14,514 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
Thomas decompositions of parametric nonlinear control systems
This paper presents an algorithmic method to study structural properties of
nonlinear control systems in dependence of parameters. The result consists of a
description of parameter configurations which cause different control-theoretic
behaviour of the system (in terms of observability, flatness, etc.). The
constructive symbolic method is based on the differential Thomas decomposition
into disjoint simple systems, in particular its elimination properties
Differentiable Genetic Programming
We introduce the use of high order automatic differentiation, implemented via
the algebra of truncated Taylor polynomials, in genetic programming. Using the
Cartesian Genetic Programming encoding we obtain a high-order Taylor
representation of the program output that is then used to back-propagate errors
during learning. The resulting machine learning framework is called
differentiable Cartesian Genetic Programming (dCGP). In the context of symbolic
regression, dCGP offers a new approach to the long unsolved problem of constant
representation in GP expressions. On several problems of increasing complexity
we find that dCGP is able to find the exact form of the symbolic expression as
well as the constants values. We also demonstrate the use of dCGP to solve a
large class of differential equations and to find prime integrals of dynamical
systems, presenting, in both cases, results that confirm the efficacy of our
approach
Algorithmic Integrability Tests for Nonlinear Differential and Lattice Equations
Three symbolic algorithms for testing the integrability of polynomial systems
of partial differential and differential-difference equations are presented.
The first algorithm is the well-known Painlev\'e test, which is applicable to
polynomial systems of ordinary and partial differential equations. The second
and third algorithms allow one to explicitly compute polynomial conserved
densities and higher-order symmetries of nonlinear evolution and lattice
equations.
The first algorithm is implemented in the symbolic syntax of both Macsyma and
Mathematica. The second and third algorithms are available in Mathematica. The
codes can be used for computer-aided integrability testing of nonlinear
differential and lattice equations as they occur in various branches of the
sciences and engineering. Applied to systems with parameters, the codes can
determine the conditions on the parameters so that the systems pass the
Painlev\'e test, or admit a sequence of conserved densities or higher-order
symmetries.Comment: Submitted to: Computer Physics Communications, Latex, uses the style
files elsart.sty and elsart12.st
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