[fricas-devel] Solving system of PDEs with polynomial power series

[Foad S Farimani]

Hello everyone,

While ago I was trying to use polynomial power series to solve a system of
partial differential and algebraic equations, when realized there is no
implementation of the idea. There is only Mathematica's
AsymptoticDSolveValue which is just for ODEs. So I decided to implement it
myself. Thanks to the Sympy community we now have some progress. I have one
implementation over here
<http://nbviewer.jupyter.org/gist/celliern/b38158d04d9dc3d8079dc44e3b747ac8>
by Nicolas CELLIER <https://github.com/celliern> and some ideas over here
<https://cs.stackexchange.com/questions/95886/algorithm-for-using-power-series-to-numerically-solve-a-partial-differential-equ>
by me and some of SymPy developers. I was wondering if we could join forces
to come of with a general algorithm, then implementionation on diffrent
languages shouldn't be that difficult. I was wondering if you could take a
look at this question
<https://cs.stackexchange.com/questions/95886/algorithm-for-using-power-series-to-numerically-solve-a-partial-differential-equ>
and help us out.

FYI I have made a issue on SymPy's github over here
<https://github.com/sympy/sympy/issues/15015#> which you can follow.

Best,
Foad

[Waldek Hebisch]

Post by Foad S Farimani
While ago I was trying to use polynomial power series to solve a system of
partial differential and algebraic equations, when realized there is no
implementation of the idea. There is only Mathematica's
AsymptoticDSolveValue which is just for ODEs.

Well, FriCAS has 'seriesSolve'. For ODEs it should work
directly. For PDE you would need some preprocessing.

Post by Foad S Farimani
So I decided to implement it
myself. Thanks to the Sympy community we now have some progress. I have one
implementation over here
<http://nbviewer.jupyter.org/gist/celliern/b38158d04d9dc3d8079dc44e3b747ac8>
by Nicolas CELLIER <https://github.com/celliern> and some ideas over here
<https://cs.stackexchange.com/questions/95886/algorithm-for-using-power-series-to-numerically-solve-a-partial-differential-equ>
by me and some of SymPy developers. I was wondering if we could join forces
to come of with a general algorithm, then implementionation on diffrent
languages shouldn't be that difficult. I was wondering if you could take a
look at this question
<https://cs.stackexchange.com/questions/95886/algorithm-for-using-power-series-to-numerically-solve-a-partial-differential-equ>
and help us out.

Well, what you write look strange. Normally, truncated series
will be only approximation to true solution so it will _not_
satisfy boundary conditions. IOW, solving in say R^n\times R
you want initial data, but no boundary conditions. If
your equation is appropriate, then Cauchy-Kovalevskaya theorem
says that for given initial data you will get unique power series
solution in neighbourhood of initial point. After little
preprocessing FriCAS 'seriesSolve' should give you this
solution.

If assumptions of Cauchy-Kovalevskaya theorem are not satisfied,
then it is possible that some transformation will give you
new system for which Cauchy-Kovalevskaya method works.
FriCAS Jet bundle package can help here.

[Foad S Farimani]

Post by Waldek Hebisch

Post by Foad S Farimani
While ago I was trying to use polynomial power series to solve a system

of

Post by Foad S Farimani
partial differential and algebraic equations, when realized there is no
implementation of the idea. There is only Mathematica's
AsymptoticDSolveValue which is just for ODEs.

Well, FriCAS has 'seriesSolve'. For ODEs it should work
directly.

This awesome. Could you be so kind to pint me to the source code? I want
to learn the algorithm.

Post by Waldek Hebisch
For PDE you would need some preprocessing.

If you have any ideas for the algorithm please share over here
<https://cs.stackexchange.com/questions/95886/algorithm-for-using-power-series-to-numerically-solve-a-partial-differential-equ>

Post by Waldek Hebisch

Post by Foad S Farimani
So I decided to implement it
myself. Thanks to the Sympy community we now have some progress. I have

one

Post by Foad S Farimani
implementation over here
<

http://nbviewer.jupyter.org/gist/celliern/b38158d04d9dc3d8079dc44e3b747ac8>

Post by Foad S Farimani
by Nicolas CELLIER <https://github.com/celliern> and some ideas over

here

Post by Foad S Farimani
<

https://cs.stackexchange.com/questions/95886/algorithm-for-using-power-series-to-numerically-solve-a-partial-differential-equ>

Post by Foad S Farimani
by me and some of SymPy developers. I was wondering if we could join

forces

Post by Foad S Farimani
to come of with a general algorithm, then implementionation on diffrent
languages shouldn't be that difficult. I was wondering if you could take

a

Post by Foad S Farimani
look at this question
<

https://cs.stackexchange.com/questions/95886/algorithm-for-using-power-series-to-numerically-solve-a-partial-differential-equ>

Post by Foad S Farimani
and help us out.

Well, what you write look strange.

disclaimer, I collected most of the pices from other places, so I'm barely
the developer. Would you be so kind to let me know which part doesn't make
sense?

Post by Waldek Hebisch
Normally, truncated series
will be only approximation to true solution so it will _not_
satisfy boundary conditions.

I'm not sure if I understand your point completely, but I mentioned some
solutions to the boundary condition incompatibility over here second comment
<https://github.com/sympy/sympy/issues/15015#>. I think integral of the
square of the error could be used to minimize the error.

Post by Waldek Hebisch
IOW, solving in say R^n\times R
you want initial data, but no boundary conditions.

I don not understand this, would you please elaborate?

Post by Waldek Hebisch
If
your equation is appropriate, then Cauchy-Kovalevskaya theorem
says that for given initial data you will get unique power series
solution in neighbourhood of initial point.

This was also mentioned here in the Sympy mailing list
<https://groups.google.com/forum/#!topic/sympy/X3h38SqQBCo>. I'm going to
learn more about this.

Post by Waldek Hebisch
After little
preprocessing FriCAS 'seriesSolve' should give you this
solution.

Looking forwards to that.

Post by Waldek Hebisch
If assumptions of Cauchy-Kovalevskaya theorem are not satisfied,
then it is possible that some transformation will give you
new system for which Cauchy-Kovalevskaya method works.
FriCAS Jet bundle package can help here.

No idea what is this. is there a good link I can read more about it?

Post by Waldek Hebisch
--
Waldek Hebisch

[Kurt Pagani]

Post by Waldek Hebisch
If assumptions of Cauchy-Kovalevskaya theorem are not satisfied,
then it is possible that some transformation will give you
new system for which Cauchy-Kovalevskaya method works.
FriCAS Jet bundle package can help here.
No idea what is this. is there a good link I can read more about it?

Have a look at

https://groups.google.com/forum/#!searchin/fricas-devel/JET%7Csort:date/fricas-devel/DCwxIiwWRC4/xiKIpC6QAQAJ

especially diss.ps and jet_Axiom.ps.

Solving BVPs numerically is usually not in the scope of a CAS, otherwise there
is hardly more to add to Waldeks statements.

Post by Waldek Hebisch
--
                              Waldek Hebisch
--