40,590 research outputs found
Status of the differential transformation method
Further to a recent controversy on whether the differential transformation
method (DTM) for solving a differential equation is purely and solely the
traditional Taylor series method, it is emphasized that the DTM is currently
used, often only, as a technique for (analytically) calculating the power
series of the solution (in terms of the initial value parameters). Sometimes, a
piecewise analytic continuation process is implemented either in a numerical
routine (e.g., within a shooting method) or in a semi-analytical procedure
(e.g., to solve a boundary value problem). Emphasized also is the fact that, at
the time of its invention, the currently-used basic ingredients of the DTM
(that transform a differential equation into a difference equation of same
order that is iteratively solvable) were already known for a long time by the
"traditional"-Taylor-method users (notably in the elaboration of software
packages --numerical routines-- for automatically solving ordinary differential
equations). At now, the defenders of the DTM still ignore the, though much
better developed, studies of the "traditional"-Taylor-method users who, in
turn, seem to ignore similarly the existence of the DTM. The DTM has been given
an apparent strong formalization (set on the same footing as the Fourier,
Laplace or Mellin transformations). Though often used trivially, it is easily
attainable and easily adaptable to different kinds of differentiation
procedures. That has made it very attractive. Hence applications to various
problems of the Taylor method, and more generally of the power series method
(including noninteger powers) has been sketched. It seems that its potential
has not been exploited as it could be. After a discussion on the reasons of the
"misunderstandings" which have caused the controversy, the preceding topics are
concretely illustrated.Comment: To appear in Applied Mathematics and Computation, 29 pages,
references and further considerations adde
Making big steps in trajectories
We consider the solution of initial value problems within the context of
hybrid systems and emphasise the use of high precision approximations (in
software for exact real arithmetic). We propose a novel algorithm for the
computation of trajectories up to the area where discontinuous jumps appear,
applicable for holomorphic flow functions. Examples with a prototypical
implementation illustrate that the algorithm might provide results with higher
precision than well-known ODE solvers at a similar computation time
On universality of critical behaviour in Hamiltonian PDEs
Our main goal is the comparative study of singularities of solutions to the
systems of first order quasilinear PDEs and their perturbations containing
higher derivatives. The study is focused on the subclass of Hamiltonian PDEs
with one spatial dimension. For the systems of order one or two we describe the
local structure of singularities of a generic solution to the unperturbed
system near the point of "gradient catastrophe" in terms of standard objects of
the classical singularity theory; we argue that their perturbed companions must
be given by certain special solutions of Painleve' equations and their
generalizations.Comment: 59 pages, 2 figures. Amer. Math. Soc. Transl., to appea
Regular polynomial interpolation and approximation of global solutions of linear partial differential equations
We consider regular polynomial interpolation algorithms on recursively
defined sets of interpolation points which approximate global solutions of
arbitrary well-posed systems of linear partial differential equations.
Convergence of the 'limit' of the recursively constructed family of
polynomials to the solution and error estimates are obtained from a priori
estimates for some standard classes of linear partial differential equations,
i.e. elliptic and hyperbolic equations. Another variation of the algorithm
allows to construct polynomial interpolations which preserve systems of linear
partial differential equations at the interpolation points. We show how this
can be applied in order to compute higher order terms of WKB-approximations of
fundamental solutions of a large class of linear parabolic equations. The error
estimates are sensitive to the regularity of the solution. Our method is
compatible with recent developments for solution of higher dimensional partial
differential equations, i.e. (adaptive) sparse grids, and weighted Monte-Carlo,
and has obvious applications to mathematical finance and physics.Comment: 28 page
Source Galerkin Calculations in Scalar Field Theory
In this paper, we extend previous work on scalar theory using the
Source Galerkin method. This approach is based on finding solutions to
the lattice functional equations for field theories in the presence of an
external source . Using polynomial expansions for the generating functional
, we calculate propagators and mass-gaps for a number of systems. These
calculations are straightforward to perform and are executed rapidly compared
to Monte Carlo. The bulk of the computation involves a single matrix inversion.
The use of polynomial expansions illustrates in a clear and simple way the
ideas of the Source Galerkin method. But at the same time, this choice has
serious limitations. Even after exploiting symmetries, the size of calculations
become prohibitive except for small systems. The calculations in this paper
were made on a workstation of modest power using a fourth order polynomial
expansion for lattices of size ,, in , , and . In
addition, we present an alternative to the Galerkin procedure that results in
sparse matrices to invert.Comment: 31 pages, latex, figures separat
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