Abel ODEs: Equivalence and integrable classes

Abstract

A classification, according to invariant theory, of non-constant invariant Abel ODEs known as solvable and found in the literature is presented. A set of new integrable classes depending on one or no parameters, derived from the analysis of the works by Abel, Liouville and Appell [2,3,4], is also shown. Computer algebra routines were developed to solve ODEs members of these classes by solving their related equivalence problem. The resulting library permits a systematic solving of Abel type ODEs in the Maple symbolic computing environment. Keywords: Abel type first order ordinary differential equations (ODEs), equivalence problem, integrable cases, symbolic computation. PROGRAM SUMMARY Nature of mathematical problemAnalytical solving of Abel type first order ODEs having non-constant invariant. Methods of solutionSolving the equivalence problem between a given ODE and representatives of a set of non-constant invariant Abel ODE classes for which solutions are available.Restrictions concerning 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 timeThe 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 nonconstant 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 ODEtoo...

On the integrable rational Abel differential equations

“…Abel equations appear in the reduction of order of many second-and high-order families, and hence are frequently found in the modelling of real problems in varied areas. In [5] it is given a classification according to invariant theory of the integrable Abel differential equations, i.e. of the Abel equations known as solvable in the literature.…”

“…During these last 130 years all the integrable rational Abel differential equations that have been found have been reduced by Cheb-Terrab and Roche [5] to four classes depending on one parameter and seven classes formed by a unique equation. All these classes are summarized in Appendix.…”

“…In [6], generalizing all the integrable classes collected in [5], it was presented a single multi-parameter non-constant invariant class of Abel ordinary differential equations denoted by Abel Inverse-Abel (AIA) type in the notation of [6]. In addition the class AIA contains a new subclass of Abel Inverse-Riccati (AIR) type depending on six parameters all of whose members can be systematically transformed into Riccati-type equations.…”

Generalized Weierstrass Integrability of the Abel Differential Equations

“…For such systems the notion of integrability is based on the existence of a first integral, and we want to characterize when the differential equations (1) have either a Weierstrass first integral or a Weierstrass inverse integrating factor. The integrability of the Abel differential equations has been studied by several authors, see [3,4,8,11,8] to cite just a few. For instance in [3], [4] and [8], the authors provide a list of the known integrable Abel differential equations with a, b, c and d rational functions.…”

A New Class of Symmetry Reductions for Parameter Identification Problems

“…A collection of the known cases of solvable Abel ODEs of the form (46) is presented in [21, pp. 107-120] and new results can be found, for example, in [8]. Each of these ODEs corresponds to a nonclassical equivalence transformation related to (5).…”