

In [1]:

    
using Plots, ApproxFun

We demonstrate numerical solution of the examples in [Hale & Olver 2016]:

Example 1

$$y(x) + {}_{-1} {\cal Q}_x^{1/2} y(x) = 1$$



In [4]:

    
S=Legendre()⊕JacobiWeight(0.5,0.,Ultraspherical(1))
Q½=LeftIntegral(S,0.5)

y=(I+Q½)\1
plot(y)









    



[Plots.jl] Initializing backend: gr






    



INFO: Precompiling module GR...






    Out[4]:

We can compare with the exact solution:



In [5]:

    
x=Fun()
norm(exp(1+x)*erfc(sqrt(1+x))-y)









    Out[5]:





6.16319575427255e-15

Example 2

$$y + e^{-{1+x \over 2}} {}_{-1}Q_x^{1/2}[e^{1+x \over 2} y] = e^{-{1+x \over 2}}$$



In [6]:

    
S=Legendre()⊕JacobiWeight(0.5,0.,Ultraspherical(1))
x=Fun()
Q½=LeftIntegral(S,0.5)

y=(I+exp(-(1+x)/2)*Q½[exp((1+x)/2)])\exp(-(1+x)/2)
plot(y)









    Out[6]:

We can compare with the exact solution:



In [7]:

    
norm(y-exp((1+x)/2)*erfc(sqrt(1+x)))









    Out[7]:





3.3104392104716646e-15