2,070 research outputs found
Meromorphic solutions of nonlinear ordinary differential equations
Exact solutions of some popular nonlinear ordinary differential equations are
analyzed taking their Laurent series into account. Using the Laurent series for
solutions of nonlinear ordinary differential equations we discuss the nature of
many methods for finding exact solutions. We show that most of these methods
are conceptually identical to one another and they allow us to have only the
same solutions of nonlinear ordinary differential equations
Order reduction methods for solving large-scale differential matrix Riccati equations
We consider the numerical solution of large-scale symmetric differential
matrix Riccati equations. Under certain hypotheses on the data, reduced order
methods have recently arisen as a promising class of solution strategies, by
forming low-rank approximations to the sought after solution at selected
timesteps. We show that great computational and memory savings are obtained by
a reduction process onto rational Krylov subspaces, as opposed to current
approaches. By specifically addressing the solution of the reduced differential
equation and reliable stopping criteria, we are able to obtain accurate final
approximations at low memory and computational requirements. This is obtained
by employing a two-phase strategy that separately enhances the accuracy of the
algebraic approximation and the time integration. The new method allows us to
numerically solve much larger problems than in the current literature.
Numerical experiments on benchmark problems illustrate the effectiveness of the
procedure with respect to existing solvers
Fuchs versus Painlev\'e
We briefly recall the Fuchs-Painlev\'e elliptic representation of Painlev\'e
VI. We then show that the polynomiality of the expressions of the correlation
functions (and form factors) in terms of the complete elliptic integral of the
first and second kind,
and , is a straight consequence of the fact that the differential
operators corresponding to the entries of Toeplitz-like determinants, are
equivalent to the second order operator which has as solution (or,
for off-diagonal correlations to the direct sum of and ). We show
that this can be generalized, mutatis mutandis, to the anisotropic Ising model.
The singled-out second order linear differential operator being replaced
by an isomonodromic system of two third-order linear partial differential
operators associated with , the Jacobi's form of the complete elliptic
integral of the third kind (or equivalently two second order linear partial
differential operators associated with Appell functions, where one of these
operators can be seen as a deformation of ). We finally explore the
generalizations, to the anisotropic Ising models, of the links we made, in two
previous papers, between Painlev\'e non-linear ODE's, Fuchsian linear ODE's and
elliptic curves. In particular the elliptic representation of Painlev\'e VI has
to be generalized to an ``Appellian'' representation of Garnier systems.Comment: Dedicated to the : Special issue on Symmetries and Integrability of
Difference Equations, SIDE VII meeting held in Melbourne during July 200
New contiguity relation of the sixth Painlev\'e equation from a truncation
For the master Painlev\'e equation P6(u), we define a consistent method,
adapted from the Weiss truncation for partial differential equations, which
allows us to obtain the first degree birational transformation of Okamoto. Two
new features are implemented to achieve this result. The first one is the
homography between the derivative of the solution and a Riccati
pseudopotential. The second one is an improvement of a conjecture by Fokas and
Ablowitz on the structure of this birational transformation. We then build the
contiguity relation of P6, which yields one new second order nonautonomous
discrete equation.Comment: LaTex 2e. To appear, Physica
Formulas for Continued Fractions. An Automated Guess and Prove Approach
We describe a simple method that produces automatically closed forms for the
coefficients of continued fractions expansions of a large number of special
functions. The function is specified by a non-linear differential equation and
initial conditions. This is used to generate the first few coefficients and
from there a conjectured formula. This formula is then proved automatically
thanks to a linear recurrence satisfied by some remainder terms. Extensive
experiments show that this simple approach and its straightforward
generalization to difference and -difference equations capture a large part
of the formulas in the literature on continued fractions.Comment: Maple worksheet attache
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