I have the following equation
$-[(1+u^{4})u_x]_x = sin(x)+sin(5x)$, where the domain is $[0,2\pi]$
$u(0)=u(2\pi)=0$ for boundaries.
How to find a numerical solution for $u$ using numerical method?
My idea is to replace all the derivatives with fixed-point approximation, and then do some iterations until convergence. But I'm stuck with the differentiation approximation vector...
The equation can be written in the form $$ (1+u^4)u_{xx} + 4 u^3 (u_x)^2 = f(x). $$
Using centred finite differences, we get $$ (1+u_i^4)(u_{i+1}-2u_i+u_{i-1}) + u_i^3 (u_{i+1}-u_{i-1})^2 = h^2 f_i. $$
Finally, you must solve this nonlinear system using a fixed point method. One possibility could be
$$ u_i^{(k+1)}=\frac{u_{i+1}^{(k)}+u_{i-1}^{(k)}}{2}+\frac{(u_i^{(k)})^3(u_{i+1}^{(k)}-u_{i-1}^{(k)})^2}{2(1+(u_{i}^{(k)})^4)}-\frac{h^2 f_i}{2(1+(u_i^{(k)})^4)} $$
Now, you need to check if it converges and, if it does not converge, rewrite the nonlinear system into another fixed point problem.