delierium 0.9.0.dev10

Symmetry Analysis for ODEs/PDEs using SageMath

delierium

Differential Equations' LIE symmetries Research InstrUMent

Searching for symmetries in ODEs using Python/SageMath/sympy

Status

• still playing around with Janet bases
• Lie output form a alpha

Release 0.9.0.dev1

• Just constructing a Janet basis from a list of homogenuous linear PDEs (for grevlex and degrevlex order, lex is dubious)

Literature (and inspiration):

• Werner M. Seiler: Involution. The Formal Theory of Differential Equations and its Applications in Computer Algebra, Spinger Berlin 2010, ISBN 978-3-642-26135-0.
• Gerd Baumann: Symmetry Analysis of Differential Equations with Mathematica, Springer New York Berlin Heidelberg 2000, ISBN 0-387-98552-2.
• Fritz Schwarz: Algorithmic Lie Theory for Solving Ordinary Differential Equations, CRC Press 2008, ISBN 978-1-58488-889-5
• Fritz Schwarz: Loewy Decomposition of Linear Differential Equations, Springer Wien 2012, ISBN 978-3-7091-1687-6
• Daniel J. Arrigo: Symmetry Analysis of Differential Equations, Wiley Hoboken/New Jersey 2015, ISBN 978-1-118-72140-7
• John Starrett: Solving differential equations by Symmetry Groups (e.g https://www.researchgate.net/publication/233653257_Solving_Differential_Equations_by_Symmetry_Groups)
• Alexey A. Kasatkin, Aliya A. Gainetdinova: Symbolic and Numerical Methods for Searching Symmetries of Ordinary Differential Equations with a Small Parameter and Reducing Its Order, https://link.springer.com/chapter/10.1007%2F978-3-030-26831-2_19 (if you are able and willing to pay the 27 bucks)
• Vishwas Khare, M.G. Timol: New Algorithm In SageMath To Check Symmetry Of Ode Of First Order, https://www.researchgate.net/publication/338388495_New_Algorithm_In_SageMath_To_Check_Symmetry_Of_Ode_Of_First_Order

Goals:

• Short term:
  • All kinda stuff for symmetry analysis of ODE/PDE , doing is step by step, whatver comes to my mind
• Mid term:
  • Make it a valuable package
• Long term:
  • Maybe integration into SciPy|SymPy|SageMath

Release History

Release 0.9.0.dev10

• 'infinitesimalsODE' has been renamed to 'overdeterminedSystemODE' as described below.
• 'Janet_Basis' discarded until real working

Release 0.0.1.dev1

• just alphas for 'infinitesimalsODE' and 'Janet_Basis'

Documentation(work in progress)

How to use

Get the overdetermined equations for the infinitesimals of an third order ODE:

>>> from delierium.Infinitesimals import overdeterminedSystemODE
>>> from sage.calculus.var import var, function
>>> from sage.calculus.functional import diff
>>> x   = var('x')
>>> y   = function('y')
>>> ode = diff(y(x), x, 3) + y(x) * diff(y(x), x, 2)
>>> inf = overdeterminedSystemODE(ode, y, x)
>>> for _ in inf:
>>>     print(_)
-3*D[0](xi)(y(x), x)
-6*D[0, 0](xi)(y(x), x)
y(x)*D[0](xi)(y(x), x) + 3*D[0, 0](phi)(y(x), x) - 9*D[0, 1](xi)(y(x), x)
y(x)*D[1](xi)(y(x), x) + phi(y(x), x) + 3*D[0, 1](phi)(y(x), x) - 3*D[1, 1](xi)(y(x), x)
-D[0, 0, 0](xi)(y(x), x)
-y(x)*D[0, 0](xi)(y(x), x) + D[0, 0, 0](phi)(y(x), x) - 3*D[0, 0, 1](xi)(y(x), x)
y(x)*D[0, 0](phi)(y(x), x) - 2*y(x)*D[0, 1](xi)(y(x), x) + 3*D[0, 0, 1](phi)(y(x), x) - 3*D[0, 1, 1](xi)(y(x), x)
2*y(x)*D[0, 1](phi)(y(x), x) - y(x)*D[1, 1](xi)(y(x), x) + 3*D[0, 1, 1](phi)(y(x), x) - D[1, 1, 1](xi)(y(x), x)
y(x)*D[1, 1](phi)(y(x), x) + D[1, 1, 1](phi)(y(x), x)    

If you are using JupyterLab, you can print the results in a more human readable way(and the easiest way to install) %pip install delierium --upgrade

from IPython.display import Math from delierium.helpers import latexer

display(Math(latexer(ode))) from delierium.Infinitesimals import overdeterminedSystemODE from sage.calculus.var import var, function from sage.calculus.functional import diff x = var('x') y = function('y') ode = diff(y(x), x, 3) + y(x) * diff(y(x), x, 2) inf = overdeterminedSystemODE(ode, y, x)

for _ in inf:

display(Math(latexer(_)))

In this mode a derivative like d^2y/dx^2 is shown as y_x(superscript x)