PROGRAM SUMMARY
Title of program:
Extension to ODEtools package
Catalogue identifier:
ADMB
Ref. in CPC:
130(2000)204
Distribution format: zip file
Operating system: UNIX, Macintosh, Windows (95/98/NT)
High speed store required:
16MK words
Number of lines in distributed program, including test data, etc:
Keywords:
Computer algebra, Abel type first order ordinary differential
equations (ODEs), Equivalence problem, Integrable cases,
Symbolic computation.
Programming language used: Maple V Release 4 and 5
Computer:
Pentium II 400 -128Mb .
Nature of physical problem:
Analytical solving of Abel type first order ODEs having non-constant
invariant.
Method of solution
Solving the equivalence problem between a given ODE and representatives
of a set of non-constant invariant Abel ODE classes for which solutions
are avaliable.
Restrictions on the complexity of the problem
The computational routines presented work when the input ODE belongs to
one of the Abel classes considered in this work. This set of Abel
classes can be extended, but there are classes - depending intrinsically
on many parameters - for which the solution of the equivalence problem,
as presented here, may lead to large and therefore untractable
expressions. When the invariants of a given Abel ODE depend on analytic
functions, the success of the routines depends on Maple's ability to
normalize these invariants and recognize zeros (this is well implemented
in Maple, but it may nevertheless not work as expected in some cases).
Also, when the solution for the class parameter depends on other
algebraic symbols entering the ODE being solved, the routines can
determine this dependency only when it has rational form.
Typical running time
The methods being presented here have been implemented in the framework
of the ODEtools Maple package. On the average, over Kamke's [1] first
order Abel examples (see sec. 6), the ODE-solver of ODEtools is now
spending ~ 6 sec. per ODE when successful, and ~ 11 sec. when
unsuccessful. The timings in this paper were obtained using Maple R5 on
a Pentium II 400 - 128 Mb. of RAM - running Windows98.
Unusual features of the program
These computational routines are able - in principle - to integrate the
infinitely many members of all the non-constant invariant Abel ODE
classes considered in this work. Concretely, when a given Abel ODE
belongs to one of these classes, the routines can determine this fact,
by solving the related equivalence problem, and then use that
information to return a closed form solution without requiring further
participation from the user. The ODE families that are covered include,
as particular cases, all the Abel solvable cases presented in Kamke's
and Murphy's books, as well as the Abel ODEs member of other classes not
previously presented in the literature to the best of our knowledge.
After incorporating the new routines, the ODE solver of the ODEtools
package succeeds in solving 97 per cent of Kamke's first order examples.
References
[1] E. Kamke, Differentialgleichungen: Losungsmethoden und Losungen,
(Chelsea, New York, 1959).