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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?

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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.

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    $\begingroup$ I had been thinking about it for a little while and something clicked me.. To solve an equation with two dependent variables we need two equations. Does it apply same here? Do we need two differential equations to solve for a differential equation with two dependent variables? $\endgroup$
    – Hari Sam
    May 26, 2023 at 0:54
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    $\begingroup$ Even I’m not an expert but I would think like this, at-least in FEM or linear algebra context, let’s take the vector laplacian, we have three unknowns in 3D at each DOF and we can write three equations for each DOF from the weak form of vector laplacian. So in that way maybe the number of unknowns and equations maybe applicable. $\endgroup$ May 26, 2023 at 8:09

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