Hi all,

I am a first time user of Mathematica. What I am trying to do is quite simple to do by hand, but not so much using Mathematica.

Suppose I have a partial differential equation in u(x,t)

\partial_{t} u + u \partial_{x} u = 0

I want to be able to manipulate this equation as such,

\partial_{t} u = - u \partial_{x} u

and

\partial_{tt} u = \partial_{xt}(-u^2/2) = \partial{x}(u*\partial_{t}u) = \partial_{x}(u^2*\partial_{x}u)

As you can see, a Taylor expansion of u entirely made of t can be achieved.

u(x,t) ~ u(x0,t0) + t*\partial u_{t} + t^2/2*\partial u_{tt}

I would like some guidance in Mathematica to achieve this process. I would appreciate any help from experienced users.