DELiA 1.3
A. V. Bocharov
Wolfram Research
100 Trade Center Dr.
Urbana-Champaign, IL 61820-7237
alexei@wri.com
DELiA is a special-purpose computer-algebraic system designed
for investigation of differential equations. The goal of the DELiA
project is to create an integrated intelligent system to be used for
investigation and online solving of differential equations, as well as
for generating standalone numeric, seminumeric and symbolic d.e.
solvers.
Presently DELiA contains the following items: Symmetry
analyzer, Conservation laws handler and a Simplifier for differential
systems. The symmetry and invariance analysis is implemented in most
general setting. The simplifier includes a general passivization
algorithm together with a set of integration rules well tested on
linear and quasilinear systems of p.d.e.
DELiA has been implemented on a totally original algorithmic
and computer-algebraic basis: Standard Pascal has been used for the
implementation, the present MS DOS releases (1.2x, 1.3x) make
restricted usage of Turbo Pascal 5.5 facilities. DELiA's symmetry
analyzer (less than 35% of the system) essentially covers what has
been done in the REDUCE SPDE package but it compares favorably with
the SPDE in that it makes optimum usage of the scarce MS DOS resources
thus gaining in speed and memory efficiency.
DELiA's further evolution
In the course of the present evolution DELiA will acquire
special facilities for ordinary differential equations, some knowledge
on integro-differential equations, general symbolic-numeric interface
facilities, means for analyzing initial-value and boundary-value
problems. Still more important is the project to incorporate into
DELiA knowledge, know-how and heuristics relevant to hunting for
solutions. With this DELiA will tend to become an intelligent expert
system on differential equations.
It is well-known since the time of S. Lie that if a
finite-type differential system admits a sufficiently ample abelian or
solvable symmetry algebra, then its general solutions may be obtained
in quadratures in terms of characteristic functions of that algebra.
A version of an algorithm for building such quadratures will
be added in DELiA together with an algorithmic test for finiteness of
type. This will enhance the power of DELiA's simplifier/integrator.
Ordinary differential equations are always of finite type - so
the algorithm mentioned in the previous section always applies to
sufficiently symmetrical o.d.e.'s.
Otherwise powerful invariance methods may be used to test for
linearizability of an o.d.e., or, what is more general, to test
whether the o.d.e. under investigation is equivalent to one of the
"model" ones, well-described in textbooks and reference books.
If all the rigorous methods fail for an o.d.e., then it comes
to heuristics that is to hunting for lucky substitutions. A knowledge
base on such matters is to be added to DELiA.
Why using AI methods for differential equations
All the above-mentioned novelties are still in the scope of
traditional rigorous mathematics.
The crucial thing however is the strategy of applying rigorous
math methods to a system of differential equations, because even if
solutions of a differential system in a closed form are guaranteed by
a theorem they are seldom provided by this theorem in a constructive
way. The cases when nothing is guaranteed exactly are still more
numerous.
The expert knowledge how to deal with such cases is an
important part of "differential science".
A no-joke job is ahead of selecting a reasonable and flexible
set of expert parameters providing an adequate description of a
differential system. (Conservation laws, differential invariants,
classical point symmetries without doubt are the first candidates for
the role of such parameters).
Thus exact algebraic methods to compute invariance properties
of differential systems implemented in DELiA now may be considered as
a basis and prolegomena to its future intelligent editions.