For plate includes both geometric non-linearity and nonlinear stiffness, use arc-length method.

A secant based implementation is done by .@Chennakesava-Kadapa

https://github.com/chennachaos/ArcLengthMethod

I have done geometrical nonlinear bending analysis

using tangent based arc-length.

Here is the code https://github.com/suri2501/NonLinear-Bending-analysis-FEM

Feel free to contact me regarding my implementation.

Thank you

QUALITATIVE PROPERTIES OF NONLINEAR PDEs. An international conference on occasion of J.I. Díaz's 70th birthday

https://eventos.ucm.es/62140/dates/qualitative-properties-of-nonlinear-pdes.-an-international-conference-on-occasion-of-j.i.-diazs-70t.html

if you want to solve the Eq (21) or Eq(22), you could use conventional forth order RK with tiny time step.

However, by experience, Implicit versions of RK get more accurate results than forth-order RK, in less cpu time(for linear and nonlinear dynamic systems).

of course it can.

firstly，express governing equation and boundary conditions in discreted form with RK method and you will obtain a set of nonlinear equations.

secondly, apply Newton method to solve the nonlinear equations.

The DQ, GDQ,DR and... methods all are for solving nonlinear equ. specially when you have a system of equations

My advise: stay away from nonlinearity! Expand for linear equations and solve those...the results will be far more useful and impactful.

Try using some inbuilt solvers in MAPLE. I have worked on it with someone before but was not able to publish due to some reason

There are many methods, can be used to solve nonlinear differential equation such as RK, Gdq,Gdqr,....

I used HPM and VIM to solve nonlinear vibration of circular plate based on nonlocal elasticity and you can find them in my page.

With numeric methods. For example, you can try the binary search algorithm, or Newton.