Package: delierium
delieriumDifferentialEquations'LIEsymmetriesResearchInstrUMentSearching for symmetries in ODEs using Python/SageMath/sympyStatusstill playing around with Janet basesLie output form a alphaRelease 0.9.0.dev1Just 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 978-1-58488-889-5Fritz Schwarz: Loewy Decomposition of Linear Differential Equations, Springer Wien 2012, ISBN 978-3-7091-1687-6Daniel J. Arrigo: Symmetry Analysis of Differential Equations, Wiley Hoboken/New Jersey 2015, ISBN 978-1-118-72140-7John Starrett: Solving differential equations by Symmetry Groups  (e.ghttps://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_OrderGoals:Short term:All kinda stuff for symmetry analysis of ODE/PDE , doing is step by step, whatver comes to my mindMid term:Make it a valuable packageLong term:Maybe integration into SciPy|SymPy|SageMathRelease HistoryRelease 0.9.0.dev10'infinitesimalsODE' has been renamed to 'overdeterminedSystemODE' as described below.'Janet_Basis' discarded untilrealworkingRelease 0.0.1.dev1just alphas for 'infinitesimalsODE' and 'Janet_Basis'Documentation(work in progress)How to useGet 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 --upgradefrom IPython.display import Mathfrom delierium.helpers import latexerdisplay(Math(latexer(ode)))from delierium.Infinitesimals import overdeterminedSystemODEfrom sage.calculus.var import var, functionfrom sage.calculus.functional import diffx   = 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 liked^2y/dx^2is shown asy_x(superscript x)