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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?

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  • $\begingroup$ What are the domain and the boundary conditions? $\endgroup$ – Julián Aguirre Mar 20 at 18:13
  • $\begingroup$ 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. $\endgroup$ – User Mar 20 at 18:25
  • $\begingroup$ It is obviously not going to be very well posed without boundary conditions.. $\endgroup$ – DaveNine Mar 20 at 18:27
  • $\begingroup$ 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 $a, c <0$, $b,d >0$. $\endgroup$ – User Mar 20 at 18:28
  • $\begingroup$ Mathematica has the function NDSolve, which solves PDEs in two variables with boundary conditions on rectangular regions. $\endgroup$ – Julián Aguirre Mar 20 at 18:49

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