# Symbolics.jl | Lie series expansion for PDEs

**URL:** <https://discourse.julialang.org/t/symbolics-jl-lie-series-expansion-for-pdes/84885>\
**Category:** Modelling & Simulations\
**Tags:** question, symbolic\
**Created:** [July 27, 2022, 6:24pm UTC](https://discourse.julialang.org/t/symbolics-jl-lie-series-expansion-for-pdes/84885 "2022-07-27T18:24:00Z")\
**Posts on this page:** 3\
**Page:** 1

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**Author:** ![Audrius-St](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/audrius-st/32/24175_2.png) [@Audrius-St](https://discourse.julialang.org/u/Audrius-St)\
**Post date:** [July 27, 2022, 6:24pm UTC](https://discourse.julialang.org/t/symbolics-jl-lie-series-expansion-for-pdes/84885/1 "2022-07-27T18:24:00Z")

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Hello,

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](http://dx.doi.org/10.1080/00207729208949241)

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

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

2. 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?

 ![Lie_series_PDE_simple_example](https://global.discourse-cdn.com/julialang/original/3X/d/3/d31e7c0ec29999982eb31426e4e104609365de54.jpeg)

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<div class="post-metadata">

**Author:** ![ChrisRackauckas](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/chrisrackauckas/32/77_2.png) [@ChrisRackauckas](https://discourse.julialang.org/u/ChrisRackauckas)\
**Post date:** [July 27, 2022, 7:18pm UTC](https://discourse.julialang.org/t/symbolics-jl-lie-series-expansion-for-pdes/84885/2 "2022-07-27T19:18:27Z")

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Open an issue, that might be the easiest way to pull everyone in

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<div class="post-metadata">

**Author:** ![Audrius-St](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/audrius-st/32/24175_2.png) [@Audrius-St](https://discourse.julialang.org/u/Audrius-St)\
**Post date:** [August 1, 2022, 7:43pm UTC](https://discourse.julialang.org/t/symbolics-jl-lie-series-expansion-for-pdes/84885/4 "2022-08-01T19:43:31Z")

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Done.

[https://github.com/JuliaSymbolics/Symbolics.jl/issues/672](https://github.com/JuliaSymbolics/Symbolics.jl/issues/672)

Also posted a naïve solution along with additional comments and questions there.