The Orchestra of Partial Differential Equations. Adam Larios

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1 The Orchestra of Partial Differential Equations Adam Larios 19 January 2017 Landscape Seminar

2 Outline 1 Fourier Series 2 Some Easy Differential Equations 3 Some Not-So-Easy Differential Equations

3 Outline Fourier Series 1 Fourier Series 2 Some Easy Differential Equations 3 Some Not-So-Easy Differential Equations Adam Larios (UNL) Orchestra of PDEs 19 January / 24

4 Frequency Fourier Series u t Adam Larios (UNL) Orchestra of PDEs 19 January / 24

5 Frequency Fourier Series u t u(t) = 0.5 cos(2t) cos(8t) cos(32t) +1.0 sin(1t) sin(4t) sin(16t) Adam Larios (UNL) Orchestra of PDEs 19 January / 24

6 Frequency Fourier Series u t u(t) = 0.5 cos(2t) cos(8t) cos(32t) +1.0 sin(1t) sin(4t) sin(16t) Adam Larios (UNL) Orchestra of PDEs 19 January / 24

7 Frequency Fourier Series u t u(t) = 0.5 cos(2t) cos(8t) cos(32t) +1.0 sin(1t) sin(4t) sin(16t) Adam Larios (UNL) Orchestra of PDEs 19 January / 24

8 Frequency Fourier Series u t u(t) = 1.5 cos(2t) cos(8t) cos(32t) +1.0 sin(1t) sin(4t) sin(16t) Adam Larios (UNL) Orchestra of PDEs 19 January / 24

9 Fourier Series Fourier Series u(t) = (a k cos(kt) + b k sin(kt)) k=0 Adam Larios (UNL) Orchestra of PDEs 19 January / 24

10 Fourier Series Fourier Series u(t) = (a k cos(kt) + b k sin(kt)) k=0 e ikt = cos(kt) + i sin(kt) cos(kt) = eikt + e ikt 2 sin(kt) = eikt e ikt 2i Adam Larios (UNL) Orchestra of PDEs 19 January / 24

11 Fourier Series Fourier Series u(t) = (a k cos(kt) + b k sin(kt)) k=0 e ikt = cos(kt) + i sin(kt) cos(kt) = eikt + e ikt 2 sin(kt) = eikt e ikt 2i u(t) = k= c k e ikt c k = 1 2 (a k ib k ), k > 0, c k = 1 2 (a k + ib k ), k < 0. Adam Larios (UNL) Orchestra of PDEs 19 January / 24

12 Fourier Series Adam Larios (UNL) Orchestra of PDEs 19 January / 24

13 Fourier Series Multi-dimensional Fourier series u(x) = k Z û k e ikx u( x) = û k e i k x k Z n Adam Larios (UNL) Orchestra of PDEs 19 January / 24

14 Fourier Series Multi-dimensional Fourier series u(x) = k Z û k e ikx u( x) = û k e i k x k Z n û k = 1 π u( x)e i k x d x 2π π Adam Larios (UNL) Orchestra of PDEs 19 January / 24

15 Fourier Series (a) Original image Adam Larios (UNL) Orchestra of PDEs 19 January / 24

16 Fourier Series (a) Original image (b) Fourier transform (magnitude) Adam Larios (UNL) Orchestra of PDEs 19 January / 24

17 Fourier Series (a) Original image (b) Fourier transform (magnitude) (c) Zero-out Fourier coeffcients Adam Larios (UNL) Orchestra of PDEs 19 January / 24

18 Fourier Series (a) Original image (b) Fourier transform (magnitude) (c) Zero-out Fourier coeffcients (d) Inverse Fourier transform Adam Larios (UNL) Orchestra of PDEs 19 January / 24

19 Derivatives Fourier Series u(x) = k Z û k e ikx Adam Larios (UNL) Orchestra of PDEs 19 January / 24

20 Derivatives Fourier Series u(x) = k Z û k e ikx d dx u(x) = ikû k e ikx k Z Adam Larios (UNL) Orchestra of PDEs 19 January / 24

21 Derivatives Fourier Series u(x) = k Z û k e ikx d dx u(x) = ikû k e ikx k Z d 2 dx 2 u(x) = ( k 2 )û k e ikx k Z Adam Larios (UNL) Orchestra of PDEs 19 January / 24

22 Derivatives Fourier Series u(x) = k Z û k e ikx d dx u(x) = ikû k e ikx k Z d 2 dx 2 u(x) = ( k 2 )û k e ikx k Z Idea Can the Fourier transform be used to understand differential equations? Adam Larios (UNL) Orchestra of PDEs 19 January / 24

23 Outline Some Easy Differential Equations 1 Fourier Series 2 Some Easy Differential Equations 3 Some Not-So-Easy Differential Equations Adam Larios (UNL) Orchestra of PDEs 19 January / 24

24 Some Easy Differential Equations The Simplest Partial Differential Equation dρ dt = position = x = x(t) velocity = v = v(t, x(t)) = dx dt density = ρ = ρ(t, x(t)) Adam Larios (UNL) Orchestra of PDEs 19 January / 24

25 Some Easy Differential Equations The Simplest Partial Differential Equation dρ dt = 0 position = x = x(t) velocity = v = v(t, x(t)) = dx dt density = ρ = ρ(t, x(t)) Adam Larios (UNL) Orchestra of PDEs 19 January / 24

26 Some Easy Differential Equations The Simplest Partial Differential Equation dρ dt = 0 dρ dt = ρ t + dx ρ dt x = 0 position = x = x(t) velocity = v = v(t, x(t)) = dx dt density = ρ = ρ(t, x(t)) Adam Larios (UNL) Orchestra of PDEs 19 January / 24

27 Some Easy Differential Equations The Simplest Partial Differential Equation dρ dt = 0 dρ dt = ρ t + dx ρ dt x = ρ t + v ρ x = 0 position = x = x(t) velocity = v = v(t, x(t)) = dx dt density = ρ = ρ(t, x(t)) Adam Larios (UNL) Orchestra of PDEs 19 January / 24

28 Some Easy Differential Equations The Simplest Partial Differential Equation position = x = x(t) velocity = v = v(t, x(t)) = dx dt density = ρ = ρ(t, x(t)) dρ dt = 0 dρ dt = ρ t + dx ρ dt x = ρ t + Transport Equation ρ t + vρ x = 0 v ρ x = 0 Adam Larios (UNL) Orchestra of PDEs 19 January / 24

29 Some Easy Differential Equations The Simplest Partial Differential Equation dρ dt = 0 dρ dt = ρ t + dx ρ dt x = ρ t + v ρ x = 0 position = x = x(t) velocity = v = v(t, x(t)) = dx dt density = ρ = ρ(t, x(t)) Transport Equation ρ t + vρ x = 0 Transport Equation in R n ρ t + ( v )ρ = 0 ρ ( v )ρ = v 1 x + v ρ 2 y + v ρ 3 z Adam Larios (UNL) Orchestra of PDEs 19 January / 24

30 Some Easy Differential Equations Idea What about the water itself? What if we set ρ = v = u? Adam Larios (UNL) Orchestra of PDEs 19 January / 24

31 Some Easy Differential Equations Idea What about the water itself? What if we set ρ = v = u? Burgers Equation u t + uu x = 0 Burgers Equation in R n u t + ( u ) u = 0 Adam Larios (UNL) Orchestra of PDEs 19 January / 24

32 Some Easy Differential Equations Computer Time! Figure : Programmers working on ENIAC, one of the first computers (c. 1946) Adam Larios (UNL) Orchestra of PDEs 19 January / 24

33 Some Easy Differential Equations Diffusion Equation Adam Larios (UNL) Orchestra of PDEs 19 January / 24

34 Some Easy Differential Equations Diffusion Equation concentration = θ = θ(x, t) flux = f = f (x, t) = f (x) Adam Larios (UNL) Orchestra of PDEs 19 January / 24

35 Some Easy Differential Equations Diffusion Equation concentration = θ = θ(x, t) flux = f = f (x, t) = f (x) a Adam Larios (UNL) b x Orchestra of PDEs 19 January / 24

36 Some Easy Differential Equations Diffusion Equation concentration = θ = θ(x, t) flux = f = f (x, t) = f (x) θ(x, t0 ) f (a) f (b) a Adam Larios (UNL) b x Orchestra of PDEs 19 January / 24

37 Some Easy Differential Equations Diffusion Equation concentration = θ = θ(x, t) d dt flux = f = f (x, t) = f (x) θ(x, t) dx = f (a) f (b) a b = f (a) f (b) Adam Larios (UNL) b Z θ(x, t0 ) a Z b a f dx x x Orchestra of PDEs 19 January / 24

38 Some Easy Differential Equations Diffusion Equation concentration = θ = θ(x, t) d dt flux = f = f (x, t) = f (x) b Z θ(x, t) dx = f (a) f (b) a = f (a) a f (b) a b x Z a Adam Larios (UNL) b Z θ(x, t0 ) Orchestra of PDEs b θ dx = t Z a b f dx x f dx x 19 January / 24

39 Some Easy Differential Equations Diffusion Equation Fourier s law: b a θ b t dx = f a x dx f = ν θ x, ν > 0 Adam Larios (UNL) Orchestra of PDEs 19 January / 24

40 Some Easy Differential Equations Diffusion Equation Fourier s law: b a θ b t dx = f a x dx f = ν θ x, ν > 0 b a θ t dx = b a ν 2 θ x 2 dx Adam Larios (UNL) Orchestra of PDEs 19 January / 24

41 Some Easy Differential Equations Diffusion Equation Fourier s law: b a θ b t dx = f a x dx f = ν θ x, ν > 0 b a θ t dx = Diffusion Equation b a θ t = νθ xx Diffusion Equation in R 3 ν 2 θ x 2 dx θ t = ν(θ xx + θ yy + θ zz ) = ν θ Adam Larios (UNL) Orchestra of PDEs 19 January / 24

42 Some Easy Differential Equations Diffusion Equation and the Fourier Transform u t = νu xx u(x, t) = k Z û k (t)e ikx Adam Larios (UNL) Orchestra of PDEs 19 January / 24

43 Some Easy Differential Equations Diffusion Equation and the Fourier Transform u t = νu xx u(x, t) = k Z û k (t)e ikx u xx = k Z ( k 2 )û k e ikx Adam Larios (UNL) Orchestra of PDEs 19 January / 24

44 Some Easy Differential Equations Diffusion Equation and the Fourier Transform u t = νu xx u(x, t) = k Z û k (t)e ikx u xx = k Z ( k 2 )û k e ikx u t = k Z d ikx ûke dt Adam Larios (UNL) Orchestra of PDEs 19 January / 24

45 Some Easy Differential Equations Diffusion Equation and the Fourier Transform u t = νu xx u(x, t) = k Z û k (t)e ikx u xx = k Z ( k 2 )û k e ikx d dt ûk = νk 2 û k, u t = k Z d ikx ûke dt Fourier Coefficients k Z Adam Larios (UNL) Orchestra of PDEs 19 January / 24

46 Some Easy Differential Equations Diffusion Equation and the Fourier Transform u t = νu xx u(x, t) = k Z û k (t)e ikx u xx = k Z ( k 2 )û k e ikx u t = k Z d ikx ûke dt Fourier Coefficients d dt ûk = νk 2 û k, k Z û k (t) = e νk 2tû k (0) Adam Larios (UNL) Orchestra of PDEs 19 January / 24

47 Some Easy Differential Equations Diffusion Equation and the Fourier Transform u t = νu xx u(x, t) = k Z û k (t)e ikx u xx = k Z ( k 2 )û k e ikx u t = k Z d ikx ûke dt Fourier Coefficients d dt ûk = νk 2 û k, k Z û k (t) = e νk 2tû k (0) u(x, t) = k Z e νk2tû k (0)e ikx Adam Larios (UNL) Orchestra of PDEs 19 January / 24

48 Some Easy Differential Equations Computer Time Again! Figure : Woman working on a Cray supercomputer. (c. 1986) Adam Larios (UNL) Orchestra of PDEs 19 January / 24

49 Some Easy Differential Equations Backward Diffusion u t = νu xx Adam Larios (UNL) Orchestra of PDEs 19 January / 24

50 Some Easy Differential Equations Backward Diffusion u t = νu xx u(x, t) = k Z e +νk2tû k (0)e ikx Adam Larios (UNL) Orchestra of PDEs 19 January / 24

51 Some Easy Differential Equations Backward Diffusion u t = νu xx u(x, t) = k Z e +νk2tû k (0)e ikx Massively unstable! Adam Larios (UNL) Orchestra of PDEs 19 January / 24

52 Some Easy Differential Equations Fourth-Order Diffusion u t = νu xxxx Adam Larios (UNL) Orchestra of PDEs 19 January / 24

53 Some Easy Differential Equations Fourth-Order Diffusion u t = νu xxxx u xxxx = k Z (k 4 )û k e ikx Adam Larios (UNL) Orchestra of PDEs 19 January / 24

54 Some Easy Differential Equations Fourth-Order Diffusion u t = νu xxxx u xxxx = k Z (k 4 )û k e ikx u(x, t) = k Z e νk4tû k (0)e ikx Massively stable! Adam Larios (UNL) Orchestra of PDEs 19 January / 24

55 Some Easy Differential Equations Third-Order Dispersion u t = νu xxx Adam Larios (UNL) Orchestra of PDEs 19 January / 24

56 Some Easy Differential Equations Third-Order Dispersion u t = νu xxx u xx = k Z (ik 3 )û k e ikx Adam Larios (UNL) Orchestra of PDEs 19 January / 24

57 Some Easy Differential Equations Third-Order Dispersion u t = νu xxx u xx = k Z (ik 3 )û k e ikx u(x, t) = k Z e iνk3tû k (0)e ikx Adam Larios (UNL) Orchestra of PDEs 19 January / 24

58 Some Easy Differential Equations Transport-Diffusion Equation Transport-Diffusion Equation ρ t + uρ x = νρ xx Transport-Diffusion Equation in R n ρ t + ( u )ρ = ν ρ Adam Larios (UNL) Orchestra of PDEs 19 January / 24

59 Outline Some Not-So-Easy Differential Equations 1 Fourier Series 2 Some Easy Differential Equations 3 Some Not-So-Easy Differential Equations Adam Larios (UNL) Orchestra of PDEs 19 January / 24

60 Some Not-So-Easy Differential Equations Nonlinear Equations Burgers Equation [Shock Waves, Traffic] u t + uu x = νu xx Adam Larios (UNL) Orchestra of PDEs 19 January / 24

61 Some Not-So-Easy Differential Equations Nonlinear Equations Burgers Equation [Shock Waves, Traffic] u t + uu x = νu xx Korteweg-de Vries (KdV) Equation [Water Waves] u t + uu x = u xxx Kuramoto-Sivashinsky (KS) Equation [Flames] u t + uu x = λu xx u xxxx Navier-Stokes Equations [Incompressible Fluids] { ut + ( u ) u = ν u p u = 0 Adam Larios (UNL) Orchestra of PDEs 19 January / 24

62 Some Not-So-Easy Differential Equations Nonlinear Equations Burgers Equation [Shock Waves, Traffic] u t + uu x = νu xx Korteweg-de Vries (KdV) Equation [Water Waves] u t + uu x = u xxx Kuramoto-Sivashinsky (KS) Equation [Flames] u t + uu x = λu xx u xxxx Navier-Stokes Equations [Incompressible Fluids] { ut + ( u ) u = ν u p u = 0 Adam Larios (UNL) Orchestra of PDEs 19 January / 24

63 Some Not-So-Easy Differential Equations Computer Time Once More! Figure : Hopper Cray XE6 at NERSC, named after American computer scientist Dr. Grace Hopper, Adam Larios (UNL) Orchestra of PDEs 19 January / 24

64 Some Not-So-Easy Differential Equations Figure : Simulation of a solution to the 3D Navier-Stokes equations. Adam Larios (UNL) Orchestra of PDEs 19 January / 24

65 Some Not-So-Easy Differential Equations The Incompressible Navier-Stokes Equations Momentum Equation u + ( u ) u = p + ν u }{{} t }{{}}{{}}{{} Advection Pressure Viscous Acceleration Gradient Diffusion Claude L.M.H. Navier Continuity Equation (Divergence-Free Condition) u = 0 Unknowns u := Velocity (vector) p := Pressure (scalar) Parameter ν := Kinematic Viscosity George G. Stokes Problem (Leray 1933) Existence and uniqueness of strong solutions in 3D for all time. ($1,000,000 Clay Millennium Prize Problem) Adam Larios (UNL) Orchestra of PDEs 19 January / 24

66 Some Not-So-Easy Differential Equations Thank you! Adam Larios (UNL) Orchestra of PDEs 19 January / 24

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