180,859 research outputs found
Improved estimates for nonoscillatory phase functions
Recently, it was observed that solutions of a large class of highly
oscillatory second order linear ordinary differential equations can be
approximated using nonoscillatory phase functions. In particular, under mild
assumptions on the coefficients and wavenumber of the equation, there
exists a function whose Fourier transform decays as and
which represents solutions of the differential equation with accuracy on the
order of . In this article, we establish an
improved existence theorem for nonoscillatory phase functions. Among other
things, we show that solutions of second order linear ordinary differential
equations can be represented with accuracy on the order of using functions in the space of rapidly decaying Schwartz
functions whose Fourier transforms are both exponentially decaying and
compactly supported. These new observations play an important role in the
analysis of a method for the numerical solution of second order ordinary
differential equations whose running time is independent of the parameter
. This algorithm will be reported at a later date.Comment: arXiv admin note: text overlap with arXiv:1409.438
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
Fast computation of power series solutions of systems of differential equations
We propose new algorithms for the computation of the first N terms of a
vector (resp. a basis) of power series solutions of a linear system of
differential equations at an ordinary point, using a number of arithmetic
operations which is quasi-linear with respect to N. Similar results are also
given in the non-linear case. This extends previous results obtained by Brent
and Kung for scalar differential equations of order one and two
Transformations of ordinary differential equations via Darboux transformation technique
A new approach for obtaining the transformations of solutions of nonlinear
ordinary differential equations representable as the compatibility condition of
the overdetermined linear systems is proposed. The corresponding
transformations of the solutions of the overdetermined linear systems are
derived in the frameworks of the Darboux transformation technique.Comment: 7 pages, LaTeX2
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