I’ve used Sympy for several years now and am interested in giving Symbolics.jl a go with an eye to switching over completely. I’ve previously used Sympy to solve systems of ODES using the symbolic Lie series method, I would now like to use the generalized symbolic Lie series method to solve a particular PDE.

The test example below, of the first part of the method, is from the reference paper

Banks, S. P. (1992) Infinite-dimensional Carleman linearization, the Lie series, and optimal control of non-linear partial differential equations, International Journal of Systems Science, 23:5, 663 675. http://dx.doi.org/10.1080/00207729208949241

After reading the Symbolics.jl documentation, I am stuck on two points:

How best to represent Equation 1 to be substituted into Equations 6, 9, . . . ?

How to rearrange the order of the differential operators of x, t as in Equations 8, 14, . . . ? Can this be done in Symbolics.jl or would I need to “roll my own” using SymbolicUtils.jl?