(, t); t T } {(1, t); t T } {(x, ); x 1}, Q = Q T := Q T \ Γ T = (, 1) (, T ] Γ Q (parabolic boundary) 1 u u t = u xx Q T u Q T (1) u Q T = [, 1] [, T ] (2) u t, u x, u xx Q T = (, 1) (, T ] (3) Q T u t = u xx 1 (parabolic equation) 63

65 (H-IBP) t [, ) Q T ( (maximum principle)) v = v(x, t) Q T (2.4.1) max v(x, t) = max v(x, t), (x,t) Q T (x,t) Γ T (2.4.2) min v(x, t) = min v(x, t). (x,t) Q T (x,t) Γ T Q T R 2 v Q T ( ) λ := max v(x, t) (x,t) Γ T (2.4.3) v(x, t) λ ((x, t) Q T ) w(x, t) := e t (v(x, t) λ) w (Γ T ) (2.4.4) w t + w = w xx (Q T ) ( v = e t w +λ v t = e t v +e t v t, v xx = e t w xx v t = v xx e t w t + e t w = e t w xx. e t > e t (2.4.4) ) : Q T w. ( (2.4.3) ) µ := max (x,t) Q T w(x, t) µ (x, t ) Q T : µ = w(x, t ) >. 11 : Q T Q T (, 1) (, T ) ( ) [, 1] [, ) (, 1) (, ) 64

66 w (Γ T ) (x, t ) Γ (x, t ) Q T (2.4.4) (x, t) = (x, t ) w t (x, t ) + µ = w xx (x, t ) 3 (1) 1. ( < t T w(x, ): t w(x, t) t = t < t < T w t (x, t ) = 12 t = T w t (x, t ) t = t = T t [, T ) w(x, t) w(x, t ) h ( T, ) w(x, t + h) w(x, t ). (h < ) w(x, t + h) w(x, t ). w t (x, t ) 13 ) h (2) 2 >. ( ) (3). ( < x < 1 w(, t ): x w(x, t ) x w x (x, t ) = w xx (x, t ) w xx (x, t ) > w(, t ) x x ) >, µ Q T w [ ] (2.4.3) 14 max v(x, t) λ = max v(x, t). (x,t) Q T (x,t) Γ T max v(x, t) = max v(x, t). (x,t) Q T (x,t) Γ T (, ) u, w Q T Γ u w Q T u w. 12 f a f (a) =. 13 () 14 Q T Γ T max (x,t) Q T v(x, t) max (x,t) Γ T v(x, t). 65

67 v := u w v Q T v t = t v = t (u w) = u t v t = u xx w xx = 2 x 2 (u w) = v xx. min v(x, t) = min v(x, t) (x,t) Q T (x,t) Γ T Γ T v Q T u w. min v(x, t). (x,t) Q T () v Q T Γ T v v Q T v ( ) Q T v. min v(x, t) = min v(x, t) (x,t) Q T (x,t) Γ T ( (H-IBP) ) (H-IBP) u 1, u 2 (H-IBP) T > Q T 15 v := u 1 u 2 v Q T v t = v xx Γ T v = max v(x, t) = max v(x, t) =, (x,t) Q T (x,t) Γ T min v(x, t) = min v(x, t) = (x,t) Q T (x,t) Γ T Q T v =. Q T u 1 = u 2. T u 1 = u 2 on [, 1] [, ). 1: Q Γ Q ( Q ) v Q Q ( ) 15 u 1 u 2, u 1 u 2 u 1 = u 2 ( ) 66

68 2: 2 16 ( ) ( ) ( ) D C f : D C f D max f(z) = max f(z), z D z D min f(z) = min f(z). z D z D D f. ( n 1 ) u t = u + n j=1 b j (x, t) u x j ( ) c(x, t)u u t = u + n j=1 b j (x, t) u x j + c(x, t)u (c ) 17 u t = u + n j=1 b j (x, t) u x j + F (x, t, u(x, t)) ( ) 2 u ( ) Protter-Weinberger [78] 2.5 u f u f T (f) T [, 1] X = C[, 1] (2.5.1) f := max x [,1] f(x) (f X). 16 ( ) 17 u w = e λt u 67

69 2.5.1 (2.5.1) X = C[, 1] X Banach 4 (1) () (a) f X f. (b) f X f = f =. (2) f X, λ R λf = λ f. (3) ( ) f X, g X f + g f + g. (4) (: Cauchy ) lim f n f m = f n,m lim f n f =. n X s.t. ( ) () (H-IBP) u f (2.5.2) max u(x, t) max f(x) x 1 x 1 ( t [, )). u(, t) f ( t [, )). f = M M f(x) M (x [, 1]). u T > T (2.5.2) M u(x, t) M ((x, t) Γ T ). M u(x, t) M ((x, t) Q T ). M u(x, t) M ((x, t) [, 1] [, )) (H-IBP) f = f j u = u j (j = 1, 2) : u 1 (, t) u 2 (, t) f 1 f 2 ( t [, )). 68

70 2.6 Fourier (1) (H-IBP) (HE), (DBC), (IC) u = u(x, t) : (HE) u t (x, t) = u xx (x, t) (x (, 1), t > ), (DBC) u(, t) = u(1, t) = (t > ), (IC) u(x, ) = f(x) (x [, 1]). Fourier (Fourier ) Fourier Fourier (, 1) Fourier Fourier Fourier Fourier Fourier Fourier (Jean-Baptiste-Joseph Fourier) ( ) (Jean François Champollion, ) Fourier Fourier 6 d Alembert (175 ) Fourier (Fourier [25] 91 ) 18 NHK 5 1, (1974) [65] ( ) Fourier 69

71 Fourier 19 (Fourier ) Fourier 1 sin, cos 2 () Fourier Fourier : (IC) (HE), (DBC) u(x, t) = ζ(x)η(t) (HE), (DBC) 23 u(x, t) = ζ(x)η(t) (DBC) ζ()η(t) = ζ(1)η(t) = ( t > ). (2.6.1) ζ() = ζ(1) = η(t) = ( t > ) u(x, t) = ζ(x)η(t) u(x, t) = ζ(x)η(t) (HE) ζ(x)η (t) = ζ (x)η(t). η (t) η(t) = ζ (x) ζ(x) sin cos 21 Fourier Fourier 22 Carleson-Hunt (1966, 1968) p > 1 p f L p f Fourier f Fourier (?) () 23 7

72 x t x t λ : (2.6.2) η (t) η(t) = ζ (x) ζ(x) = λ. (2.6.3) (2.6.4) ζ (x) = λζ(x), η (t) = λη(t) (2.6.3) s (2.6.3) s = ± λ. s 2 = λ (i) λ = s = ( ) (2.6.3) ζ(x) = Ax + B (A, B ). A, B (2.6.1) A = B =. ζ(x) (ii) λ (2.6.3) ζ(x) = Ae λx + Be λx (A, B ). A, B (2.6.1) ζ() = ζ(1) = A + B = and Ae λ + Be λ = B = A and A(e λ e λ ) =. A = B = ζ(x) A e λ e λ =, (2.6.5) e 2 λ = 1. ( ) n Z s.t. 2 λ = 2nπi, i = 1. λ = nπi, λ = n 2 π 2. ζ ζ(x) = A(e nπix e nπix ) = 2iA sin nπx. A n n ζ, λ λ ( ) n N 71

73 η η(t) = Ce λt (C ). (HE), (DBC) u(x, t) = ζ(x)η(t) = c n u n (x, t), u n (x, t) := e n2 π 2t sin nπx (c n, n N) (c n = 2iAC A, C c n ) (HE), (DBC) 24 (2.6.5) C e z = 1 z = 2nπi (n Z). i z C, r >, θ R exp z = re iθ n Z s.t. z = log r + i (θ + 2nπi). ( log r e x = r unique x R log r ) z = x + iy (x, y R) e z = e x e iy = e x (cos y + i sin y) e z = e x e z = 1 e x = 1 and cos y + i sin y = 1 x = and n Z s.t. y = 2nπ n Z s.t. z = 2nπi ((2.6.2) = ) (2.6.6) η (t)ζ(x) = η(t)ζ (x) ((x, t) (, 1) (, )) η (t) η(t) = ζ (x) ζ(x) 24 ( ) 72

74 ( ) ( ) A, B ζ 1, ζ 2 : A C, η 1, η 2 : B C ζ 1 (x)η 1 (t), (2.6.7) ζ 2 (x)η 1 (t) = ζ 1 (x)η 2 (t) ((x, t) A B) λ C s.t. ζ 2 (x) = λζ 1 (x) (x A) η 2 (t) = λη 1 (t) (t B). x A t B s.t. ζ 1 (x )η 1 (t ). (2.6.7) x = x, t = t ζ 1 (x ), η 1 (t ) ζ 2 (x ) ζ 1 (x ) = η 2(t ) η 1 (t ). λ (2.6.7) x = x η 2 (t) = ζ 2(x ) ζ 1 (x ) η 1(t) = λη 1 (t) (t B). (2.6.7) t = t ζ 2 (x) = η 2(t ) η 1 (t ) ζ 1(x) = λζ 1 (x) (x A). [64] 2 21 ( ) 25 (ζ(x) = C sin nπx, n N) ( ) 26 73

75 : (HE), (DBC) (HE), (DBC) c 1, c 2,, c n c 1 e π2t sin πx + c 2 e 4π2t sin 2πx + c 3 e 9π2t sin 3πx + + c n e n2 π 2t sin nπx (HE), (DBC) ( ) {c n } n N (2.6.8) u(x, t) := c n e n2 π 2t sin nπx u (HE), (DBC) (HE), (DBC) ( ) u {c n } n N {c n } n N 27 (HE) : (IC) {c n } (2.6.8) t = u(x, ) = c n sin nπx f(x) (2.6.9) f(x) = c n sin nπx (x [, 1]) {c n } n N u (HE), (DBC), (IC) (H-IBP) () Fourier f Fourier f() = f(1) = f ( C 1 ) {c n } n N c n = 2 1 f(x) sin nπx dx (2.6.9) ( B.2.7 ) 27 n ( ) f n (x) f n (x) = f n(x) n n 74

76 Fourier (2.6.9) sin mπx [, 1] 1 f(x) sin mπx dx = 1 f(x) sin mπx dx = 1 c n sin nπx sin mπxdx (x [, 1]). 1 c n sin nπx sin mπx dx = c m = 2 1 f(x) sin mπx dx c n δ nm 2 = c m 2. c m c m (2.6.9) ( Fourier ) (f ) (2.6.1) c n := 2 {c n } u(x, t) := 1 f(x) sin nπx dx c n e n2 π 2t sin nπx u (H-IBP) () ( ) (1) f(x) = sin πx u(x, t) (2) f(x) = sin 3 πx u(x, t) (1) f(x) = sin πx f Fourier u(x, t) = e π2t sin πx. (2) f(x) = sin 3 πx = (3 sin πx sin 3πx)/4 f Fourier 28 u(x, t) = (3e π2t sin πx e 9π2t sin 3πx)/ Fourier (2) sin 3θ = 3 sin θ 4 sin 3 θ (Google 183 )sin 3θ = Im(cos 3θ+i sin 3θ) = Im e 3iθ = Im [ (cos θ + i sin θ) 3] = 3 cos 2 θ sin θ sin 3 θ = 3(1 sin 2 θ) sin θ sin 3 θ 75

77 2.7.1 f (H-IBP) S f (x, t) (2.7.1) S f (x, t) := c n e n2 π 2t sin nπx, c n := 2 1 f(x) sin nπx dx (n N) ( C 1 (H-IBP) ) f [, 1] C 1 f() = f(1) = u := S f (H-IBP) ( f() = f(1) = ) f() = f(1) = t > u(, t) = u(1, t) = u [, 1] [, ) t u(, ) = u(1, ) = u(x, ) = f(x) (x [, 1]) f f() = f(1) = (H-IBP) f f() = f(1) = (compatibility condition) ( : ) (1) (x, t) [, 1] [, ) S f S f [, 1] [, ) (2) S f t S f, x S f, 2 x 2 S f (, 1) (, ) (3) (a) S f (HE) (b) S f Dirichlet (DBC) (c) S f (IC) 76

78 (HE), Dirichlet (DBC) S f S f (HE), (DBC) (3b) S f (DBC) (3a) S f (HE) (1) (2) S f (3c) S f (IC) f C 1 (f S f (x, ) ) f 1 (2) (3a) (HE) 2 f C 1 (1) (3c) (IC) : Fourier S f S f S f x 2 t 1 u = S f f C[, 1] S f (x, t) [, 1] (, ) S f [, 1] (, ) t S f = 2 x S f (t = ) [, 1] [, ) t = [, 1] (, ) t = [, 1] [, ) f S f x l t m ( c n ( n 2 π 2 ) m e n2 π 2t (nπ) l sin nπx + lπ ) 2 77

79 29 n c n (nπ) l+2m e n2 π 2 t 1 c n = 2 1 f(x) sin nπx dx 2 f(x) sin nπx dx 2 1 f(x) dx M. δ > (x, t) [, 1] [δ, ) e n2 π 2t e n2 π 2δ n Mπ l+2m e n2 π 2δ n l+2m. x, t Mπ l+2m e n2 π 2δ n l+2m < 3 Weierstrass M 31 [, 1] [δ, ) S f C δ > S f [, 1] (, ) C S f : Fourier t = S f S f (x, ) = f(x) (x [, 1]) f(x) = c n sin nπx, c n = 2 1 f(x) sin nπx dx (n N) f Fourier f Fourier f f C 1 f() = f(1) = f [, 1] C 1 f() = f(1) = f(x) = b n sin nπx ( ), b n = 2 ( B.2.7 ) 1 f(x) sin nπx dx (n = 1, 2, ). 29 sin x k sin(x + kπ/2) 3 r = e π2δ, α = 2m + l r (, 1), α >. n α r n2 n α r n a n := n α r n lim a n+1 / a n = r < 1 a n < (d Alembert ) n n n α r n2 n α r n <. n l+2m e n2 π 2δ <. n n n 31 B

80 S f [, 1] [, ) () c n < c n ( [ ] 1 f(x) cos nπx c n = ) 1 f (x) cos nπx dx nπ nπ = 2 nπ A n := f (x) cos nπx dx = 1 nπ A n, f (x) cos nπx dx A n f Fourier Fourier Bessel ( B.2.7 ) 1 A n 2 2 f (x) 2 dx 32 1 c n = n n c n ( n 2 ( 2 π S f (x, t) ) 1/2 ( ) 1/2 ( 1 n 2 c n 2 = 1 n 2 1 n 2 ) 1/2 ( 1 1/2 f (x) dx) 2 <. n c n ((x, t) [, 1] [, )) ) 1/2 ( ( ) ) 2 1/2 An π Weierstrass M S f (x, t) [, 1] [, ) S f (x, t) [, 1] [, ) : f C 1 Green ( 95% ) f C 1 f f Fourier 32 2 cos nπx (n N) L 2 (, 1) 79

81 Yes ( (H-IBP) ) f [, 1] f() = f(1) = { u(x, t) := S f (x, t) ((x, t) [, 1] (, )) f(x) (x [, 1], t = ) u (H-IBP) S f (x, t) (2.7.2) S f (x, t) f(x) (t x [, 1] ) u [, 1] [, ) (2.7.2) (H-IBP) Green G(x, y, t) ( ) G(x, y, t) := e λnt φ n (x)φ n (y) (x, y [, 1], t > ), λ n := n 2 π 2, φ n (x) := 2 sin nπx (n N) G S f : S f (x, t) = 1 ( ) 1 G(x, y, t)f(y) dy = G(x, y, t)f(y) dy (x [, 1], t > ). = = = 1 e λnt φ n (x)φ n (y)f(y) dy ( e λnt φ n (x) 1 ( e λnt sin nπx 2 c n e λnt sin nπx = S f (x, t). ) f(y)φ n (y) dy 1 ) f(y) sin nπy dy 8

82 G = G(x, y, t) [, 1] [, 1] (, ) C y [, 1] x, t (HE) (DBC) 2 G(x, y, t) = G(x, y, t) ((x, t) [, 1] (, )), t x2 G(, y, t) = G(1, y, t) = (t (, )). (2.7.3) f C 1 ([, 1]), f() = f(1) = (H-IBP) Fourier C([, 1] [, )) u(x, ) = f(x) (x [, 1]) G(x, y, t) t = () 1 G(x, y, t)f(y) dy t = f(x) [, 1] [, ) (a) ( ) u 1 G(x, y, t)f(y) dy ((x, t) [, 1] (, )) u(x, t) := f(x) (x [, 1], t = ) u C([, 1] [, )). (b) 1 G(x, y, t)f(y) dy t f(x) [, 1] (2.7.4) lim sup t x [,1] 1 G(x, y, t)f(y) dy f(x) =. (a) (b) (4) ( (2.7.2)) f C 1 81

83 2.7.7 (Green ) G (1) G(x, y, t) = G(y, x, t). (2) G(x, y, t). (3) 1 G(x, y, t) dy 1. (4) f C ([, 1]), f() = f(1) = 1 G(x, y, t)f(y) dy f(x) (t x [, 1] ). (2.7.4) (1) G(x, y, t) x y (2) y [, 1] G(x, y, t) (x [, 1], t > ) y =, 1 G(x, y, t) = y (, 1) ( G(x, y, t) δ( y ) ) (molifier) ρ C (R), ρ, ρ(x) = ( x 1), ρ ρ ε (x) := 1 ε ρ ( x ε ) (x R, ε > ) ρ(x) dx = 1 ρ ε (y) dy = ε ε ρ ε (y) dy = 1, ρ ε (y y ) dy = y +ε y ε ρ ε (y y ) dy = 1 f : R R y lim ε ρ ε (y y )f(y) dy f(y ) = = ρ ε (y y )f(y) dy = f(y ) y +ε y ε y +ε y ε ρ ε (y y )f(y) dy f(y ) ρ ε (y y )(f(y) f(y ))dy 82 y +ε y ε ρ ε (y y )dy

84 ρ ε (y y )f(y) dy f(y ) sup f(y) f(y ) ρ ε (y y ) dy y y ε y y ε sup f(y) f(y ) (ε ). y y ε < ε < y ε f ε (y) := ρ ε (y y ) f ε [, 1] C 1 f ε () = f ε (1) = 1 1 G(x, y, t)f u ε (x, t) := ε (y) dy = G(x, y, t)ρ ε (y y ) dy (x [, 1], t > ) f ε (x) (x [, 1], t = ) u ε f ε (H-IBP) f ε ( 2.4.3) x [, 1], t > u ε (x, t). ε u ε (x, t) G(x, y, t) G(x, y, t). ( : f ε δ( y ) Green ) (3) () δ( y ) G(, y, ) ( ) ( ) f ε = ρ ε ( y ) u ε ( ) f n C 1 ([, 1]), f n () = f n (1) =, f n 1, x (, 1) lim n f n (x) = 1 {f n } 1 G(x, y, t)f u n (x, t) := n (y) dy (t > ) f n (x) (t = ) u n f n (H-IBP) ( [, 1] [, ) ) ũ(x, t) := 1 ((x, t) [, 1] [, )) 83

85 [, 1] [, ) t = x =, 1 u n ũ ( 2.4.2) n (x, t) [, 1] [, ) u n (x, t) ũ(x, t) = 1. (4) 33 u n (x, t) = 1 1 G(x, y, t)f n (y) dy 1 G(x, y, t) dy 1. f n C 1 ([, 1]), f n () = f n (1) =, lim {f n } 34 sup n x [,1] G(x, y, t) dy f(x) f n (x) = u(x, t) := u n (x, t) := u(x, t) u n (x, t) = G(x, y, t)f(y) dy, G(x, y, t)f n (y) dy G(x, y, t) (f(y) f n (y)) dy max y [,1] f(y) f n(y) = max y [,1] f(y) f n(y). 1 G(x, y, t)dy n N f n C 1 ([, 1]), f n () = f n (1) = x, t, n lim sup t x [,1] u n (x, t) f n (x, t) =. u(x, t) f(x) = u(x, t) u n (x, t) + u n (x, t) f n (x) + f n (x) f(x) u(x, t) u n (x, t) + u n (x, t) f n (x) + f n (x) f(x) 2 max y [,1] f(y) f n(y) + max y [,1] u n(y, t) f n (y). 33 Lebesgue f n [1/n, 1 1/n] f n = 1 1 G(x, y, t)f n (y) dy 1/n G(x, y, t) dy /n 1 G(x, y, t) dy G(x, y, t) dy. y G(x, y, t) n 34 {f n } ( ) 84

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