   
                                  DELiA
   
   
                 
                               A. V. Bocharov
                Program Systems Institute, USSR Academy of Science
               Pereslavl, P.O. Box 11, 152140 USSR, Tlx: 412531 BOAT
   
   
   
   
   	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 dirrerential 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 funcitons 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.
   
   
                               Cooperation
   
   
   	A cooperation is desirable for transferring DELiA to
   workstations and alternative computer-algebraic environments.
   
   	A cooperation in creating a knowledge-base on investigation
   strategies is also probable.
   
   	The author kindly requests to contact him primarily through
   telex 412531 BOAT SU or FAX (095) 2002216.
