Galerkin Finite element for solving third order time dependent PDE into weak form

[Hooi]

How to solve the third order time dependent partial differential equation (i.e. u_t + 6uu_x + u_xxx = 0) into weak form using galerkin finite different method?

 

[Deleted member 4993]

How to solve the third order time dependent partial differential equation (i.e. u_t + 6u_x + u_xxx = 0) into weak form using galerkin finite different method?

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[Hooi]

The attached image below shows my workout to solving a time dependent PDE, am i solve it correctly? How to put the right handside into weak form especially the third order derivative term ? Thank you.