Initial commit

figs/numerical.fig/simpleq/simpleq.jl

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+# \eta
+function eta(x,t,weights,d,v)
+  if d==2
+    return integrate_chebyshev(y->4*((x+t)*y+abs(x-t)*(1-y))*v((x+t)*y+abs(x-t)*(1-y))/sqrt(((x+t)*y+abs(x-t)*(2-y))*((x+t)*(1+y)+abs(x-t)*(1-y))),0,1,length(weights))
+  elseif d==3
+    return (x>t ? 2*t/x : 2)* integrate_legendre(y->2*pi*((x+t)*y+abs(x-t)*(1-y))*v((x+t)*y+abs(x-t)*(1-y)),0,1,weights)
+  end
+end
+
+# initialize V and Eta
+function init_veta(weights,d,v)
+  order=length(weights[1])
+  V=Array{Complex{Float64}}(undef,order)
+  Eta=Array{Array{Complex{Float64}}}(undef,order)
+  Eta0=Array{Complex{Float64}}(undef,order)
+  V0=v(0)
+  for i in 1:order
+    k=(1-weights[1][i])/(1+weights[1][i])
+    V[i]=v(k)
+    Eta[i]=Array{Complex{Float64}}(undef,order)
+    for j in 1:order
+      y=(weights[1][j]+1)/2
+      Eta[i][j]=eta(k,(1-y)/y,weights,d,v)
+    end
+    y=(weights[1][i]+1)/2
+    Eta0[i]=eta(0,(1-y)/y,weights,d,v)
+  end
+  return(V,V0,Eta,Eta0)
+end
+
+# inverse Fourier transform
+function u_x(x,u,weights,d)
+  order=length(weights[1])
+  if d==2
+    out=integrate_legendre_sampled(y->(1-y)/y^3*besselj(0,x*(1-y)/y)/(2*pi),u,0,1,weights)
+  elseif d==3
+    out=integrate_legendre_sampled(y->(1-y)/y^3*sin(x*(1-y)/y)/x/(2*pi^2),u,0,1,weights)
+  end
+  return out
+end