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E. S. Cheb-Terrab

Publications and source records attributed to E. S. Cheb-Terrab.

14 recordsLinked to original sources
arXiv2004

Solutions for the General, Confluent and Biconfluent Heun equations and their connection with Abel equations

In a recent paper, the canonical forms of a new multi-parameter class of Abel differential equations, so-called AIR, all of whose members can be mapped into Riccati equations, were shown to be related to the differential equations for the hypergeometric 2F1, 1F1 and 0F1 functions. In this paper, a connection between the AIR canonical forms and the Heun General (GHE), Confluent (CHE) and Biconfluent (BHE) equations is presented. This connection fixes the value of one of the Heun parameters, expresses another one in terms of those remaining, and provides closed form solutions in terms of pFq functions for the resulting GHE, CHE and BHE, respectively depending on four, three and two irreducible parameters. This connection also turns evident what is the relation between the Heun parameters such that the solutions admit Liouvillian form, and suggests a mechanism for relating linear equations with N and N-1 singularities through the canonical forms of a non-linear equation of one order less.

math-ph↗
arXiv2004

Non-Liouvillian solutions for second order linear ODEs

There exist sound literature and algorithms for computing Liouvillian solutions for the important problem of linear ODEs with rational coefficients. Taking as sample the 363 second order equations of that type found in Kamke's book, for instance, 51 % of them admit Liouvillian solutions and so are solvable using Kovacic's algorithm. On the other hand, special function solutions not admitting Liouvillian form appear frequently in mathematical physics, but there are not so general algorithms for computing them. In this paper we present an algorithm for computing special function solutions which can be expressed using the 2F1, 1F1 or 0F1 hypergeometric functions. The algorithm is easy to implement in the framework of a computer algebra system and systematically solves 91 % of the 363 Kamke's linear ODE examples mentioned.

math-ph↗
arXiv2004

A connection between Abel and pFq hypergeometric differential equations

In a recent paper, a new 3-parameter class of Abel type equations, so-called AIR, all of whose members can be mapped into Riccati equations, is shown. Most of the Abel equations with solution presented in the literature belong to the AIR class. Three canonical forms were shown to generate this class, according to the roots of a cubic. In this paper, a connection between those canonical forms and the differential equations for the hypergeometric functions 2F1, 1F1 and 0F1 is unveiled. This connection provides a closed form pFq solution for all Abel equations of the AIR class.

math-ph↗
arXiv2004

An Abel ordinary differential equation class generalizing known integrable classes

We present a multi-parameter non-constant-invariant class of Abel ordinary differential equations with the following remarkable features. This one class is shown to unify, that is, contain as particular cases, all the integrable classes presented by Abel, Liouville and Appell, as well as all those shown in Kamke's book and various other references. In addition, the class being presented includes other new and fully integrable subclasses, as well as the most general parameterized class of which we know whose members can systematically be mapped into Riccati equations. Finally, many integrable members of this class can be systematically mapped into an integrable member of a different class. We thus find new integrable classes from previously known ones.

math.GM↗
arXiv2000

First order ODEs, Symmetries and Linear Transformations

An algorithm for solving first order ODEs, by systematically determining symmetries of the form [ xi = F(x), eta = P(x) y + Q(x) ], where xi d/dx + eta d/dy is the symmetry generator - is presented. To these {\it linear} symmetries one can associate an ODE class which embraces all first order ODEs mappable into separable through linear transformations {t = f(x), u = p(x) y + q(x)}. This single ODE class includes as members, for instance, 78% of the 552 solvable first order examples of Kamke's book. Concerning the solving of this class, a restriction on the algorithm being presented exists only in the case of Riccati type ODEs, for which linear symmetries {\it always} exist but the algorithm will succeed in finding them only partially.

math-ph↗
arXiv2000

Integrating factors for second order ODEs

A systematic algorithm for building integrating factors of the form mu(x,y), mu(x,y') or mu(y,y') for second order ODEs is presented. The algorithm can determine the existence and explicit form of the integrating factors themselves without solving any differential equations, except for a linear ODE in one subcase of the mu(x,y) problem. Examples of ODEs not having point symmetries are shown to be solvable using this algorithm. The scheme was implemented in Maple, in the framework of the "ODEtools" package and its ODE-solver. A comparison between this implementation and other computer algebra ODE-solvers in tackling non-linear examples from Kamke's book is shown.

math-ph↗
arXiv2000

Abel ODEs: Equivalence and Integrable Classes

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, 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.

math-ph↗
arXiv1998

Integrating Factors and ODE Patterns

A systematic algorithm for building integrating factors of the form mu(x,y') or mu(y,y') for non-linear second order ODEs is presented. When such an integrating factor exists, the algorithm determines it without solving any differential equations. Examples of ODEs not having point symmetries are shown to be solvable using this algorithm. The scheme was implemented in Maple, in the framework of the ODEtools package and its ODE-solver. A comparison between this implementation and other computer algebra ODE-solvers in tackling non-linear examples from Kamke's book is shown.

physics.comp-ph↗
arXiv1998

Computer Algebra Solving of Second Order ODEs Using Symmetry Methods

An update of the ODEtools Maple package, for the analytical solving of 1st and 2nd order ODEs using Lie group symmetry methods, is presented. The set of routines includes an ODE-solver and user-level commands realizing most of the relevant steps of the symmetry scheme. The package also includes commands for testing the returned results, and for classifying 1st and 2nd order ODEs.

gr-qc↗
arXiv1996

Computer Algebra Solving of First Order ODEs Using Symmetry Methods

A set of Maple V R.3/4 computer algebra routines for the analytical solving of 1st. order ODEs, using Lie group symmetry methods, is presented. The set of commands includes a 1st. order ODE-solver and routines for, among other things: the explicit determination of the coefficients of the infinitesimal symmetry generator; the construction of the most general invariant 1st. order ODE under given symmetries; the determination of the canonical coordinates of the underlying invariant group; and the testing of the returned results.

gr-qc↗
arXiv1995

Poincare Sections of Hamiltonian Systems

A set of Maplev R.3 software routines, for plotting 2D/3D projections of Poincaré surfaces-of-section of Hamiltonian dynamical systems, is presented. The package consists of a plotting-command plus a set of facility-commands for a quick setup of the Hamilton equations of motion, initial conditions for numerical experiments, and for the zooming of plots.

gr-qc↗
arXiv1995

Self-Similar Collapse of Conformally Coupled Scalar Fields

A massless scalar field minimally coupled to the gravitational field in a simplified spherical symmetry is discussed. It is shown that, in this case, the solution found by Roberts, describing a scalar field collapse, is in fact the most general one. Taking that solution as departure point, a study of the gravitational collapse for the self-similar conformal case is presented.

gr-qc↗