MATLAB Solving Nonlinear PDEs in MATLAB: FDM or FEM Method? | Code Included

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The discussion centers on solving the partial differential equation u_{tt}=12{u_x}^2+12u_{xx}+{u_x}^4+6u_{xx}{u_x}^2+4u_{xxx}u_x+3{u_{xx}}^2+u_{xxxx} using numerical methods. The user is exploring the feasibility of employing Finite Difference Method (FDM) or Finite Element Method (FEM) in MATLAB for this purpose. Key inquiries include the best approach to implement these methods in MATLAB and the availability of existing code that can be adapted for this specific equation. The ultimate goal is to find the solution u(x,t).
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I have the following equation to solve:

u_{tt}=12{u_x}^2+12u_{xx}+{u_x}^4+6u_{xx}{u_x}^2+4u_{xxx}u_x+3{u_{xx}}^2+u_{xxxx}

I have been told to look into FDM or FEM. My question, is it possible to code something in MATLAB to solve this and if so what is the best method to use and how do I do it? It would also be useful if there is some code already that I can change to fit the problem.

Cheers
 
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I am looking for solution u(x,t) btw
 

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