delierium

Differential Equations' LIE symmetries Research InstrUMent

Searching for symmetries in ODEs using Python/SageMath/sympy

Status

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

Release 0.0.1.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 0.1.0

offers two functions 'Janet_Basis' and 'infinitesimalsODE' as described below.

Note that 'infinitesimalsODE' does only return the overdetermined system stemming from the prolongation of the original ODE. The real infinitesimal are part of the next release.

Documentation(work in progress)

How to use

Get the determining equations for the symmetry of an third order ODE:

>>> from delierium.Infinitesimals import infinitesimalsODE
>>> 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 = infinitesimalsODE(ode, y, x)
>>> print("determining system:")
>>> 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:

from IPython.display import Math

from delierium.helpers import latexer

from delierium.helpers import latexer

from delierium.helpers import latexer

x = var('x')

y = function('y')

ode = diff(y(x), x, 3) + y(x) * diff(y(x), x, 2)

display(Math(latexer(ode)))

inf = infinitesimalsODE(ode, y, x)

print("determining system:")

for _ in inf:

display(Math(latexer(_)))

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

Janet Basis

>>> import sage.all
>>> from delierium.JanetBasis import Janet_Basis
>>> from sage.calculus.var import var, function
>>> from sage.calculus.functional import diff
>>> vars = var ("x y")
>>> z = function("z")(*vars)
>>> w = function("w")(*vars)
>>> f1 = diff(w, y) + x*diff(z,y)/(2*y*(x**2+y)) - w/y
>>> f2 = diff(z, x, y) + y*diff(w,y)/x + 2*y*diff(z, x)/x
>>> f3 = diff(w, x, y) - 2*x*diff(z, x, 2)/y - x*diff(w,x)/y**2
>>> f4 = diff(w, x, y) + diff(z, x, y) + diff(w, y)/(2*y) - diff(w,x)/y + x* diff(z, y)/y - w/(2*y**2)
>>> f5 = diff(w,y,y) + diff(z,x,y) - diff(w, y)/y + w/(y**2)
>>> system_2_24 = [f1,f2,f3,f4,f5]
>>> jb=Janet_Basis(system_2_24, (w,z), vars)
>>> jb.show()
>>> diff(z(x, y), y)
>>> diff(z(x, y), x) + (1/2/y) * w(x, y)
>>> diff(w(x, y), y) + (-1/y) * w(x, y)
>>> diff(w(x, y), x)