— Ekaterina Shemyakova 2012/07/25 14:21
This is a package for computer algebra system MAPLE. It contains symbolic algorithms for Linear Partial Differential Operators (LPDOs) and for linear Partial Differential Equations (PDEs) with parametric coefficients. We consider arbitrary LPDOs in arbitrary many independent variables and of arbitrary orders and allow coefficients to be symbolic. Numerous tools are available with the focus on the Darboux, generalized Darboux and related invariant algorithms.
The package is in txt format: lpdo_.txt.
Use “read” to read this file at the top of your Maple sheet. In Linux it should look like this:
In Windows it should look like this:
Solves E by Laplace method (not to be confused with Laplace integral method) if this is possible to do so by less or equal to n iterations. Works significantly better than existing in MAPLE PDEtools[Laplace], which fails in many examples.
or a PDE, a Partial Differential Equation in the form
Here a,b,c may depend on the independent variables. One can use any names for independent variables. When work with LPDOs, specify them using LPDO__set_vars at the top of the worksheet. For example, LPDO__set_vars([s,t]): would declare two independent variables s and t.
Procedure ginv(L) returns a set of generating differential invariants for L with respect to the gauge transformations
The same procedure but with two arguments, ginv(L,M) returns a set of generating joint differential invariants for L and M with respect to the gauge transformations. Works for the following operators and pairs of operators:
Package can work with LPDOs of arbitrary orders and depended on arbitrary many independent variables. The independent variables are to be declared at the start of each worksheet using function
where vars is a list of independent variables. For example,
sets 4 independent variables x,y,z,t.
means that using some of the function available in the package, you may get some additional information (e.g. solutions of which PDEs must belong to the field). By default
where J is a multi-index,
are kept as a one dimensional array of its coefficients.
ordered lexicographically. E.g.
All differential invariants of operators
with respect to the gauge transformations
can be expressed in terms of so-called Laplace invariants
If h is not zero then define a +1 Laplace transformation by equality
If k is not zero then define a -1 Laplace transformation by equality
Thus until h or k are not zero, we have chain
There are some formulas relating those invariants, which implies that each new iteration of a Laplace
transformation brings us only one new invariant (only h or k is new). So, basically, we have the following chain of invariants:
The algorithm works as follows:
The following functions are available:
Here n is the number of iterations of Laplace transformations.
The property of the existence of a factorization of an LPDO is invariant with respect to gauge transformations. Thus, differential invariants can be very useful for search of a factorization of an LPDO. For each type of factorization (e.g. for LPDOs with symbol (pX+qY)XY there are 12 different types of factorizations) return the conditions for the existence of a factorization for an LPDO are implemented for all bivariate LPDOs of third order in terms of generating invariants of this LPDO. More specialized notions as obstacles to factorizations are implemented too.
Classical Laplace transformations (do not confuse with Laplace transform involving integrals) are implemented. Namely, the funcitons return the resulting operators, as well as the chain of invariants for Laplace LPDOs in classical for Darboux form, that is with symbol XY, as well as for Laplace LPDOs in the form more familiar for Physicists, that is with symbol XX+YY. Construction of a Darboux transformation of order one (two) by one (two) linearly independent solutions.
Cancellation of the mixed derivatives in M from the left and from the right, for a pair (M,L) defining a Darboux transformation. Generating sets of invariants for pairs (M,L) defining a Darboux transformation unders the gauge transforamtiosn of the pairs and under the gauged evolutions. Expressions of the later in terms of the former. Invariant description of pairs (M,L) defining any at all Darboux transformation.