How can I use FFT to solve this function?

My problem is to solve the treamfunction satisfication use FFT method and I do not have ideas how to do it, can anyone help me improve my codes? Or give me some hints please?

tspan=0:0.5:4;

v=0.001;

L=20;n=64;

x2=linspace(-L/2,L/2,n+1);

x=x2(1:n);

y=x;

kx=(2*pi/L)*[0:(n/2-1) (-n/2):-1];

kx(1)=10^-6;

ky=kx;

[X,Y]=meshgrid(x,y);

[KX,KY]=meshgrid(kx,ky);