3,588 research outputs found
Symbolic computation of exact solutions expressible in hyperbolic and elliptic functions for nonlinear PDEs
Algorithms are presented for the tanh- and sech-methods, which lead to
closed-form solutions of nonlinear ordinary and partial differential equations
(ODEs and PDEs). New algorithms are given to find exact polynomial solutions of
ODEs and PDEs in terms of Jacobi's elliptic functions.
For systems with parameters, the algorithms determine the conditions on the
parameters so that the differential equations admit polynomial solutions in
tanh, sech, combinations thereof, Jacobi's sn or cn functions. Examples
illustrate key steps of the algorithms.
The new algorithms are implemented in Mathematica. The package
DDESpecialSolutions.m can be used to automatically compute new special
solutions of nonlinear PDEs. Use of the package, implementation issues, scope,
limitations, and future extensions of the software are addressed.
A survey is given of related algorithms and symbolic software to compute
exact solutions of nonlinear differential equations.Comment: 39 pages. Software available from Willy Hereman's home page at
http://www.mines.edu/fs_home/whereman
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.Comment: 14 pages, LaTeX, submitted to Computer Physics Communications.
Soft-package (On-Line Help) and sample MapleV session available at:
http://dft.if.uerj.br/symbcomp.htm or ftp://dft.if.uerj.br/pdetool
Symbolic Software for the Painleve Test of Nonlinear Ordinary and Partial Differential Equations
The automation of the traditional Painleve test in Mathematica is discussed.
The package PainleveTest.m allows for the testing of polynomial systems of
ordinary and partial differential equations which may be parameterized by
arbitrary functions (or constants). Except where limited by memory, there is no
restriction on the number of independent or dependent variables. The package is
quite robust in determining all the possible dominant behaviors of the Laurent
series solutions of the differential equation. The omission of valid dominant
behaviors is a common problem in many implementations of the Painleve test, and
these omissions often lead to erroneous results. Finally, our package is
compared with the other available implementations of the Painleve test.Comment: Published in the Journal of Nonlinear Mathematical Physics
(http://www.sm.luth.se/math/JNMP/), vol. 13(1), pp. 90-110 (Feb. 2006). The
software can be downloaded at either http://www.douglasbaldwin.com or
http://www.mines.edu/fs_home/wherema
Finding Liouvillian first integrals of rational ODEs of any order in finite terms
It is known, due to Mordukhai-Boltovski, Ritt, Prelle, Singer, Christopher
and others, that if a given rational ODE has a Liouvillian first integral then
the corresponding integrating factor of the ODE must be of a very special form
of a product of powers and exponents of irreducible polynomials. These results
lead to a partial algorithm for finding Liouvillian first integrals. However,
there are two main complications on the way to obtaining polynomials in the
integrating factor form. First of all, one has to find an upper bound for the
degrees of the polynomials in the product above, an unsolved problem, and then
the set of coefficients for each of the polynomials by the
computationally-intensive method of undetermined parameters. As a result, this
approach was implemented in CAS only for first and relatively simple second
order ODEs. We propose an algebraic method for finding polynomials of the
integrating factors for rational ODEs of any order, based on examination of the
resultants of the polynomials in the numerator and the denominator of the
right-hand side of such equation. If both the numerator and the denominator of
the right-hand side of such ODE are not constants, the method can determine in
finite terms an explicit expression of an integrating factor if the ODE permits
integrating factors of the above mentioned form and then the Liouvillian first
integral. The tests of this procedure based on the proposed method, implemented
in Maple in the case of rational integrating factors, confirm the consistence
and efficiency of the method.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
- …