Starting from Heat Equation

Transcription

1 Department of Applied Mathematics National Chiao Tung University Hsin-Chu 30010, TAIWAN 20th August 2009

2 Analytical Theory of Heat The differential equations of the propagation of heat express the most general conditions, and reduce the physical questions to problems of pure analysis, and this is the proper object of theory. (Analytical Theory of Heat) J. Fourier ( ) u t = ku xx, u(0, t) = u(l, t) = 0 u(x, 0) = f (x). t > 0, 0 < x < L Fourier is a mathematical poem. Lord Kelvin ( )

3 Analytical Theory of Heat The differential equations of the propagation of heat express the most general conditions, and reduce the physical questions to problems of pure analysis, and this is the proper object of theory. (Analytical Theory of Heat) J. Fourier ( ) u t = ku xx, u(0, t) = u(l, t) = 0 u(x, 0) = f (x). t > 0, 0 < x < L Fourier is a mathematical poem. Lord Kelvin ( )

4 Separation of Variables: Fourier Series The profound study of nature is the most fertile source of mathematical discoveries. J. Fourier ( ) Wave equation: First P.D.E. u tt c 2 u xx = 0, u(x, 0) = f (x), u t (x, 0) = g(x) [c] = L T, propagating speed J. d Alembert ( ) Enlightenment u(x, t) = 1 1 x+ct (f (x ct) + f (x + ct)) + g(y)dy 2 2c x ct [u] = [f ], [g] = [u]/t L. Euler ( )

5 Separation of Variables: Fourier Series The profound study of nature is the most fertile source of mathematical discoveries. J. Fourier ( ) Wave equation: First P.D.E. u tt c 2 u xx = 0, u(x, 0) = f (x), u t (x, 0) = g(x) [c] = L T, propagating speed J. d Alembert ( ) Enlightenment u(x, t) = 1 1 x+ct (f (x ct) + f (x + ct)) + g(y)dy 2 2c x ct [u] = [f ], [g] = [u]/t L. Euler ( )

6 Heat Equation: Fourier Series Daniel Bernoulli ( ) (separation of variables)???? u(x, t) = b n = 2 L Dimensional Analysis e λnt b n ϕ n (x), λ n = ( nπ L )2, n=1 L 0 f (x)ϕ n (x)dx f, ϕ n [u t ] = [ku xx ] = [k] = L2 T u(x, 0) = f (x) = [f ] = [u] [ [ϕ n ] = sin nπx ] = 1 = [x] = [L] = L L [e λnkt ] = 1 = [λ n ] = [kt] 1 = L 2 [u] = [b n ] = 1 [f ][x] = [f ] [L]

7 Fourier series f (x) a ( a n cos nπx L a n = 1 L b n = 1 L L L L L Dimensional Analysis [a n ] = [b n ] = n=1 + b n sin nπx ) L f (x) cos nπx dx, n = 0, 1, 2, 3, L f (x) sin nπx dx, n = 1, 2, 3 L [ 1 L f (x) cos nπx ] L L L dx = L 1 [f ]L = [f ] [ 1 L f (x) cos nπx ] L L L dx = L 1 [f ]L = [f ]

8 Fourier Series: Convolution Euler formula f (x) n= Convolution appears! e iθ = cos θ + i sin θ c n e i nπx L = a ( a n cos nπx L n=1 c n = 1 L f (x)e 2L L f (x) = 1 L f (y)dy + 2L L i nπx L dx + b n sin nπx ) L (space or time translation invariant) n=1 1 L f (y) cos L L nπ(x y) dy L

9 Fourier Series: Convergence? f (x) = a ( a n cos nπx L n=1 + b n sin nπx ) L Question: Topology, Function Space pointwise convergence, uniform convergence, absolute convergence, L p spaces, summability methods and Cesaro mean. Caucy and Weierstrass :rigor of analysis Dirichlet ( ) Riemann ( ) Cantor ( ) 1876, There are continuous periodic functions whose Fourier series diverge at some point. Kolmogrov ( ) There exists Fourier series diverging everywhere. L. Carleson (1966) L 2 O.K. Hunt L p, p > 1 O.K. (interpolation theorem)

10 Fourier Integral Formula: L Question? L Riemann-Lebesgue Lemma: f (x) sin nxdx 0, Fourier integral formula (Idea: Riemann sum) n 1[ ] 1 f (x+) + f (x ) = 2 π π: DNA of the evolution! Dimensional balance 0 dξ f (y) cos ξ(x y)dy [y] = [x] = L, [ξ] = L (dual variable) [cos ξ(x y)] = 1 = [ξ] = L = L 1 f (x) = 1 2π [f ] = L 1 [f ]L 0 L f (y)e iξ(x y) dξ dy

11 Fourier Transform f (ξ) = f (x)e iξx dx, f (x) = 1 f (ξ)e ixξ dξ 2π [ ] [ f ] = f (x)e iξx dx = [f ]L, [xξ] = [x][ξ] = L L = 1 f (ξ) = f (x)e iξx dx, f (x) = 1 f (ξ)e ixξ dξ 2π f (ξ) = f (x)e 2πiξx dx, f (x) = f (ξ) = 1 2π f (x)e iξx dx, f (x) = 1 2π f (ξ)e 2πixξ dξ f (ξ)e ixξ dξ

12 Fundamental Solution ρc u t k 2 u = Qδ(x)δ(t), (x, t) R [0, ) x 2 u(x, 0) = 0, u(±, t) 0 Variables [u] = deg [x] = L [t] = T (temperature) (space coordinate) (time coordinate)

13 Heat Equation (Dimensional Analysis 1) Physical quantities: [ρ] = M L 3 (density) [C] = cal deg M (specific heat) [k] = cal L TL 2 deg = cal deg TL (thermal conductivity) [Q] = cal L 2 (heat add per cross section) Dimensional Balance: [ ρc u ] = t [ ] k 2 u x 2 = [Qδ(x)δ(t)] = cal L 3 T [δ(x)] = L 1, [δ(t)] = T 1

14 Heat Equation (Dimensional Analysis 2) New Variable [κ] = [ ] k = L2 ρc T (thermal diffusivity) Dimensionless Variables z x, [z] = 1 no characteristic length(time) scale 4κt u κt Q/ρC, [ ] u κt = 1 no characteristic temperature Q/ρC u = Q ρc κt f (z), z = x, 4κt Ordinary Differential Equation d 2 f df + 2z dz2 dz f : dimensionless + 2f = 0, f (± ) 0 u(x, t) = Q/ρC 4πκt e x2 4κt

15 Heat Equation (Scaling Group, Lie Group 1) u(x, t) u (x, t ), u = γu, x = αx, t = βt u t = β u γ t, u x = α u γ x, 2 u x 2 = α2 2 u γ (x ) 2 β u γ t k α2 γ 2 u (x ) 2 = Q ρc δ(x Invariance implies α = β, γ = 1/ β α )δ(t β ) = Q ρc αβδ(x )δ(t ) u = 1 β u, x = βt, t = βt u( βx, βt) = 1 β u(x, t) = u(x, t) = 1 t f (x/ t)

16 Heat Equation (Scaling Group, Lie Group 2) β-variation [ β u( βx, βt) = 1 ] u(x, t) β=1 β 1-order partial differential equation characteristic equation ζ = 1 2 x u x + t u t = 1 2 u dx 1 2 x = dt t = du 1 2 u x 2 t, log u = 1 log t + log F (ζ) 2 u(x, t) = 1 t F (ζ), ζ = x 2 t

17 Gaussian (Heat) Kernel Theorem G(x, t) G(x, t) = δ(x, t) = δ(x)δ(t) t G(x, t) = H(t) exp (4πt) n/2 ( ) x 2 4t [G]T 1 L n T = [G]L 2 L n T = 1 = [G] = L n, T = L 2 natural space: L 1 δ: radial, : rotational invariant = G: radial G(x, t) 1 x n E 1 ( E, E: dimensionless! t) n

18 Gaussian Kernel (continuous) Dimensional analysis suggests G(x, t) = 1 ( x ( t) g ), r = x = (x n xn) 2 1/2 t Ordinary Differential Equation g: dimensionless! n 2 g rg + g + n 1 g = 0, r 1 2 (r n g) + (r n 1 g ) = 0 lim g = lim g = 0 = g = 1 r r 2 rg g = be r2 4 = g(r/ t) = be r 2 /4t

19 Heat Equation: Duhamel formula 1 u t = u + h(x, t), t u(x, 0) = f (x) u(x, t) = e t f + e (t s) h(, s)ds 0 t = G(x y, t)f (y)dy + R n 0 G(x y, t s)h(y, s)ds R n Riemann Sum f (x) = h(x, t) = j= k=1 j= f j (x), f j (x) = δ(x j x)f (j x) x. h(j x, k t) x tδ(x j x)δ(t k t)

20 Heat Equation: Duhamel formula 2 δ(x j x) G(x j x) δ(x j x)δ(t k t) G(x j x, t k t) u(x, t) = u j (x, t) = j= = G(x j x, t)f (j x) x j= R n G(x y, t)f (y)dy u(x, t) = = h(j x, k t) x tg(x j x, t k t) k=1 j= t 0 R n G(x y, t s)h(y, s)dyds

21 Heat Equation: Scaling f (x) f (λx), u(x, t) u(λx, λ 2 t) u(λx, λ 2 t) = u(x, t) = G(x y, t)f (λy)dy R n λ n G(λ 1 (x y), λ 2 t)f (y)dy R n λ n G(λ 1 (x y), λ 2 t) = G(x y, t) or G(x, t) = λ n G(λx, λ 2 t) fundamental solution: λ 2 t = 1 ( λ n G(λx, λ 2 t) = t n x 2 ) 2 g t

22 Laplace Equation Theorem E n = δ, x R n E n (x) = x 2 n (2 n) S n 1, n 3 E 2 (x) = 1 ln x, n = 2 2π (1) δ: radial, : rotational invariant = E n (x) = E n ( x ). [E (2) dimensional analysis: n] L n = 1 L 2 = [E n ] = L 2 n (3) (1)+(2) = E n (x) = C x 2 n = C x 2 n 2 n (4) Divergence theorem = C = 1 S n 1 (5) E n(r) = 1/ S n 1 (r)

23 Laplace Equation: n = 2 Difficulty: E 2 dimensionless! x 2 n = x 2 n 1 Reason x 2 n 1 E 2 (x) = lim n 2 (2 n) S n 1 = 1 ln x. 2π r 2 n = r ɛ = exp( ɛ ln r) = 1 ɛ ln r + O(ɛ 2 )

24 Gaussian vs Newtonian potential H(t)G(x, t)dt = (4πt) n 2 e x 2 4t 0 = x n+2 4π n/2 0 dt e u u n 2 2 du ( ) n x n+2 = Γ 2 1 4π n/2 1 = (n 2) S n 1 x n+2 = E n (x) t u = u u = 0, t (Heat flow)!

25 Second Order Uniformly Elliptic Equation Theorem n i,j=1 a ij 2 u = δ, [a ij ] > 0, a ij R x i x j u(x) = ρ(x) 2 n (2 n) S n 1, n 3 u(x) = 1 ln ρ(x), n = 2 2π ( ) 1/2 ρ(x) = A ij x i x j A ij cofactor of [a ij ] i,j Key: x = ρ(x) (distance function)

26 General Second Parabolic Equation Theorem u t = n i,j=1 a ij 2 u, [a ij ] > 0 x i x j ( ) H(t) Z(x, t) = (4πt) n/2 det(a ij ) exp ρ(x)2 4t ( ) 1/2 ρ(x) = A ij x i x j A ij cofactor of [a ij ] i,j

27 Helmholtz Equation u + k 2 u = δ(x), [u] = L 2 n, [kx] = 1 u: radial = u(x) = ψ( x ) = ψ(r) ψ + n 1 ψ + k 2 ψ = 0 r integrating factor = ψ = r n 2 2 φ(kr), η = kr φ + 1 ( ) n 2 2 φ η φ + φ 2 η = 0 u(x) = x n 2 u(x) = x n 2 2 J n N n 2 2 (k x ), (k x ), Bessel equation n : odd n : even J n 2 (k x ) x n N n 2 2 (k x ) x n 2 2

28 Schrödinger Equation Imaginary Time t it You have reached an imaginary number,... Theorem u i u = δ(x)δ(t), t K(x, t) = G(x, it) = H(t) e x 2 (4πit) n 4it = H(t) e i x 2 2 (4πit) n 4t 2 natural space: L 2 Isometry Hermitian Hausdorff-Young inequality [t] = [x] 2 (diffusion scaling) L 2 wave nature dispersive (oscillation)

29 Airy Equation Theorem u t = u 3 x 3 E ai (x, t) = 1 ( ) x e itξ3 +ixξ dξ = (3t) 1/3 Ai 2π (3t) 1/3 Ai(x) = 1 2π e i(s3 /3+sx) ds [x] = [ξ] 1, [t] = [ξ] 3, [E ai ] = [ξ] = [x] 1 = [t] 1/3 Restriction Fourier transform itξ 3 + ixξ = constant

30 Lie Group Motivation Symmetry: if performing certain operations on it leaves it looking the same (H. Weyl). S. Lie: (transformation group) (1) construct an integrating factor for 1st order O.D.E. (2) reduce 2nd order O.D.E to 1st order by a change of variables. L. Boltzmann: (diffusion equation) algebraic symmetry of the P.D.E. To study diffusion with concentration dependent diffusion coefficient G. Birkhoff (following Boltzmann) P.D.E. O.D.E. (Similar solution)

31 Lie Group Scaling Integral curve Scaling group dy dx = x, [y] [x] = [x] [y] = [x]2 x λx, y λ 2 y (ˆx, ŷ) = (λx, λ 2 y) y = x 2 ˆx 2 + C ŷ = λ2 C (x, y, C) (λx, λ 2 y, λ 2 C)

32 Integrating Factor M(x, y)dx + N(x, y)dy = 0 integral curve φ(x, y) = C dφ = φ x dx + φ y dy = 0 µ(x, y) := φ x M = φ y N dφ = µmdx + µndy = 0 (total differential) invariant group ˆx = λx, ŷ = λ β y, 0 < λ < φ(λx, λ β y) = C(λ) integral curve d dλ φ(λx, λ β y) = d λ=1 dλ C(λ) λ=1

33 Integrating Factor integrating factor xφ x (x, y) + βyφ y (x, y) = d dλ C(λ) λ=1 x(µm) + βy(µn) = d dλ C(λ). λ=1 µ = d dλ λ=1 C(λ) xm + βyn µ = 1 xm + βyn ẏ = y(x y 2 ) x 2 y(y 2 x)dx + x 2 dy = 0 [y] [x] = [y][x] [x] 2 = [y]3 [x] 2 = [x] = [y] 2 = x λx, y λ 1 2 y µ = 1 xy(y 2 x) yx 2 = 1 xy x 2 y.