13,408 research outputs found
Efficient numerical solution of the time fractional diffusion equation by mapping from its Brownian counterpart
The solution of a Caputo time fractional diffusion equation of order
is expressed in terms of the solution of a corresponding integer
order diffusion equation. We demonstrate a linear time mapping between these
solutions that allows for accelerated computation of the solution of the
fractional order problem. In the context of an -point finite difference time
discretisation, the mapping allows for an improvement in time computational
complexity from to , given a
precomputation of . The mapping is applied
successfully to the least-squares fitting of a fractional advection diffusion
model for the current in a time-of-flight experiment, resulting in a
computational speed up in the range of one to three orders of magnitude for
realistic problem sizes.Comment: 9 pages, 5 figures; added references for section
Generating functional analysis of Minority Games with real market histories
It is shown how the generating functional method of De Dominicis can be used
to solve the dynamics of the original version of the minority game (MG), in
which agents observe real as opposed to fake market histories. Here one again
finds exact closed equations for correlation and response functions, but now
these are defined in terms of two connected effective non-Markovian stochastic
processes: a single effective agent equation similar to that of the `fake'
history models, and a second effective equation for the overall market bid
itself (the latter is absent in `fake' history models). The result is an exact
theory, from which one can calculate from first principles both the persistent
observables in the MG and the distribution of history frequencies.Comment: 39 pages, 5 postscript figures, iop styl
Closed form summation of C-finite sequences
We consider sums of the form
in which each is a sequence that satisfies a linear recurrence of
degree , with constant coefficients. We assume further that the
's and the 's are all nonnegative integers. We prove that such a
sum always has a closed form, in the sense that it evaluates to a linear
combination of a finite set of monomials in the values of the sequences
with coefficients that are polynomials in . We explicitly
describe two different sets of monomials that will form such a linear
combination, and give an algorithm for finding these closed forms, thereby
completely automating the solution of this class of summation problems. We
exhibit tools for determining when these explicit evaluations are unique of
their type, and prove that in a number of interesting cases they are indeed
unique. We also discuss some special features of the case of ``indefinite
summation," in which
NumGfun: a Package for Numerical and Analytic Computation with D-finite Functions
This article describes the implementation in the software package NumGfun of
classical algorithms that operate on solutions of linear differential equations
or recurrence relations with polynomial coefficients, including what seems to
be the first general implementation of the fast high-precision numerical
evaluation algorithms of Chudnovsky & Chudnovsky. In some cases, our
descriptions contain improvements over existing algorithms. We also provide
references to relevant ideas not currently used in NumGfun
Connection problem for the sine-Gordon/Painlev\'e III tau function and irregular conformal blocks
The short-distance expansion of the tau function of the radial
sine-Gordon/Painlev\'e III equation is given by a convergent series which
involves irregular conformal blocks and possesses certain periodicity
properties with respect to monodromy data. The long-distance irregular
expansion exhibits a similar periodicity with respect to a different pair of
coordinates on the monodromy manifold. This observation is used to conjecture
an exact expression for the connection constant providing relative
normalization of the two series. Up to an elementary prefactor, it is given by
the generating function of the canonical transformation between the two sets of
coordinates.Comment: 18 pages, 1 figur
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