I am struggling to numerically solve the second-order nonlinear PDE for $f(x_1, x_2)$ of the form: $$0 = f + f^2 + f_1 + f_2 + f_{11} + f_{22}$$ I tried several functions from Matlab and Mathematica, but none of them seem suited for my equation. $x_1 \text{ and } x_2$ represent spacial dimensions.
My reference suggests to solve it by approximating the function $f(x_1, x_2)$ with Chebyshev polynomials, which is probably going to take me couple of days to code it by hand. Is it possible to solve this PDE with built-in functions from Matlab, Mathematica or other software?