The given equation is $$ \Delta(-\phi+\phi^3-\Delta\phi)=\frac{\partial \phi}{\partial t}. $$
I think it is equivalent to $\Delta\phi = \phi^3-\phi$
so I arrived to:
$$ \frac{\phi_{i+1,j}+\phi_{i-1,j}+\phi_{i,j-1}+\phi_{i,j+1}-4\phi_{i,j}}{\delta^2}=\phi_{i,j}^3-\phi_{i,j} $$
I should have $\phi_{i,j}= ...$ (function not depending on $\phi_{i,j}$) so that I could apply the methods of Jacobi, Gauss-Seidel and SOR that are the theme of the worksheet this question is in.
Can you help me?
Thanks
You can not just drop the time derivative and laplacian. That would be a completely different equation which share some solutions with yours.
If you're absolutely sure you need to solve $\Delta \phi = \phi^3 - \phi$ using some interational method for SLAE you need to linearize your problem first.
Start with some initial approximation for the solution $\phi_0$. Assuming it is close to $\phi$ let's introduce new unknown function $\psi = \phi - \phi_0$. $$ \Delta \psi = \Delta (\phi - \phi_0) = \phi^3 - \phi - \Delta \phi_0 = (\phi_0 + \psi)^3 - (\phi_0 + \psi) - \Delta \phi_0 \approx \phi_0^3 + 3\phi_0 \psi - \phi_0 - \psi - \Delta \phi_0 $$ So we've arrived to $$ \Delta \psi + (1 - 3\phi_0)\psi = \phi_0^3 + \phi_0 - \Delta \phi_0 $$ Boundary conditions should be adjusted accordingly (That's what I do not understand. The original problem should have a pair of conditions in each point of the boundary while the new one should have only a single one. That's an argument against dropping the outer laplacian). Discretizing that problem will result in SLAE for $\psi$ which has meaning of a correcting term for $\phi_0$, so the new approximation for the solution $\phi_1$ would be $\phi_1 = \phi_0 + \psi$. Repeat until convergence. If this method diverges you need to choose a better (closer to the solution) $\phi_0$.