# SingularIntegralEquations.jl

An experimental Julia package for solving singular integral equations.

# Acoustic Scattering

HelmholtzDirichlet.jl and HelmholtzNeumann.jl calculate the solution to the Helmholtz equation with Dirichlet and Neumann boundary conditions. The essential lines of code are:

```
k = 50 # Set wavenumber and fundamental solution for Helmholtz equation
g1 = (x,y) -> -besselj0(k*abs(y-x))/2
g2 = (x,y) -> x == y ? -(log(k/2)+γ)/2/π + im/4 : im/4*hankelh1(0,k*abs(y-x)) - g1(x,y).*logabs(y-x)/π
ui = (x,y) -> exp(im*k*(x-y)/sqrt(2)) # Incident plane wave at 45°
dom = ChebyshevInterval() # Set the domain
sp = Space(dom) # Canonical space on the domain
⨍ = DefiniteLineIntegral(dom) # Line integration functional
uiΓ = Fun(t->ui(real(t),imag(t)),sp) # Incident wave on Γ
# Instantiate the fundamental solution
G = GreensFun(g1,CauchyWeight(sp⊗sp,0)) + GreensFun(g2,sp⊗sp)
# Solve for the density
∂u∂n = ⨍[G]\uiΓ
# What is the forward error?
norm(⨍[G]*∂u∂n-uiΓ)
# Represent the scattered field
us = (x,y) -> -logkernel(g1,∂u∂n,complex(x,y))-linesum(g2,∂u∂n,complex(x,y))
```

Here is an example with 10 sources at the roots of unity scaled by 2 and scattered by multiple disjoint intervals and circles.

GravityHelmholtz.jl calculates the solution to the gravity Helmholtz equation with Dirichlet boundary conditions.

# The Faraday Cage

Laplace.jl calculates the solution to the Laplace equation with the origin shielded by infinitesimal plates centred at the Nth roots of unity. The essential lines of code are:

```
ui = (x,y) -> logabs(complex(x,y)-2) # Single source at (2,0) of strength 2π
N,r = 10,1e-1
cr = exp.(im*2π*(0:N-1)/N)
crl,crr = (1-2im*r)cr,(1+2im*r)cr
dom = ∪(Segment.(crl,crr)) # Set the shielding domain
sp = Space(dom) # Canonical space on the domain
⨍ = DefiniteLineIntegral(dom) # Line integration functional
uiΓ = Fun(t->ui(real(t),imag(t)),sp) # Action of source on shields
# Instantiate the fundamental solution
G = GreensFun((x,y)->0.5,CauchyWeight(sp⊗sp,0))
# The first column augments the system for global unknown constant charge φ0
# The first row ensure constant charge φ0 on all plates
φ0,∂u∂n=[0 ⨍;1 ⨍[G]]\[0.;uiΓ]
# Represent the scattered field
us = (x,y) -> -logkernel(∂u∂n,complex(x,y))/2
```

# Riemann–Hilbert Problems

SingularIntegralEquations.jl has support for Riemann–Hilbert problems and Wiener–Hopf factorizations. Wiener-Hopf.jl uses the Winer–Hopf factorization to calculate the UL decomposition of a scalar and a block Toeplitz operator. The essential lines of code in the matrix case are:

```
G=Fun(z->[-1 -3; -3 -1]/z +
[ 2 2; 1 -3] +
[ 2 -1; 1 2]*z,Circle())
C = Cauchy(-1)
V = (I+(I-G)*C)\(G-I)
L = ToeplitzOperator(inv(I+C*V))
U = ToeplitzOperator(I+V+C*V)
```

# Nonlocal Diffusion

Construct the nonlocal Laplacian acting on Fourier series by computing the spectrum on-the-fly:

```
α = 2.5 # ∈ [0, d+2), where d is the number of dimensions
# α is the strength of the singularity of the algebraic kernel
δ = 0.1 # the horizon of the nonlocal integral operator
L = NonlocalLaplacian(Fourier(), α, δ)
```

Afterward, you are free to treat it as any other banded (diagonal) operator.

# References

R. M. Slevinsky & S. Olver, A fast and well-conditioned spectral method for singular integral equations, *J. Comp. Phys.*, **332**:290--315, 2017.

Y. Li & R. M. Slevinsky. Fast and accurate algorithms for the computation of spherically symmetric nonlocal diffusion operators on lattices, arXiv:1810.07131, 2018.