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I am trying to apply Newton's method for non-linear systems to this equation:

$$\frac{\partial u}{\partial t}=\frac{\partial ^{2} u}{\partial x^{2}}+(1-u^{2})u+f(x,t) , x \in [-1,1], t>0$$ $$f(x,t)=\frac{25}{4}\pi^{2}\epsilon^{2}cos(t)cos(5\pi(x-1)/2)-sin(t)cos(5\pi(x-1)/2)-[1-cos^{2}(t)cos^{2}(5\pi(x-1)/2)]cos(t)cos(5\pi(x-1)/2)$$ given boundary and initial conditions and the discretization:

$$\frac{3u_{i}^{n+1}-4u_{i}^{n}+u_{i}^{n-1}}{2 \Delta t}= \frac{u_{i+1}^{n+1}-2u_{i}^{n+1}+u_{i-1}^{n+1}}{ \Delta x^{2}}+(1-(u_{i}^{n+1})^{2})u_{i}^{n+1}+f(x_{i},t_{n+1})$$

I understand how to implement Newton's method for non-linear systems for ODE's, but I'm not sure how to change this to an ODE and apply the method. I set $\omega=\Delta t/ \Delta x^{2}$ to get

$$-\omega(u_{i+1}^{n+1}+u_{i-1}^{n+1})+(\frac{3}{2}+2\omega-\Delta t+ \Delta t(u_{i}^{n+1})^{2})u_{i}^{n+1}=2u_{i}^{n}-\frac{1}{2}u_{i}^{n-1}+\Delta tf_{i}^{n+1}$$

I am stuck here though. Can anybody help me?

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