Effective ways of solving some PDEs with complicated Boundary Conditions on Unbounded domains

Anastasia V. Kisil, Royal Society Dorothy Hodgkin Fellow, University of Manchester, partly in collaboration with Matthew Priddin, Dr Lorna Ayton and Matthew Colbrook. Overview

• Introduction to Wiener–Hopf technique, pole removal and factorisation • Conformal mapping and Rational approximations using Chebfun • Iterative method for n by n matrix equations. • Numerically effective by expanding in vanishing basis of Chebyshev polynomials, using APPROXFUN.JL and SINGULARINTEGRALEQUATIONS.JL

• Separation of variables in elliptic coordinates, Mathieu functions • Numerically effective solution which works for varying porosity and elasticity Open Problem by Sheehan Olver

• Numerically solving oscillatory Wiener-Hopf problems like the ones that will be shortly introduced. • Computing oscillatory Hilbert (or equivalently Cauchy) transforms • The answer to adequately separate out oscillatory and non-oscillatory behaviour The Classic Wiener–Hopf method

Find the functions Φ+(α) and Ψ−(α) such that

A(α)Φ+(α) + Ψ−(α) + C(α) = 0, on a strip around the real axis, where A(α) and C(α) are given and are analytic on the strip.

Classic Wiener–Hopf method relies on multiplicative factorising −1 A(α) = A+(α)(A−(α)) followed by additive decomposing − + C(α)A−(α) = (C(α)A−(α)) + (C(α)A−(α)) the forcing term.

+ − A+(α)Φ+(α) + (C(α)A−(α)) = −A−(α)Ψ−(α) − (C(α)A−(α))

Cannot solve matrix Wiener–Hopf equations in this way. The W–H Pole Removal (Singularities Matching) Suppose that A(α) is rational, then we can solve

A(α)Φ+(α) + Ψ−(α) + C(α) = 0, without multiplicative factorisation.

− + Perform an additive split A(α)Φ+(α) = (A(α)Φ+(α)) + (A(α)Φ+(α)) and C(α) = C+(α) + C−(α).

− n Ai Since A(α) is rational can write (A(α)Φ+(α)) = where αi i=0 α−αi are the poles in the upper half-plane, and Ai constants (residues at αi). P n n Ai + Ai − A(α)Φ+(α) − + C (α) = −Ψ−(α) − − C (α), α − αi α − αi i= i= X0 X0 Reduced the problem to a linear system of equations (by setting α = αi). Idea: do a rational approximation of A(α) and then apply pole removal

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Figure: Showing the error in approximating F(x) by [5,5] done by cf in Chebfun Error on the real line of the factors x 10−12 14

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Figure: The error in approximating F(y) by [10,10] (solid green curves) and the error after it is split in factors, |F+(y) − F˜+(y)| (dotted red curve). Poles and Zeros

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Figure: Contour lines for the real and imaginary are superimposed on a full colour image using a colour scheme developed by John Richardson. Red is real, blue is positive imaginary, green is negative imaginary, black is small magnitude and white is large magnitude. Branch cuts appear as colour discontinuities and coalescent contour lines. Produced using zviz.m 0 L

Acoustic plane wave incident pi = exp(−ik0x cos θi − ik0y sin θi) The scattered pressure is governed by Helmholtz (reduced wave) equation  ∂ ∂  + + k2 φ = 0. ∂x2 ∂y2

On the rigid wall and porous wall ∂φ = ik sin θ exp(−ik cos θ ), y = 0, − < x < 0, ∂y 0 i 0 i ∂φ ρ0 − ik0 sin θi exp(−ik0 cos θi) = 2 φ(x, 0),∞ y = 0, 0 < x < L, ∂y m Fourier Transforms Define half range and full range Fourier transform with respect to x = 0:

0 Φ(α, y) = φ(ξ, y)eiαξdξ + φ(ξ, y)eiαξdξ (8) Z− Z0∞ =Φ(0)(α, y) + Φ(0)(α, y) (9) −∞ + And the Fourier transform with respect to the point x = L:

L Φ(L)(α, y) = φ(ξ, y)eiα(ξ−L)dξ + φ(ξ, y)eiα(ξ−L)dξ (10) Z− ZL∞ =Φ(L)(α, y) + Φ(L)(α, y) (11) −∞ + The relation between the transforms is:

Φ(L)(α, y) = Φ(α, y)eiαL The Wiener–Hopf Equation

By taking Fourier transform of the governing equation and the boundary condition a matrix Wiener-Hopf equation can be obtained:

! −γ(α)P(α) ! ! Φ(0)(α) 1 P(α)eiαL Φ0(0)(α) f (α) − = − γ(α) + + 1 , (L) 1 −iαL 0(L) f (α) Φ− (α) γ(α) e 0 Φ+ (α) 2 q −m 2 2 Here P = 2ρ and γ(α) = α − k0. The Wiener–Hopf Factorisation A partial factorisation is

−iαL ! ! −e 1−γP Φ(0) P− P− − (12) 1 0 (L) P− Φ−

! 0(0)!   0 −P+ Φ+ f3 = P+ iαL + , (13) − P P e 0(L) f γP + + Φ+ 4 or re-arranged

−e−iαL 1 ! (0)! (L) ! Φ −γP+Φ− P− P− − + (14) 1 0 Φ(L) − 1 Φ0(0) P− − γP− + !  0 −P  Φ0(0) f  = + + + 3 , (15) −P P eiαL 0(L) f + + Φ+ 4 General Setting

This method could be used for a general W-H equation of the form ! ! Φ(0)(α)  A(α) B(α)eiαL Ψ(0)(α) f (α) − = + + 1 , (L) C(α)e−iαL 0 (L) f (α) Φ− (α) Ψ+ (α) 2 like before we would like to consider

−  −  − (L)n −  + (L)n −iαL  − (0)n − K1 Φ− =f3 − K1 Φ− + e K3 Ψ+ + f1 ,

+  +  + (0)n +  − (0)n iαL  + (L)n−1 + K3 Ψ+ =f1 − K3 Ψ+ + e K1 Φ− + f3 , where K = A and K = A . 1 CB− 3 B− Convergence

Theorem For sufficiently large L there exists a constant q < 1 such that

(L)n+1 (L)n a (L)n (L)n−1 a kΦ− − Φ− k2 6 qkΦ− − Φ− k2 , for all n and, hence, the error at the n-th iteration is qn kΦ(L)n − Φ(L)ka kΦ(L)0 − Φ(L)1ka. − − 2 6 1 − q − − 2

(0)n Analogous statements are true for Ψ+ . Example: Integral Equation

Consider the following (one-sided) integral equation (first considered by Wickham & Abrahams and recently by Antipov)

u(x) = λ k(x − t)u(t)dt + f(x), 0 < x < (28) Z0∞ with the matrix kernel given by ∞

 e−|x| e−|x−L| (29) e−|x+L| e−|x|

This integral equation is reduced to a Wiener–Hopf equation by extending the range of x and applying the Fourier transform. 0.3

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(1) (1)0 Figure: Showing the real part of U+ (solid black line), U+ (red dotted line) (1)1 and U+ (blue dashed line). The parameters are λ = −15 and L = 0.04. Two rigid strips

The boundary value problem for φ is governed by Helmholz equation  ∂ ∂  + + k2 φ = 0, ∂x2 ∂y2

∂φt with Newmann conditions (hard) ∂y = 0 valid on the strips. The boundary conditions for φs are ∂φ (x, 0±) s = ik sin θ exp(−ik cos θ ) a < x < a a < x < a , ∂y 0 i 0 i 1 2 3 4

φs(x, 0) = 0 for, − < x < a1, a2 < x < a3 and a4 < x < , and Sommerfeld radiation∞ condition at infinity. ∞ Fourier transforms

The unknown Fourier transforms are

a1 ∂φ(ξ, y) V (α, y) = eiαξdξ, 1 ∂y Z− a2 iαξ V2(α, y) = ∞φ(ξ, y)e dξ, Za1 a3 ∂φ(ξ, y) V (α, y) = eiαξdξ, 3 ∂y Za2 a4 iαξ V4(α, y) = φ(ξ, y)e dξ, Za3 ∂φ(ξ, y) V (α, y) = eiαξdξ. 5 ∂y Za∞4 The 4 by 4 Wiener–Hopf Equation

The kernel of Wiener–Hopf Equation is

 Ke i(a3−a4)α Kei(a2−a4)α ei(a1−a4)α  00 ei(a2−a1)α 0     0 ei(a3−a2)α 00  ei(a4−a3)α 000 √ with K = α2 − k2. And the plus and minus functions defined as

( ) ( )  1   −ia α   1   −ia α  Ψ− e 4 V4 Ψ+ e 4 V5 (2) −ia α (2) −ia α Ψ  e 3 V3 Ψ  e 1 V2  −  =   ,  +  =   .  (3) e−ia2αV  (3) e−ia2αV Ψ−  2 Ψ+  3   −ia α   −ia α (4) e 1 V1 (4) e 3 V4 Ψ− Ψ+ Re-arrangement

In a systematic way can re-arrange

(a −a ) (a −a ) (a −a )  (1) KE 3 4 KE 2 4 E 1 4  Ψ−  F  (2) 01 KE(a2−a3) E(a1−a3) Ψ  E(a4−a3)F    −  =   − KE (a1−a2)  (3) E(a4−a2)F 00  Ψ−    0001 (4) E(a4−a1)F Ψ−  (1)  −1000  Ψ+ (4) −E(a4−a3) K 00  Ψ  −    +  −E(a4−a2) KE(a3−a2) 10  (3)   Ψ+  −E(a4−a1) KE(a3−a1) E(a2−a1) K (2) Ψ+ where Eb = eibα The ‘vanishing basis’ of Chebyshev polynomials

Expanding h(s) in Chebyshev polynomials of the first kind Tj:

nc−1 h(s) = ajTj(s) (33) j= X0 The ‘vanishing basis’ of Chebyshev polynomials of the first kind, given by

z z z T0 (s) = 1, T1 (s) = s, Tj (s) = Tj(s) − Tj−2(s), for j > 2 (34) whose Cauchy transforms against the weight w(s) take the simple form

1 1 T z(s)  j−1 j w(s)s = i J−1(α) j 2 (35) 2πi s − α . + > Z−1 √ √ −1 where J+ (z) = z − z − 1 z + 1 is one of the right inverses of the Joukowsky map and w(s) = (1 − s)−1/2(1 + s)−1/2 N plates

Figure: Real part of the total field φ + φi for a plane wave of wavenumber k = 5 incident at an angle θi = 3π/4. Direct comparison with Hewett DP, Langdon S, Chandler-Wilde SN. 2015 paper. Code took under 6 minutes to produce on a 2018 Macbook Pro. Frequency dependence

105 kL 0.01 Φ

E 0.1 1 5 10 10− 100 relative error

10 15 − 0 20 40 60 80 100 iteration Figure: Convergence for various wavenumbers (finite plate). Key Feature of Proposed Method

• Works for cases where currently cannot directly compute Cauchy transform (open problem)

• An approximate method with explicit new bounds in the L2. • Exploit the known solutions of simpler sub-problems link to Schwarzschild series. • A replacement of multiplicative factorisation by series of additive factorisation. • Easy to use and already employed by other researchers in crack propagation problems, porosity models and silent flight of owls. Helmholtz differential equation in Elliptic coordinates

• Helmholtz differential equation in three dimensions can be solved by separation in 11 coordinate systems

• Boundary conditions given on a line the Cartesian coordinate systems most appropriate.

• Boundary conditions given on interval Elliptic coordinates are a natural choice.

• Leads to Special Functions called angular Mathieu equation and radial Mathieu equation. 15

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−15 −15−10 −50 5 10 15 To transform from the Cartesian coordinate to the elliptic coordinates we use the formula d d x = cosh ξ cos η y = sinh ξ sin η (38) 2 2 where d is the focal distance of the ellipses. We separate the variables and look for solution of the form G(ξ)H(η) where G(ξ) satisfies the radial Mathieu equation ∂G − (a − 2q cosh 2ξ)G = 0, (39) ∂ξ2

2 2 where q = d k0 and a is a separation constant. The function H(η) satisfies angular Mathieu equation ∂H + (a − 2q cosh 2η)H = 0. (40) ∂η2 Angular Mathieu Functions Angular Mathieu function have to be periodic with period π or2 π leading to countable number of different a for which a solution exists (am’s and bm’s). Associated to those am’s and bm’s the even and the odd Mathieu functions are denoted by

cem(η; q), m = 0, 1, 2 . . . Hm(η; q) = sem(η; q), m = 1, 2, 3 . . .

Angular Mathieu are orthogonal

2π 2π π if p = q, cepceqdη = sepseqdη = 0 if p 6= q. Z0 Z0  and 2π sepceqdη = 0 for all p, q (41) Z0 Radial Mathieu Functions

The solutions to radial Mathieu equation for q positive (which is always the case in this application) are even and odd Mathieu functions of first and second order.

Jem(ξ; q), Jom(ξ; q), first kind, Gm(ξ; q) = Nem(ξ; q), Nom(ξ; q), second kind.

We will need only the odd solutions and in particular define Mathieu-Hankel function,

(1) Hom (ξ; q) = Jom(ξ; q) + iNom(ξ; q). (42)

Asymptotic formula available for far-field. Rigid Plate

The scattered field can be expressed as (all the even Mathieu functions disappear)

φ1 = SrHor(ξ)ser(η), (45) ∞ r= X0 This gives the exact solution

√ r −i Jon0 (0) φ1 = 8π sen(α)Hor(ξ)ser(η), (46) ∞ πHo (0) r= n0 X0 Exact solution is not possible for more complicated boundary conditions the same expression for scattered field is possible. Poro-elastic Plate

The boundary condition on the plate will now be

∂φ ∂φI 2h i + = ρfω (1 − αH)η + αHηa . (52) ∂y y=0 ∂y y=0 the plate deformation is given by η(x)e−ıωt (the time dependence is again assumed) and η(x) satisfies the thin-plate equation

∂4η(x)   4α  (1 − α )B¯ − k¯ 4 η(x) = − 1 + H [φ](x), (53) H ∂x4 p π where [φ](x) = (φ(x, 0+) − φ(x, 0−)) Far-Field Directivity

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10-4 10-10 102 103 104 100 101 102 103 A. V. G. Cavalieri, W. R. Wolf, and J. W. Jaworski 2016 Conclusion

• General Theme: use analytical methods, adapt them numerically to solve more complicated problems. • Two ways to do it for Wiener-Hopf problem: rational approximation and iteration of scalar factorisation. • Using the correct expansion involving Mathieu Functions to use it for plates with varying porosity and elasticity. • Iterative method is fastest for high frequency • Mathieu Functions is fastest for low frequency • Both are very fast for far-field directivity • Both methods compare favourably with other methods available for these problems. Open vacancies: PhD and PostDoc

I have a funded PhD place https://www.findaphd.com/phds/project/mathematical-analysis-of- acoustic-wave-scattering-in-finite-metamaterials/?p113172

I have an opening for a 2 year position https://www.jobs.ac.uk/job/BXE912/research-associate

Happy to support a Fellowship Application to work at University of Manchester. Thank you Bibliography

1. Matthew J. Priddin, Anastasia V. Kisil and Lorna J. Ayton 2019. Applying an iterative method numerically to solve n by n matrix Wiener–Hopf equations with exponential factors, 378, Phil. Trans. R. Soc. Lond. A 2. A. Kisil, 2018. An Iterative Wiener–Hopf method for triangular matrix functions with exponential factors, SIAM J. Appl. Math. 3. A. Kisil, L. J. Ayton, 2018. Aerodynamic noise from rigid trailing edges with finite porous extensions J. Fluid Mech. 4. A. Kisil, 2015. The relationship between a strip Wiener–Hopf problem and a line Riemann–Hilbert problem, IMA J. Appl. Math. 5. A. Kisil, 2013. A Constructive method for approximate solution to scalar Wiener-Hopf Equations, Proc. R. Soc. A