Warm-Up Problem Problem: Find the exact roots of x2 + ✏x 1=0 − and then perform a series expansion in ✏. Solution: The roots are 1 1 2 1 4 1 ✏2 1 2 ✏ + 8 ✏ 128 ✏ + x = ✏ 1+ = − − ··· −2 ± r 4 ( 1 1 ✏ 1 ✏2 + 1 ✏4 + − − 2 − 8 128 ··· converging for ✏ < 2. | | Asymptotics, APPM 5480 Approximation Accuracy 0 1 10 0.8 −5 10 0.6 + x −10 0.4 10 first order 0.2 Exact error in approximation second order −15 Approx 10 fourth order 0 −2 0 −2 0 10 10 10 10 ϵ ϵ Asymptotics, APPM 5480 My name is • Mark Hoefer • Course webpage: https://goo.gl/DmCKIb (amath->course pages) • Office: ECOT 325 • Email: [email protected] • My Office Hours: M 10am-11am, W 12pm-1pm or by appt • TAs: Michelle Maiden, Patrick Sprenger (emails: fi[email protected]) • Office Hours in APPM Conf Room ECOT 226: M 12pm-1pm Asymptotics, APPM 5480 Applications of Asymptoics • Approximate solutions: algebraic/ transcendental/differential equations • Approximate equations: simplify ODEs/PDEs • Dynamical stability of solutions to ODEs/PDEs • Numerical methods: approximation theory, convergence, ... • Pure math: prime number theorem, Riemann hypothesis, ... Asymptotics, APPM 5480 Asymptotics: Physical Example dyed, diluted corn syrup light corn syrup computer piston pump nozzle Asymptotics, APPM 5480 Interfacial Waves • Solitary waves • Dispersive shock waves (DSWs) • How to model? ★ Solitary waves are long ★ DSWs are longer Asymptotics, APPM 5480 Conduit Modeling Model u(i,e) =0 r · @u(i,e) Two-fluid incompressible Re(i,e) +(u(i,e) )u(i,e) = p(i,e) + 2u(i,e) @t · r r r Navier-Stokes equations ✓ ◆ (REALLY HARD) + free surface boundary conditions + far field boundary conditions u(i,e) =0 r · ⇢(i) Two-fluid incompressible P (i) + z = 2u(i) r ∆⇢ r Stokes equations ✓ ◆ ⇢(e) µ(i) (HARD) Re(i,e) 0 P (e) + z = 2u(e) ! r ∆⇢ µ(e) r ✓ ◆ + free surface boundary conditions + far field boundary conditions Non-dimensional equations Asymptotics, APPM 5480 Conduit Modeling (cont) Model 2 2 1 A˜ +(A ) (A (A− A˜) ) =0 Long wave conduit equation t z˜ − t z˜ z˜ (not so hard) buoyancy viscous response nonlinear self-steepening nonlinear dispersion Cross-sectional| {z } | conduit{z area: A}(˜z,t˜) Concepts: scaling, maximal April 2014 Ph.D. Defense, balance µ(i) = 1 µ(e) (L/✏1/2) O 8 Rescale vertical length:z ˜ = 1/2z 1/2 ˜ Rescale time: t = t modulation scale wave scale experiments (L) O Asymptotics, APPM 5480 How does a wave train evolve? How do long wavelength perturbations propagate? How do long wavelength condition along free interface Velocity continuity, stress matching and kinematic continuity, Velocity - - - Modeling Approach - Multiple Scales - Multiple Approach Modeling Nonlinear Dispersive Conduit Waves Nonlinear Dispersive Conduit - True physical scale: True - - Macroscale description: - Macroscale - Mesoscale description: - “Exact” microscale mathematical description: Conduit Modeling (cont) Model Whitham modulation equations: slowly varying 2 2 periodic waves (A)T + A 2k!A✓ =0 − Z 1/2 (not so hard) (L/(✏ ✏˜)) 1 2 2 ⇣2 ⌘ A− + k A− A✓ 2ln A =0 O T − Z ⇣ ⌘ Concepts: singular kT + !Z =0 perturbation, kZ !T Periodic traveling wave: A = A (✓) , ✓ = − solvability condition ✏˜ k = k(Z, T),a= a(Z, T), A = A(Z, T), ! = !(k, a, A) Rescale vertical length: Z =˜z˜ Rescale time: T =˜t˜ (L) O Asymptotics, APPM 5480 Warm-Up Problem Problem: Find approximate expansions for the two roots of ✏x2 + x 1=0. − Solution: 2 3 1 p1+4✏ 1 ✏ +2✏ 5✏ + x = − ± = − − ··· ± 2✏ ( 1 1+✏ 2✏2 +5✏3 + − ✏ − − ··· Asymptotics, APPM 5480 My name is • Mark Hoefer • Course webpage: https://goo.gl/DmCKIb (amath->course pages) • Office: ECOT 325 • Email: [email protected] • My Office Hours: M 10am-11am, W 12pm-1pm or by appt • TAs: Michelle Maiden, Patrick Sprenger (emails: fi[email protected]) • Office Hours in APPM Conf Room ECOT 226: M 12pm-1pm Asymptotics, APPM 5480 “Asymptotology” - Martin Kruskal • M. Kruskal, Math Models in Phys. Sci., ed. Drobot, Prentice Hall, pp. 17-42, 1962 • Asymptotics: science of the evaluation of integrals, equations, etc. in various limiting cases • Asymptotology: art (quasi-science) of dealing with applied mathematical systems in limiting cases 1. Simplification (in spite of resulting complications) 2. Recursion: e.g., iteration 3. Interpretation: e.g., change of variables 4. Wild Behavior: solutions behave “wildly” in the limit 5. Annihilation: complete set of persistent annihilators 6. Maximal (Dominant) Balance: keep both terms for flexibility, ... 7. Mathematical Nonsense: if result doesn’t make sense, try again Asymptotics, APPM 5480 Warm-Up Problem Problem: Given the eigenvalue-eigenvector pair (λ0, v0) for a square matrix A (Av0 = λ0v0), formulate an ansatz (guess) to approximate an eigenvalue- eigenvector pair (λ, v) for the perturbed matrix C = A + B (Cv = λv), assuming that is small? Solution: Try λ = λ + λ + 2λ + , v = v + v + 2v + . 0 1 2 ··· 0 1 2 ··· Then, Cv = λv (A + B)(v + v + )=(λ + λ + )(v + v + ) ⇒ 0 1 ··· 0 1 ··· 0 1 ··· Av + (Av + Bv )+ = λ v + (λ v + λ v )+ ⇒ 0 1 0 ··· 0 0 0 1 1 0 ··· Equating like powers of , we already have Av0 = λ0v0 so we require (A λ I)v = (B λ I)v . − 0 1 − − 1 0 But we need both v1 and λ1!?! Asymptotics, APPM 5480 Warm-Up Problem T Problem: Given the eigenvalue-eigenvector pair (1, 10) for the matrix 11 11 A = , estimate the eigenvalues for the matrix C= for small. 01 1 Solution: Because A is degenerate, expect non-standard asymptotic expansion 1 λ =1+pλ + 2pλ + , v = + pv + 2pv + . 1 2 ··· 0 1 2 ··· Evaluating like powers of , we already satisfy 0 equation. If p = 1, then 01 λ 1 :(A λ I)v = v = (B λ I)v = 1 − 0 1 00 1 − − 1 0 1 − NOT SOLVABLE! (P7: Mathematical nonsense) Need p<1, then 1/p 01 λ1 0 :(A λ0I)v1 = v1 = λ1v0 = v1 = − 00 0 ⇒ λ1 Asymptotics, APPM 5480 Warm-Up Problem Solution (continued): Proceeding to the next order 2/p and assuming 2/p < 1 01 0 λ 2/p :(A λ I)v = v = λ v + λ v = + 2 − 0 2 00 2 1 1 2 0 λ2 0 1 NOT SOLVABLE! Need 2/p =1(p =1/2) so that the matrix perturbation enters at this order 01 0 λ 2/p = 1 :(A λ I)v = v = λ v Bv + λ v = + 2 − 0 2 00 2 1 1 − 0 2 0 λ2 1 0 1 − 2 ( ) Therefore, we must have λ1 = 1 or λ1± = 1 so that the approximate eigen- values are λ =1 1/2. Actually, these are± the exact eigenvalues! ± ± More generally, if geom(λ0) = 1, alg(λ0) = 2, then 1/2 w0∗Bv0 λ = λ0 ± ± w0∗u0 where u is a generalized eigenvector (A λ I)u = v and (A∗ λ∗I)w = 0. 0 − 0 0 0 − 0 0 Asymptotics, APPM 5480 Warm-Up Problem Problem: Find an asymptotic series approximation to x 1 e f(!)=! − dx, ! . ! + x !1 Z0 Hint: consider the Taylor expansion of 1 for z < 1. 1+z | | Solution: We have ! 1 x x2 x3 = =1 + + , converging for x < !. ! + x 1+x/! − ! !2 − !3 ··· | | Then, formally n n 1 ( 1) 1 n x 1 ( 1) n! 1 2! 3! f(!)= − x e− dx = − =1 + + !n !n − ! !2 − !3 ··· n=0 0 n=0 X Z X 1 n x because 0 x e− dx = n!. This series does not converge. Consider the ratio test R a (n + 1)! !n n +1 n+1 = = ,n . a !n+1 n! ! !1 !1 n Asymptotics, APPM 5480 Warm-Up Problem Problem: Find the first two terms in an asymptotic series approximation to the elliptic integral of the first kind π/2 1 K(m)= dθ 2 0 1 m sin θ − as m 0+. → Solution: Taylor expand the integrand in m 1 m 1+ sin2 θ,m 0+, 1 m sin2 θ ∼ 2 → − uniformly valid for θ [0, π/2]. Then ∈ π/2 m π/2 π m K(m) 1dθ + sin2 θ dθ = 1+ . ∼ 0 2 0 2 8 Asymptotics, APPM 5480 Warm-Up Problem Problem: Evaluate 1 ax2+bx+c e− dx, a > 0. Z1 This integral arises in many asymptotic problems. Solution: Complete the square 2 2 1 ax2+bx+c c 1 b b e− dx = e exp a x + dx. − − 2a 4a Z1 Z1 " ✓ ◆ # Let y = pa(x b/2a), then − 2 1 ax2+bx+c 1 c+ b 1 y2 e− dx = e 4a e− dy pa Z1 Z1 2 2 1 c+ b 1 y2 1 y2 1 c+ b x 2 = e 4a e− dy e− dy = e 4a e−| | dx pa s pa s 2 Z1 Z1 ZR 2 2⇡ 2 1 c+ b 1 r2 1 c+ b 1 r2 = e 4a e− r dφ dr = e 4a 2⇡ e− r dr pa s pa Z0 Z0 s Z0 2 ⇡ c+ b = e 4a . a r Asymptotics, APPM 5480 Warm-Up Problem Problem: Find the leading order behavior of 1 I(x)= sin t exp x sinh4 t dt, x + . − ! 1 Z0 Solution: Dominant contribution comes endpoint t = arg max( sinh4 t) = 0. t [0,1] − 2 Expand sin t and sinh t ✏ xt4 1 xt4 I(x) te− dt te− dt. ⇠ ⇠ Z0 Z0 Let s = xt4 to obtain 1 1 1/2 s Γ(1/2) 1 ⇡ I(x) s− e− ds = = ,x + . ⇠ 4px 4px 4 x ! 1 Z0 r So long as f(t) goes to zero algebraically at the critical point, the Laplace method applies. Asymptotics, APPM 5480 Warm-Up Problem Problem: Solve the following initial value problem using the Fourier transform ut + ux + uxxx =0,u(x, 0) = u0(x),xR.