Numerical solution to 2D nonlinear PDE

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?

• What are the domain and the boundary conditions? – Julián Aguirre Mar 20 at 18:13
• This is another problem. The paper, which introduces this PDE, does not specify the boundary conditions. After I will know that some method allows to solve this PDE I will play a little bit with boundary conditions, trying to guess reasonable values. – User Mar 20 at 18:25
• It is obviously not going to be very well posed without boundary conditions.. – DaveNine Mar 20 at 18:27
• In general, it will be something like $x_1 \in [a,b]$, $x_2 \in [c,d]$, and boundary conditions for $f(a,0)$, $f(b,0)$, $f(0,c)$, $f(0,d)$, where $a, c <0$, $b,d >0$. – User Mar 20 at 18:28
• Mathematica has the function NDSolve, which solves PDEs in two variables with boundary conditions on rectangular regions. – Julián Aguirre Mar 20 at 18:49