1. PDE u(x, y, ) PDE F (x, y,, u, u x, u y,, u xx, u xy, ) = 0 (1) F x, y,,uu (solution) u (1) u(x, y, )(1)x, y, Ω (1) x, y, u (1) u Ω x, y, Ωx, y, (P

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1 Laplace Li-Yau s Harnack inequality Cauchy Cauchy-Kowalevski H. Lewy Open problems F. John, Partial Differential Equations, Springer-Verlag,

2 1. PDE u(x, y, ) PDE F (x, y,, u, u x, u y,, u xx, u xy, ) = 0 (1) F x, y,,uu (solution) u (1) u(x, y, )(1)x, y, Ω (1) x, y, u (1) u Ω x, y, Ωx, y, (PDEs): n m PDE n > m(under-determined) n < m(over-determined) PDE PDEs PDE PDEs x, y, PDE u x, y, m PDE u m m x, y, u m 2

3 m PDE u m P DE (linear) (nonlinear) : : (quasilinear) (fully nonlinear) 2. PDEs t(x 1, x 2,, x n )n = 3 x, y, zn x 2 1 x 2 n = n i=1 2 x 2 i Laplace, 2 t x 2 1 x 2 n 1. Laplace = 2 t 2 n i=1 2 x 2 i = 2 t 2 u 2 u u x 2 1 x 2 n u (harmonic function) = n i=1 2 u x 2 i = 0. (2) n = 2 x 1 = x, x 2 = y v(x, y)u vcauchy-riemann u x = v y, u y = v x (3) (3)(u, v) z = x + iy f(z) = f(x + iy) = u(x, y) + iv(x, y). (4) 3

4 (u(x, y), v(x, y)) n = 3(2) 2. wave equation u tt = c 2 u (c > 0), (5) u = u(t, x 1,, x n ) n = 1 : c n = 2 : n = 3 : 3. Maxwell Maxwell equations E = E(E 1, E 2, E 3 ) H = (H 1, H 2, H 3 ) Maxwell.. εe t = curlh, µh t = curle, dive = divh = 0, (6) ε, µ εe t = curlh, µh t = curle t = 0 dive = divh = 0 te i, H k c 2 = 1/εµ (5) (6) curl(curle) = µ(curlh) t = εµe tt, 4

5 (6) curl(curle) = (dive) E, E tt = (εµ) 1 E. H ρ 2 u i t 2 = µ u i + (λ + µ) x i (divu) (i = 1, 2, 3) (7) u i (t, x 1, x 2, x 3 ) uρ λ, µlame u i ( 2 t λ + 2µ ) ( 2 2 ρ t µ ) 2 ρ u i = 0. (8).. u t = u = 0. (9) k > 0 u t = k u, (10) 6. V (x, y, z)m Schrödinger h = 2π Planck i ψ t = 2 (ψ) + V ψ, (11) 2m 7. Tricomi u xx = xu yy. (12) u xx = yu yy. 5

6 8. 3 Euclid.... z = u(x, y) (1 + u 2 y)u xx 2u x u y u xy + (1 + u 2 x)u yy = 0. (13) n Minkowski.. x = x(t, θ) R n x θ 2 x tt 2 x t, x θ x tθ + ( x t 2 1)x θθ = 0. (14) 10. ρ... φ(x, y) (φ x, φ y ) (1 c 2 φ 2 x)φ xx 2c 2 φ x φ y φ xy + (1 c 2 φ 2 y)φ yy = 0, (15) c q = φ 2 x + φ 2 y γ p = Aρ γ (16) c 2 = 1 γ 1 q 2. (17) Navier-Stokes.. u k p u i t + 3 k=1 3 u i u k + 1 p = µ u i (i = 1, 2, 3), x k ρ x i k=1 u k x k = 0 ( divu = 0), ρ µ (18) 6

7 12. ρ t + 3 j=1 t (ρv i) + t (ρe) + x j (ρv j ) = 0, 3 j=1 3 j=1 x j (ρv i v j + δ ij p) = 0, x j (ρv j E + pv j ) = 0, (19) ρ(t, x) v = (v 1 (t, x), v 2 (t, x), v 3 (t, x))pe = E(t, x) ρ p T E p = p(ρ, E) ( p = p(ρ, T )) (20) (20) (20)(19) 13. u(t, x)korteweg-de Vries.. u t + cuu x + u xxx = 0, (21) 14. Monge-Apére S ττ = S2 τθ 1 S θθ + S. (22)... 7

8 1. u = u(t, x) u t + cu x = 0 (1.1) c > 0 (t, x)- (1.1)u dx dt = c. (1.2) x ct = const. ξ, (1.3) du dt = d dt u(t, ct + ξ) = u t + cu x = 0. (1.4) u ξ (1.1) u(t, x) = u(0, ξ) f(ξ) = f(x ct), (1.5) f(ξ) u... u u(0, x) = f(x) (1.6) fc 1 (R) (1.5) (1.1) fu(t, x) f 1

9 ξ = x ct ξ(x, t)x- u(t, x).... ξ ξ.. (1.3)u(t, x) 1.1 t (t, x) x ct = ξ 0 ξ x 1.1: t(x, u)-u t = T t = 0x-cT u(x, 0) = u(x + ct, T ) = f(x). (1.7) c. 1.2 u u(0, x) u(t, x) c x x + ct x 1.2: 2

10 x h t k (t, x)- x h t k(t, x) v(t + k, x) v(t, x) k v(t, x + h) v(t, x) + c h = 0 (1.8) (1.1)h, k 0 v t + cv x = 0. h, k (1.8) v (1.1) (1.6) (1.8) v(0, x) = f(x) (1.9) λ = k/h. v(t + k, x) = (1 + λc)v(t, x) λcv(t, x + h). (1.10) tvt + kv.... E (1.10) Ef(x) = f(x + h). (1.11) v(t + k, x) = ((1 + λc) λce)v(t, x), (1.12) t = nk (1.8) v(t, x) = v(nk, x) = ((1 + λc) λce) n v(0, x) n ( ) m( λce) = 1 + λc n m f(x) = m=0 n C m n Cn m m=0 ( 1 + λc ) m( λc) n m f(x + (n m)h). (1.13) 3

11 v(t, x) = v(nk, x) x- x, x + h, x + 2h,, x + nh = x + t λ, (1.14) x x + nh ξ = x ct = x cλnh [x, x + nh]h, k 0 vv(t, x) u(t, x)f(ξ) u(t, x) f [x, x + tλ 1 ] Courant-Friedrichs-Lewy.. (1.8).... f (1.13)f f ε v(t, x) = v(nk, x) ε n Cn m (1 + λc) m (λc) n m = (1 + 2λc) n ε (1.15) m=0 λvt n (1.17) v(t + k, x) v(t, x) k v(t, x) v(t, x h) + c h = 0, (1.16) v(t + k, x) = ((1 λc) + λce 1 )v(t, x). (1.17) v(t, x) = v(nk, x) = n Cn m m=0 ( 1 λc ) m(λc) n m f(x (n m)h). (1.18) v(t, x)f x, x h, x 2h,, x nh = x t λ (1.19) 4

12 x tλ 1 xh, k 0 λ (1.19)x [x t, x] ξ = x ct λ λ λc 1 (1.20) Courant-Friedrichs-Lewy (1.20) (1.18) f ε v(t, x) = v(nk, x) ε n Cn m (1 λc) m (λc) n m = ε((1 λc) + λc) n = ε. (1.21) m=0 (1.20)f h, k 0k/h = λ (1.18) v u(t, x) = f(x ct) u(t, x) u(t + k, x) (1 λc)u(t, x)) λcu(t, x h) = f(x ct ck) (1 λc)f(x ct) λcf(x ct h) (1.22) Kh 2, K = 1 2 (c2 λ 2 + λc) sup f. (1.23) fx cttaylorw = u v w(t + k, x) (1 λc)w(t, x)) λcw(t, x h) Kh 2. (1.24) λc 1, sup x w(t + k, x) (1 λc) sup x = sup w(x, t) + Kh 2. x w(t, x) + λc sup w(t, x h) + Kh 2 x (1.25) w(x, 0) = 0(1.25) t = nk u(t, x) v(t, x) sup x w(nk, x) sup w(0, x) + nkh 2 = Kth x λ. (1.26) 5

13 h 0w(t, x) 0(1.16)v u 1. fλ c 1 h 0(1.16) fv u. fεu v ε 2. (1.17)v v(t + k, x) (1 λc)v(t, x) λcv(t, x h) < δ. (1.20)v(0, x) = f(x)δ(1.23) K u(t, x) v(t, x) Kth λ + t λh δ. u(t, x)λ h. 3. f(x) = e αx αt, x λ = k/h n (1.13) (1.18) e α(x ct) Courant-Friedrichs-Lewyf ξ 6

14 2. Burgers Burgers u t + u u x = 0 (2.1) Burgers(2.1) u(0, x) = ϕ(x) (2.2) Cauchy ϕ(x) x R C 1 C 1 Cauchy(2.1)-(2.2) x = X(t) dx(t) dt X = X(t) = u(t, X(t)) (2.3) U(t) u(t, X(t)) (2.4) du dt = u t + u x dx dt = u t + uu x = 0, (2.5) (2.3) (2.1) (X(t), U(t)) (2.6)-(2.7) dx = U dt (2.6) du = 0 dt (2.7) (X, U) = (X(0) + tu(0), U(0)), (2.8) 1

15 X(t) = X(0) + tu(0) U(t) = U(0) (2.9) X(0) = α U(0) = u(0, X(0)) = ϕ(α) (2.10) (2.9) X(t) = α + tϕ(α), U(t) = ϕ(α) (2.11) (t, x) x = α + tϕ(α) (2.12) α α = α(t, x) (2.13) (2.13) U(t) = ϕ(α)cauchy(2.1)-(2.2) u(t, x) = ϕ(α(t, x)) (2.14) (2.11) Cauchy(2.1)-(2.2) ϕ(x) = sin x (2.15) t [0, 1) x = α + t sin α α α = α(t, x)cauchy(2.1)-(2.2) u(t, x) = sin α(t, x) (t, x) [0, 1) R (2.16) ϕ(x) = tanh x t R + x = α + t tanh α α = α(t, x) Cauchy (2.1)-(2.2) u(t, x) = tanh α(t, x) (t, x) R + R 2

16 x = X(t) ϕ (x) 0, x R (2.17) t R + (2.12) α = α(t, x) (2.17), Cauchy(2.1)- (2.2) (2.17)Cauchy(2.1)-(2.2) (2.17) t [ 0, ϕ (x) 1 C 0 ) x α = 1 + ϕ (α)t 1 ϕ (x) C 0t > 0 (2.18) t [ 0, ϕ (x) 1 C 0 ) (2.12) α = α(t, x)cauchy(2.1)-(2.2) [ 0, ϕ (x) 1 C 0 ) R (2.17)Cauchy(2.1)-(2.2) (2.17) α 1 α 2 (α 1 < α 2 ) (0, α 1 ) (0, α 2 ) ϕ(α 1 ) > ϕ(α 2 ) (2.19) X 1 (t) = α 1 + tϕ(α 1 ), X 2 (t) = α 2 + tϕ(α 2 ) (2.20) u ϕ(α 1 ) ϕ(α 2 ) (2.17)ϕ(x) = tanh x. (2.17) (2.17) (1.52) ϕ(α) α R x- 2.1(a) (2.17) 2.1(b) 3

17 t t 0 (a) x 0 (b) x 2.1: Cauchy { ut + a(u)u x = 0, (2.21) u t=0 = ϕ(x), (2.22) a(u) u C 1 ϕ(x) x R C 1 C Cauchy(2.21)-(2.22) R + R C 1 da(ϕ(x)) dx (2.21) 0, x R (2.23) dx dt = a(u) (2.24) Cauchy(2.21)-(2.22) C 1 du(t, x(t)) dt = u t + u x dx dt = u t + a(u)u x = 0 (2.25) u = u(t, x) (2.24) t x- 4

18 (2.22) (0, α) x = α + a(ϕ(α))t, (2.26) u u = ϕ(α) (2.27) : Cauchy(2.21)-(2.22) R + R C 1 (2.23) (2.23) α 1 α 2 α 1 < α 2 a(ϕ(α 1 )) > a(ϕ(α 2 )). (2.28) (0, α 1 ) x = α 1 + a(ϕ(α 1 ))t (0, α 2 ) x = α 2 + a(ϕ(α 2 ))t u ϕ(α 1 ) ϕ(α 2 ) C 1 : (2.23), t R + (2.26) α (2.23) (2.26) x α = 1 + da(ϕ(α)) t 1 > 0, t > 0 (2.29) dα t R + x α a(ϕ(α)) a ϕ(x) C 0 α ± x ± (2.30) (2.29) (2.30) t R +, (2.26) R R C 1 (2.26) αα = α(t, x) (2.27) Cauchy(2.21)-(2.22)C 1 u = a(α(t, x)) 5

19 (2.23)(2.23) (0, α) a(ϕ(α)) α R x- 2.1: (2.23)(2.21) Cauchy(2.21)-(2.22) (2.23) Cauchy(2.21)-(2.22) (2.23) (t, x ), 2.2 t (t, x ) 0 α 1 α 2 x 2.2: x i = α i + a(ϕ(α i ))t (i = 1, 2) u(t, α 2 + a(ϕ(α 2 ))t) u(t, α 1 + a(ϕ(α 1 ))t) α 2 α 1 + [a(ϕ(α 2 )) a(ϕ(α 1 ))] t, t t (t, x) (t, x ) u x (t, x) 6

20 2.1 Cauchy(2.21)-(2.22) u = u(t, x) u x t b 0 breaking time gradient catastrophe Cauchy(2.21)-(2.22) C 1 (t, x), x = ξ(τ; t, x) a C 1 ϕ C 1 x- (0, α) u(t, x) = ϕ(α) (2.31) ξ(τ; t, x) = α + a(ϕ(α))τ (2.32) τ = t x = α + a(ϕ(α))t (2.33) (2.31) x (2.33) α (2.35) (2.34) u x (t, x) = ϕ (α) α (2.34) x x α = 1 + da(ϕ(α)) t (2.35) dα u x = ϕ (α) (2.36) 1 + da(ϕ(α)) t dα u x (2.36) da(ϕ(α)) dα 0, α R, (2.37) t R + (2.36)1 7

21 α R da(ϕ(α)) dα ] 1 α t 0 (2.36) [ da(ϕ(α)) dα 0 α 0 da(ϕ(α)) dα a(ϕ(α 0 )) = min α R t b = { da(ϕ(α)) da(ϕ(α)) dα dα } (2.38) 1, (2.39) α=α0 α 0 (2.38) α Cauchy u t + uu x = 0, t = 0 : u = exp{ x 2 } (2.40) a(u) = u, ϕ(x) = exp{ x 2 } (0, α) a(ϕ(α)) = a(exp{ α 2 }) = exp{ α 2 } (2.39) f(α) d dα a(ϕ(α)) = d dα exp{ α2 } = 2α exp{ α 2 } f (α) = ( 2 + 4α 2 ) exp{ α 2 }, f(α) ± 1 2 α 0 = 1 2 f(α) Cauchy(2.40) 1 t b = 2α 0 exp{ α0} = exp{ 1 } = 2 e (2.33) x b = α 0 + a(ϕ(α 0 ))t b = 1 { e exp 1 } = 2 2 Cauchy(2.40) ( e 2, 2) u x 2.3 8

22 t t b x 2.3: x = α + exp{ α 2 }t (t b, x b ). 2.2 : Cauchy u t + uu x = 0, u t=0 = sin x, u t + uu x = 0, u t=0 = tanh x u t + u 2 u x = 0, u t=0 = (1 + x 2 ) 1. 9

23 3. u = u(x 1,, x n ) F (x 1,, x n, u, u x1,, u xn ) = 0, (3.1) F (3.1) (ODEs) 3.1 x, y a(x, y, u)u x + b(x, y, u)u y = c(x, y, u), (3.2) a, b, c x, y, u C 1 (x, y, z) z = u(x, y) u(x, y) (3.2) a(x, y, z), b(x, y, z), c(x, y, z) (x, y, z) Ω(3.2) (3.2) (u x, u y, 1)z = u(x, y) (3.2) (a, b, c) (a, b, c) 1

24 3.1: dx a(x, y, z) = dy b(x, y, z) = dz c(x, y, z) (3.3) t (3.3) dt dx dt = a(x, y, z), dy dt = b(x, y, z), dz dt = c(x, y, z). (3.4) t 3.2 (3.4)t a, b, c(x, y, z) (3.2) 3.1 Σ : z = u(x, y)σ Σp pσl lp Σ plσ p Σ Σ Σ Σ P = (x 0, y 0, z 0 )Σ : z = u(x, y)l P l Σ 2

25 (x(t), y(t), z(t)) l (3.4)t = t 0 (x, y, z) = (x 0, y 0, z 0 ) (x(t), y(t), z(t))p U(t) = z(t) u(x(t), y(t)). (3.5) P Σ U(t 0 ) = 0(3.4) (3.5) du dt = dz dt u x(x(t), y(t)) dx dt u y(x(t), y(t)) dy dt = c(x(t), y(t), z(t)) u x (x(t), y(t))a(x(t), y(t), z(t)) u y (x(t), y(t))b(x(t), y(t), z(t)). du dt = c(x(t), y(t), U(t) + u(x(t), y(t))) u x(x(t), y(t))a(x(t), y(t), U(t) + u(x(t), y(t))) u y (x(t), y(t))b(x(t), y(t), U(t) + u(x(t), y(t))). (3.6) u(x, y)(3.2) U(t) 0 (3.6) (3.6), U(t 0 ) = 0 U(t) 0. (3.5) U(t) l Σ 3.1 P P l 3.2 Σ 1 Σ 2 l l l P Σ 1 Σ 2 π 1 π 2 P (a, b, c) π 1 π 2 π 1 π 2 (a, b, c) l P T π 1 π 2 T (a, b, c) l 3

26 4. Cauchy (3.2) u z = u(x, y)... G.. G.. G u ug Cauchy (x, y, z) Γ x = f(s), y = g(s), z = h(s). (4.1) (3.2)u = u(x, y) h(s) = u(f(s), g(s)). (4.2) Cauchy(f, g, h) (3.2) (4.2) (3.2) Cauchy 4.1 Γ s = ϕ(σ) σ Cauchyu(x, y) 4.2 x 0 = f(s 0 ), y 0 = g(s 0 ) x, y Cauchy... yt x y = u(x, 0) = h(x) (4.3) u(x, y).... Cauchy Γ x = s, y = 0, z = h(s), (4.4) 1

27 (x, z) x h(x) u 4.3 Γ u (x, z) 1 s 0 Γ f(s), g(s), h(s) C 1 2 P 0 = (x 0, y 0, z 0 ) = (f(s 0 ), g(s 0 ), h(s 0 )). (4.5) P 0 (3.2) a, b, cc 1 P z = u(x, y) P s 0 s (3.4) t = 0f(s), g(s), h(s) x = X(s, t), y = Y (s, t), z = Z(s, t) (4.6) X, Y, Zs, t X t = a(x, Y, Z), Y t = b(x, Y, Z), Z t = c(x, Y, Z) (4.7) X(s, 0) = f(s), Y (s, 0) = g(s), Z(s, 0) = h(s). (4.8) X(s, t), Y (s, t), Z(s, t)(s 0, 0) C 1 (4.7) (4.8) (4.5) (4.8) x 0 = X(s 0, 0), y 0 = Y (s 0, 0). (4.9) 3 (s 0, 0) f (s 0 ) g (s 0 ) a(x 0, y 0, z 0 ) b(x 0, y 0, z 0 ) 2 0. (4.10)

28 (4.7) (4.8)(4.10) X s (s 0, 0) Y s (s 0, 0) X t (s 0, 0) Y t (s 0, 0) 0. (4.11) (x 0, y 0 ) x = X(s, t), y = Y (s, t) (4.12) s, t s = S(x, y), t = T (x, y). (4.13) (4.6)s, t Σ : z = u(x, y) z = u(x, y) = Z(S(x, y), T (x, y)) (4.14) u Σ (4.10) (4.6)Σ : z = u(x, y) (4.6) Σ Σp (X t, Y t, Z t ) Σs =Σ p(4.7) (a, b, c) Σ 4.4 (4.14) u(3.2) Cauchy p p (4.6) (4.10) Cauchy C 1 (4.10) (4.10) J f (s 0 ) g (s 0 ) a(x 0, y 0, z 0 ) b(x 0, y 0, z 0 ) = 0, 3

29 (4.2) (3.2)s = s 0, x = f(s 0 ), y = g(s 0 ) bf ag = 0, h = f u x + g u y, c = au x + bu y. (4.15) bh cg = 0, ah cf = 0. (4.16) (4.16) f, g, h a, b, c Γ s 0 J = 0 Γ Cauchy Γ p (4.10) Γ Γ Cauchy4.1 Γ Γ P Γ 4.1: a(x, y)u x + b(x, y)u y = c(x, y)u + d(x, y). (4.17) dx dt = a(x, y), dy dt = b(x, y), (4.18) dy dx = b(x, y) a(x, y). (4.19) 4

30 (4.18) (4.19)(x, y).... (x, y, z) (x, y) x(t), y(t) dz dt z(t) = c(x(t), y(t))z + d(x(t), y(t)) (4.20) n u = u(x 1, x 2,, x n ) ai (x 1,, x n, u)u xi = c(x 1,, x n, u), (4.21) a i cc 1 (4.21) dx i ds = a i(x 1,, x n, z) (i = 1,, n), dz ds = c(x 1,, x n, z). (4.21) Cauchy R n+1 (4.22) (n 1)- M z = u(x 1,, x n ) (n 1)- x i = f i (s 1,, s n 1 ) (i = 1,, n), z = h(s 1,, s n 1 ). M (s 1,, s n 1 ) t = 0 : x i = f i (s 1,, s n 1 ), z = h(s 1,, s n 1 ) (4.23) (4.22) x i = X i (s 1,, s n 1, t) (i = 1,, n) z = Z(s 1,, s n 1, t). (4.24) (4.24) ns 1,, s n 1 t(4.24) z = u(x 1,, x n ) Jacobi f 1 f n s 1 s 1.. J f 1 f 0 (4.25) n s n 1 s n 1 a 1 a n 5

31 (4.24)n s 1,, s n 1 t. Cauchy 1 Cauchy u y + cu x = 0, u(x, 0) = h(x), (4.26) ch(x)c 1 Γ x = s, y = 0, z = h(s). dx dt = c, dy dt = 1, dz dt = 0. x = X(s, t) = s + ct, y = Y (s, t) = t, z = Z(s, t) = h(s). s, t Cauchy(4.26) z = h(x ct), (1.5) u(x 1,, x n ) Euler n x k u xk = αu (4.27) k=1 α(4.25) J (4.27) u(x 1,, x n 1, 1) = h(x 1,, x n 1 ), (4.28) h C 1 Cauchy(4.27)-(4.28) 6

32 (4.28) Γ s i (i = 1,, n 1), x i = z = h(s 1,, s n 1 ). (4.29) 1 (i = n), dx i dt = x i (i = 1,, n) dz dt = αz, s i e t (i = 1,, n 1), x i = e t (i = n), z = e αt h(s 1,, s n 1 ). (4.30) (4.31) λ > 0u ( z = u(x 1,, x n ) = x α x1 nh,, x ) n 1. (4.32) x n x n u(λx 1,, λx n ) = λ α u(x 1,, x n ). (4.33) α α < 0(4.27) C 1 (4.27) u 0 t (4.27)u du dt = x i = c i t, (i = 1,, n), (4.34) n c k u xk (c 1 t,, c n t) = α u. (4.35) t k=1 ut α t 0 u u 4.5 7

33 1. u t + au x = f(t, x), t > t 0, < x < +, u(t 0, x) = ϕ(x), a f, f x C([t 0, ) R), ϕ C 1 (R). 2. u t + (x cos t)u x = 0, u(0, x) = x 2. t > 0, < x < +, 3. xu t tu x = u, t > 0, x > 0, u(0, x) = g(x), x > 0, g(x) C 1 ((0, )). 4. Cauchy u t + u x = u 2, t = 0 : u = sin x. 8

微积分授课讲义

微积分授课讲义 2018 10 aiwanjun@sjtu.edu.cn 1201 / 18:00-20:20 213 14:00-17:00 I II Taylor : , n R n : x = (x 1, x 2,..., x n ) R; x, x y ; δ( ) ; ; ; ; ; ( ) ; ( / ) ; ; Ů(P 1,δ) P 1 U(P 0,δ) P 0 Ω P 1: 1.1 ( ). Ω