how to solve a coupled nonlinear partial differential equations(Boundary Value Problem)

Question

So far, I have been looking into linear and nonlinear differential equations(Boundary value problem) that look like the following:

$\frac{d^2 \theta}{dz^2} + \sin(\theta) = 0$

I am now familiar with numerical methods of using finite difference or finite element methods to approximate the derivatives and Newton's method for solving them.

But now I have encountered a new problem (I'm not sure how such equations are called, but they are coupled nonlinear partial differential equations):

$\frac{d^2 \theta}{dz^2} + \cos(\theta)\frac{d^2 \phi}{dz^2}=0$

or

$\sin(\theta) \frac{d^2 \phi}{dz^2}=0$

say with boundary conditions:

$\phi(z=0)=0, \theta(z=0)=0 $ and $\phi(z = d)= 1, \theta(z=d)=1$

What are the numerical ways of solving such problems?

Can we make use of Newton's iteration method and finite difference method just as the way we did for a simple nonlinear ODE? Can these equations be written in weak form (variational form) for finite element method (FEM) calculations?

Answer 1

Closing eyes about the dynamics of the PDE, it is possible to write the finite difference and finite element formulations of the PDE.

In FEM, you can consider both $\theta$ and $\phi$ as unknowns and proceed by multiplying the system of equations with test functions and integrating by parts. Although due to the $\theta$ functions the resulting weak form would be nonlinear which you will have to linearize (which might get a bit ugly but should be possible I believe) and use Newton's iteration as you have mentioned or fixed point iterations.