Solving double integral numerically in matlab - algorithm

Question

In the paper "The fractional Laplacian operator on bounded domains as a special case of the nonlocal diffusion operator". Where the author has solved a fractional laplacian equation on bounded domain as a non-local diffusion equation.
I am trying to implement the finite element approximation of the one dimensional problem(please refer to page 14 of the above mentioned paper) in matlab.
I am using the following definition of $\phi_k$ as it is mentioned in the paper that $\phi$ is a $hat\;function$
\begin{equation}
\phi_{k}(x)=\begin{cases} {x-x_{k-1} \over x_k\,-x_{k-1}} & \mbox{ if } x \in [x_{k-1},x_k], \\
{x_{k+1}\,-x \over x_{k+1}\,-x_k} & \mbox{ if } x \in [x_k,x_{k+1}], \\
0 & \mbox{ otherwise},\end{cases}
\end{equation}
$\Omega=(-1,1)$ and $\Omega_I=(-1-\lambda,-1) \cup (1,1+\lambda)$ so that $\Omega\cup\Omega_I=(-1-\lambda,1+\lambda)$
For the integers K,N we define the partition of $\overline{\Omega\cup\Omega_I}=[-1-\lambda,1+\lambda]$ as,
\begin{equation}
-1-\lambda=x_{-K}<...
Finally the equations that we have to solve to get the solution $\tilde{u_N}=\sum_{i=-K}^{K+N}U_j\phi_j(x)$ for some coefficients $U_j$ is:
Where $i=1,...,N-1$.
I need pointers in order to simplify and solve the LHS double integral in matlab.It is written in the paper(page 15) that I should use four point gauss quadrature for inner integral and quadgk.m function for outer integral, but since the limits of the inner integral are in terms of x how can I apply four point gauss quadrature on it??.Any help will be appreciated.
Thanks.
You can find the original question here.(Since SO does not support Latex)

Answer 1

For a first stab at the problem, take a look at dblquad and/or quad2d.
In the end, you'll want custom quadrature methods, so you should do something like the following:
% The integrand is of course a function of both x and y
integrand = #(x,y) (phi_j(y) - phi_j(x))*(phi_i(y) - phi_i(x))/abs(y-x)^(2*s+1);
% The inner integral is a function of x, and integrates over y
inner = #(x) quadgk(#(y)integrand(x,y), x-lambda, x+lambda);
% The inner integral is integrated over x to yield the value of the double integral
dblIntegral = quadgk(inner, -(1+lambda), 1+lambda)
where I've used quadgk twice, but you can replace by any other (custom) quadrature method you please.
By the way -- what is the reason for the authors to suggest a (non-adaptive) 4-point Gauss method? That way, you have no estimation of (and/or control over) the errors made in the inner integral...

Answer 2

You can do a 4 point 1D Gaussian quadrature. You seem to assume that it means a 2D integral. Not so - this is assuming a higher-order quadrature over 1D.
If you're solving a 1D finite element problem, it makes no sense whatsoever to integrate over a 2D domain.
I didn't read the paper, but that's what I recall from FEA that I learned.