95,721 research outputs found
The Lie algebra of infinitesimal symmetries of nonlinear diffusion equations
By using developed software for solving overdetermined systems of partial differential equations, the authors establish the complete Lie algebra of infinitesimal symmetries of nonlinear diffusion equations
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
Symbolic computation of hyperbolic tangent solutions for nonlinear differential-difference equations
A new algorithm is presented to find exact traveling wave solutions of
differential-difference equations in terms of tanh functions. For systems with
parameters, the algorithm determines the conditions on the parameters so that
the equations might admit polynomial solutions in tanh.
Examples illustrate the key steps of the algorithm. Parallels are drawn
through discussion and example to the tanh-method for partial differential
equations.
The new algorithm is implemented in Mathematica. The package
DDESpecialSolutions.m can be used to automatically compute traveling wave
solutions of nonlinear polynomial differential-difference equations. Use of the
package, implementation issues, scope, and limitations of the software are
addressed.Comment: 19 pages submitted to Computer Physics Communications. The software
can be downloaded at http://www.mines.edu/fs_home/wherema
Exp-Function Method for Finding Exact Solutions of Nonlinear Evolution Equations
We applied Exp-function method to some nonlinear evolution equations to obtain its exact solution. The solution procedure of this method, by the help of symbolic computation of mathematical software, is of utter simplicity. The prominent merit of this method is to facilitate the process of solving systems of partial differential equations. These methods are straightforward and concise by themselves; moreover their applications are promising to obtain exact solutions of various partial differential equations. The obtained results show that Exp-function method is very powerful and convenient mathematical tool for nonlinear evolution equations in science and engineering
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
Composing Scalable Nonlinear Algebraic Solvers
Most efficient linear solvers use composable algorithmic components, with the
most common model being the combination of a Krylov accelerator and one or more
preconditioners. A similar set of concepts may be used for nonlinear algebraic
systems, where nonlinear composition of different nonlinear solvers may
significantly improve the time to solution. We describe the basic concepts of
nonlinear composition and preconditioning and present a number of solvers
applicable to nonlinear partial differential equations. We have developed a
software framework in order to easily explore the possible combinations of
solvers. We show that the performance gains from using composed solvers can be
substantial compared with gains from standard Newton-Krylov methods.Comment: 29 pages, 14 figures, 13 table
Symbolic computation of conservation laws for nonlinear partial differential equations in multiple space dimensions
A method for symbolically computing conservation laws of nonlinear partial
differential equations (PDEs) in multiple space dimensions is presented in the
language of variational calculus and linear algebra. The steps of the method
are illustrated using the Zakharov-Kuznetsov and Kadomtsev-Petviashvili
equations as examples. The method is algorithmic and has been implemented in
Mathematica. The software package, ConservationLawsMD.m, can be used to
symbolically compute and test conservation laws for polynomial PDEs that can be
written as nonlinear evolution equations. The code ConservationLawsMD.m has
been applied to (2+1)-dimensional versions of the Sawada-Kotera, Camassa-Holm,
and Gardner equations, and the multi-dimensional Khokhlov-Zabolotskaya
equation.Comment: 26 pages. Paper will appear in Journal of Symbolic Computation
(2011). Presented at the Special Session on Geometric Flows, Moving Frames
and Integrable Systems, 2010 Spring Central Sectional Meeting of the American
Mathematical Society, Macalester College, St. Paul, Minnesota, April 10, 201
Automating the parallel processing of fluid and structural dynamics calculations
The NASA Lewis Research Center is actively involved in the development of expert system technology to assist users in applying parallel processing to computational fluid and structural dynamic analysis. The goal of this effort is to eliminate the necessity for the physical scientist to become a computer scientist in order to effectively use the computer as a research tool. Programming and operating software utilities have previously been developed to solve systems of ordinary nonlinear differential equations on parallel scalar processors. Current efforts are aimed at extending these capabilities to systems of partial differential equations, that describe the complex behavior of fluids and structures within aerospace propulsion systems. This paper presents some important considerations in the redesign, in particular, the need for algorithms and software utilities that can automatically identify data flow patterns in the application program and partition and allocate calculations to the parallel processors. A library-oriented multiprocessing concept for integrating the hardware and software functions is described
- …