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Effective Ways of Solving Some Pdes with Complicated Boundary Conditions on Unbounded Domains

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Effective Ways of Solving Some Pdes with Complicated Boundary Conditions on Unbounded Domains 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 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 −50 −40 −30 −20 −10 0 10 20 30 40 50 q (y2+1) Figure: Showing the function to be approximated F(y) = (y2+22) . Function mapped to the interval 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 q 2 Figure: Showing the new function to be approximated F(x) = 5-3z Error of the mapped function x 10−12 6 4 2 0 −2 −4 −6 −1 −0.5 0 0.5 1 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 12 10 8 6 4 2 0 −2 −100 −50 0 50 100 Figure: The error in approximating F(y) by [10,10] (solid green curves) and the error after it is split in factors, jF+(y)- F˜+(y)j (dotted red curve). Poles and Zeros 2 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −2 −1 0 1 2 x 10−7 Figure: The poles and zeroes in approximating F(y) by [10,10] the poles are filled in red circles whereas the zeroes are green circles. The scale of the x-axis is 10-7. sqrt((z.2+1)./(z.2+4)) −3 −3 −2 −2 −1 −1 0 0 1 1 2 2 −3 −2 −1 0 1 2 −3 −2 −1 0 1 2 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),1 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- Z01 =Φ(0)(α, y) + Φ(0)(α, y) (9) -1 + 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- ZL1 =Φ(L)(α, y) + Φ(L)(α, y) (11) -1 + 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) Z01 with the matrix kernel given by 1 e-jxj e-jx-Lj (29) e-jx+Lj e-jxj This integral equation is reduced to a Wiener–Hopf equation by extending the range of x and applying the Fourier transform. 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -10 -5 0 5 10 (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 radiation1 condition at infinity. 1 Fourier transforms The unknown Fourier transforms are a1 ∂φ(ξ, y) V (α, y) = eiαξdξ, 1 @y Z- a2 iαξ V2(α, y) = 1φ(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 Za14 The 4 by 4 Wiener–Hopf Equation The kernel of Wiener–Hopf Equation is 0 Ke i(a3-a4)α Kei(a2-a4)α ei(a1-a4)α1 B 00 ei(a2-a1)α 0 C B C @ 0 ei(a3-a2)α 00 A ei(a4-a3)α 000 p with K = α2 - k2. And the plus and minus functions defined as ( ) ( ) 0 1 1 0 -ia α 1 0 1 1 0 -ia α 1 Ψ- e 4 V4 Ψ+ e 4 V5 (2) -ia α (2) -ia α BΨ C Be 3 V3C BΨ C Be 1 V2C B - C = B C , B + C = B C . B (3)C e-ia2αV B (3)C e-ia2αV Ψ- @ 2A Ψ+ @ 3A @ A -ia α @ A -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 ) 0 (1)1 0KE 3 4 KE 2 4 E 1 4 1 Ψ- 0 F 1 (2) B01 KE(a2-a3) E(a1-a3)C BΨ C BE(a4-a3)FC B C B - C = B C - KE (a1-a2) B (3)C E(a4-a2)F @00 A @Ψ- A @ A 0001 (4) E(a4-a1)F Ψ- 0 (1)1 0 -1000 1 Ψ+ (4) B-E(a4-a3) K 00 C BΨ C - B C B + C -E(a4-a2) KE(a3-a2) 10 B (3)C @ A @Ψ+ A -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 .
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