Perturbation Methods: When do Small Parameters Have Large Consequences? Dave Goulet Visiting Assistant Professor of Mathematics Rose-Hulman Institute of Technology Asymptotic Methods: Overview • Boundary Layers: “shockingly” large changes over small distances • Multiple Scales: processes occurring on different time and spatial scales • Homogenization: how to smooth out the bumps Example: secular terms • Estimating approximation errors | cos(1 + ✏)t cos t| = | cos t(cos ✏t 1) sin t sin ✏t| • Some known inequalities |a + b| |a| + |b| | sin x| x • An error bound | cos(1 + ✏)t | sin x| 1 | cos x| 1 1 2 | cos x 1| x 2 1 2 2 cos t| ✏t + ✏ t 2 • Asymptotic formula cos(1 + ✏)t = cos t + O(✏t) cos(1 + ✏)t ⇠ cos t Should I worry about small things? • Solutions can vaporize. ✏x2 + x = 1 x⇠1,x⇠ x=1 • Solutions can lose meaningful physical properties. y 00 + ✏y = 0 y 0 = y(1 y 00 = 0 y0 = y springs oscillate! population stabilizes! p y = A cos( ✏t + ) y = ✏ + (1/y0 • Solutions may exist, but limits may not. 0 y = f (y/✏) ✏y) ✏)e t 1 1/✏ Asymptotic Series • Sometimes Taylor series: y ⇠ y0 ✓ 0 y = ✏y 1 2 2 1 + ✏t + ✏ t 2 ◆ y = y0 e✏t • Sometimes not: y 0 (t) = y(t ⌧) ✏xex = 1 x⇠ ln ✏ x = W(1/✏) ln( ln ✏) ln( ln ✏) ln ✏ Boundary Layers • enzyme kinetics 0 ✏c = s S+E $C !P +E (s + k)c s0 = s + (s + k s(0) = 1 c(0) = 0 • outer s c= s+k s s = s+k 0 k s s e =e • inner ⌧ = t/✏ c⌧ = s s(s + k)c e ⌧ c(⌧ ) = 1+k 1 s⌧ = 0 s(⌧ ) = 1 t 1)c Boundary Layers: staying warm in a wetsuit • Fluid Motion: X ut + u · ru = 1 rp + u Re r·u=0 • Steady, high Re flow over a flat plate fast slow p p ⇠ k x/ Re cold hot p p 3 ⇠ c x/ P r Boundary Layers: staying warm in a wetsuit fast V slow T V p p ⇠ k x/ Re fast & hot cold T hot p p 3 ⇠ c x/ P r V v=0 & hot Boundary Layers: Multiple layers and Interior Layers • quantum mechanics ✏y 00 (1 x2 )y = O(✏ ✏y 00 + 2(1 1 1/3 x2 )y = 1 y2 ) O(✏1/3 ) Bender & Orszag, 2010 Boundary Layers: shock layers • Burgers’ equation ut + uux = 0 ut + uux = ✏uxx • solutions develop propagating discontinuities, shocks • need viscosity to determine wave speed • need viscosity to determine shock thickness u(x, t) = f (⌘) ⌘= x cf 0 + f f 0 = (✏/ )f 00 • shock layer thickness is O(ℇ) ct (✏) Boundary Layers: shock layers • Look at the phase plane. cf 0 + f f 0 = f 00 f0 = g g0 = cg + f g • integrate across the shock dg = ( c + f )df g+ g = c(f+ • find the shock speed f+ + f c= 2 1 2 f ) + (f+ 2 f2 ) Boundary Layers: infinitely many layers • the delay recruitment model ✏y 0 (t) = y(t) + F (y(t 1)) • lasers: Ikeda equation • population biology • white blood cell formation Thomas Erneux, 2009 Summary: Summary: • Boundary layers are where interesting stuff occurs. boring interesting • The outer solution can be close, but is often way off. • Looking at the layer can give physical insight. • Boundary layers don’t have to occur on boundaries or sit still. • Not all asymptotic problems can be described with boundary layer theory... Next Week: part 2, the revenge • Multiple Scales • Homogenization ? Perturbation Methods: When do Small Parameters Have Large Consequences? Dave Goulet Visiting Assistant Professor of Mathematics Rose-Hulman Institute of Technology Asymptotic Methods • Boundary Layers: “shockingly” large changes over small distances • Multiple Scales: processes occurring on different time and spatial scales • Homogenization: how to smooth out the bumps Summary asymptotic methods • Solutions can vaporize. ✏x2 + x = 1 x=1 x⇠1,x⇠ • Solutions can lose meaningful physical properties. y 00 + ✏y = 0 y 00 = 0 p y = A cos( ✏t + ) • Solutions may exist, but limits may not. y 0 = f (y/✏) 1/✏ Multiple Scales: climate vs. weather Climate Weather ratio Time Scale 102-108 days 10-1-101 days ~10-6 Spatial Scale 102-105 km ~10-3 100-102 km Multiple Scales: climate vs. weather • don’t call them weathermen ⇢ut + ⇢u · ru = rp + µ u + F ⇢t + r · (⇢u) = 0 st = Q u · rs + T Tt + u · rT = r · (D · rT ) + f (t, T, u) Multiple Scales: climate vs. weather • Don’t call them weathermen! ⇢ut + ⇢u · ru = rp + µ u + F ⇢t + r · (⇢u) = 0 st = Q u · rs + T Tt + u · rT = r · (D · rT ) + f (t, T, u) Multiple Scales: acoustic equations • fluid motion: Navier-Stokes equations ⇢ut + ⇢u · ru = rp + µ u • disturb pressure slightly ⇢t + r · (⇢u) = 0 p = p0 + ✏P • velocity and density also get perturbed ⇢ = ⇢0 + ✏D u = ✏U • get simpler equations ⇢0 U t = rP + O(✏) Dt + ⇢0 r · U = O(✏) • add science D = kP } kPtt r2 P ⇠ 0 Multiple Scales: recovering long term behavior y 00 + ✏y = 0 00 y =0 • Is there a second time scale? s= p yss + y = 0 ✏t • Just one scale; we we’re using the wrong one! y 00 + y = ✏y 0 • trick fails s = ✏t y⇠e • secular terms 2 s = t(✏ + a✏ + . . .) ✏t/2 cos t y⇠e ✏t/2 cos t + O(✏2 t) Multiple Scales: effects of weak non-linearity • Pendulum equation: 00 y + sin(y) = 0 sin(✏x) x + =0 ✏ 00 y(0) = ✏ x(0) = 1 0 y (0) = 0 y = ✏x x0 (0) = 0 • approximate solution x000 + x0 = 0 x0 = cos t • correction 2 x ⇠ cos t + ✏ x1 x001 1 + x1 = cos t 6 • secular terms --> new time scale 2 s=✏ t resonance! Multiple Scales: effects of weak non-linearity sin(✏x) x + =0 ✏ 00 x(0) = 1 0 x (0) = 0 s = ✏2 t • get a PDE @t2 x 2 + 2✏ @t @s x + ✏4 @s2 x sin(✏x) + =0 ✏ • first approximation @t2 x0 + x0 = 0 x0 = A(s) cos(t + (s)) • correction resonance! @t2 x1 + x1 = A3 /8 + 2A 0 cos t + 2A0 sin t + (A3 /24) cos 3t • suppress secular terms by avoiding resonance 0 A =0 0 = 2 A /16 frequency correction x ⇠ cos(1 2 ✏ /16)t Multiple Scales: effects of weak non-linearity x ⇠ cos(1 x ⇠ cos t 2 ✏ /16)t Multiple Scales: destabilizing a swing • manipulate length of pendulum 0 L y 00 + 2 y 0 + sin y = 0 L L = ✏ cos !t • look for an asymptotic approximation y = y0 + ✏y1 + . . . y000 + sin y0 = 0 y100 + sin y1 = 2! sin !t • choose forcing frequency to destabilize equilibrium ! = 1 + b✏ + . . . s = t(✏ + a✏2 + . . .) Homogenization: repetitive arrays of resistors • series Rtotal = nR1 + nR2 = 2nRef f Ref f R 1 + R2 = 2 • parallel Rtotal = n R11 Ref f 1 1 = 2n + n R2 Ref f = 1 R1 2 + 1 R2 Homogenization: propagation through a complex environment • velocity changes over short time scale periodically ⇣ ⌘ x dx = f x, dt ✏ x = ✏y ✏y 0 = f (✏y, y) • separate & integrate Z x 0 ds =t f (s, s/✏) • Riemann-Lebesgue Lemma Z b g(s/✏)ds = (b a Z b g(s, s/✏)ds = a a) Z b a Z Z 1 g(z)dz + O(✏) 0 1 g(s, z)dzds + O(✏) 0 0 = f (0, y) Homogenization: propagation through a complex environment Z x 0 ds =t f (s, s/✏) Z b g(s, s/✏)ds = a Z b a Z 1 g(s, z)dzds + O(✏) 0 • homogenize! Z x 0 Z 1 0 dz ds = t + O(✏) f (s, z) • differentiate! 0 x (t) Z 1 0 dz =1 f (x, z) • algebra-tize! x0 (t) = R 1 1 dz 0 f (x,z) effective wave speed ⇣ x0 (t) = f x, ⌘ x ✏ Homogenization: effective diffusivity • heat conduction in 1-D, with small-scale periodic changes in conductivity d dx ✓ du a(x/✏) dx ◆ d dx =0 • can solve this exactly du aef f dx ◆ =0 Rx u(x) = u(0) + (u(1) • compute the flux du u(1) a(x/✏) = R1 dx ✓ ds 0 a(s/✏) u(0)) R 1 ds 0 a(s/✏) u(0) dz 0 a(s/✏) u(1) ⇠ R1 u(0) dz 0 a(z) du aef f = aef f (u(1) dx • effective heat transfer constant? aef f ⇠ R 1 1 dz 0 a(z) u(0)) Homogenization: effective diffusion tensor r · (a(y/✏)ru) = 0 • solve two problems on a square of this material no flux u=1 no flux no flux u=0 â = Z r· â 0 u=1 no flux 1 a(z)dz 0 • homogenized equation ✓✓ u=0 ◆ 0 ru ā ◆ =0 ā = R 1 1 dz 0 a(z) âuxx + āuyy = 0 Summary: • Small scales can be the hideout for interesting physics. 2 r P ⇠0 kPtt • Analysis of space and time scales improves understanding. • Secular terms can give you control. • Effective (homogenized) equations give insight. y100 + sin y1 = 2! sin !t r· ✓✓ • Smudging out details instead of brute force computation. â 0 ◆ 0 ru ā ◆ =0