# Complex Integral Equation Solution in MATLAB

I need to solve an integral equation in the form:

$$A(z)+\int\limits^{z_2}_{z_1}B(z') \frac{z^N}{z^N-z'^N} \frac{e^{i\beta}}{|z|}\mathrm{d}z'=0$$

where $A(z)$ distribution is known and we are solving for $B(z')$ distribution and $z,z'$ are complex variables. I also know the value of $B(z_2)$.

Can anyone help me how I can solve this equation using MATLAB (preferably numerically)?

• This is what's called a "Fredholm integral equation of the first kind". What have you tried already to solve this? – Wolfgang Bangerth Jan 4 '18 at 15:56
• Thanks for the keyword. I did a little search on the net to see if there is a quick solution to this which seems not to be the case. I tried with MATLAB fsolve but it was not successful. If you think solving with symbolic code is easier I can switch to symbolic. – Ferferimori Jan 4 '18 at 22:21
• OK, the equation can be solved using Panel methods. However, I have worked with Panel methods and they require a relatively long implementation. Any other time-saving approach using MATLAB or Mathematica built-in functions will be appreciated. – Ferferimori Jan 10 '18 at 0:52