541,051 research outputs found
Power Series Solutions of Non-Linear q-Difference Equations and the Newton-Puiseux Polygon
Adapting the Newton-Puiseux Polygon process to nonlinear q-difference
equations of any order and degree, we compute their power series solutions,
study the properties of the set of exponents of the solutions and give a bound
for their Gevrey order in terms of the order of the original equation
Confluence of meromorphic solutions of q-difference equations
In this paper, we consider a q-analogue of the Borel-Laplace summation where
q>1 is a real parameter. In particular, we show that the Borel-Laplace
summation of a divergent power series solution of a linear differential
equation can be uniformly approximated on a convenient sector, by a meromorphic
solution of a corresponding family of linear q-difference equations. We perform
the computations for the basic hypergeometric series. Following J. Sauloy, we
prove how a fundamental set of solutions of a linear differential equation can
be uniformly approximated on a convenient domain by a fundamental set of
solutions of a corresponding family of linear q-difference equations. This
leads us to the approximations of Stokes matrices and monodromy matrices of the
linear differential equation by matrices with entries that are invariants by
the multiplication by q
The Exact Solutions of Certain Linear Partial Difference Equations
Difference equations have many applications and play an important role in
numerical analysis, probability, statistics, combinatorics, computer science,
and quantum consciousness, etc. We first prove that the partial differential
equation is equivalent to partial difference equation with an example of heat
equation. Additionally, we use generating functions to find the exact solutions
of some simple linear partial difference equations. Then we extend it to more
general partial difference equations of higher dimensions and obtain their
solutions. We conclude that using multivariable power series as generating
function is a very efficient method to solve partial difference equations
Introduction to 1-summability and resurgence
This text is about the mathematical use of certain divergent power series.
The first part is an introduction to 1-summability. The definitions rely on the
formal Borel transform and the Laplace transform along an arbitrary direction
of the complex plane. Given an arc of directions, if a power series is
1-summable in that arc, then one can attach to it a Borel-Laplace sum, i.e. a
holomorphic function defined in a large enough sector and asymptotic to that
power series in Gevrey sense. The second part is an introduction to Ecalle's
resurgence theory. A power series is said to be resurgent when its Borel
transform is convergent and has good analytic continuation properties: there
may be singularities but they must be isolated. The analysis of these
singularities, through the so-called alien calculus, allows one to compare the
various Borel-Laplace sums attached to the same resurgent 1-summable series.In
the context of analytic difference-or-differential equations, this sheds light
on the Stokes phenomenon. A few elementary or classical examples are given a
thorough treatment (the Euler series, the Stirling series, a less known example
by Poincar\'e). Special attention is devoted to non-linear operations:
1-summable series as well as resurgent series are shown to form algebras which
are stable by composition. As an application, the resurgent approach to the
classification of tangent-to-identity germs of holomorphic diffeomorphisms in
the simplest case is included. An example of a class of non-linear differential
equations giving rise to resurgent solutions is also presented. The exposition
is as self-contained as can be, requiring only some familiarity with
holomorphic functions of one complex variable.Comment: 127 page
Some notes to extend the study on random non-autonomous second order linear differential equations appearing in Mathematical Modeling
The objective of this paper is to complete certain issues from our recent
contribution [J. Calatayud, J.-C. Cort\'es, M. Jornet, L. Villafuerte, Random
non-autonomous second order linear differential equations: mean square analytic
solutions and their statistical properties, Advances in Difference Equations,
2018:392, 1--29 (2018)]. We restate the main theorem therein that deals with
the homogeneous case, so that the hypotheses are clearer and also easier to
check in applications. Another novelty is that we tackle the non-homogeneous
equation with a theorem of existence of mean square analytic solution and a
numerical example. We also prove the uniqueness of mean square solution via an
habitual Lipschitz condition that extends the classical Picard Theorem to mean
square calculus. In this manner, the study on general random non-autonomous
second order linear differential equations with analytic data processes is
completely resolved. Finally, we relate our exposition based on random power
series with polynomial chaos expansions and the random differential transform
method, being the latter a reformulation of our random Fr\"obenius method.Comment: 15 pages, 0 figures, 2 table
A toolbox to solve coupled systems of differential and difference equations
We present algorithms to solve coupled systems of linear differential
equations, arising in the calculation of massive Feynman diagrams with local
operator insertions at 3-loop order, which do {\it not} request special choices
of bases. Here we assume that the desired solution has a power series
representation and we seek for the coefficients in closed form. In particular,
if the coefficients depend on a small parameter \ep (the dimensional
parameter), we assume that the coefficients themselves can be expanded in
formal Laurent series w.r.t.\ \ep and we try to compute the first terms in
closed form. More precisely, we have a decision algorithm which solves the
following problem: if the terms can be represented by an indefinite nested
hypergeometric sum expression (covering as special cases the harmonic sums,
cyclotomic sums, generalized harmonic sums or nested binomial sums), then we
can calculate them. If the algorithm fails, we obtain a proof that the terms
cannot be represented by the class of indefinite nested hypergeometric sum
expressions. Internally, this problem is reduced by holonomic closure
properties to solving a coupled system of linear difference equations. The
underlying method in this setting relies on decoupling algorithms, difference
ring algorithms and recurrence solving. We demonstrate by a concrete example
how this algorithm can be applied with the new Mathematica package
\texttt{SolveCoupledSystem} which is based on the packages \texttt{Sigma},
\texttt{HarmonicSums} and \texttt{OreSys}. In all applications the
representation in -space is obtained as an iterated integral representation
over general alphabets, generalizing Poincar\'{e} iterated integrals
On the summability and convergence of formal solutions of linear -difference-differential equations with constant coefficients
We consider the Cauchy problem for homogeneous linear
-difference-differential equations with constant coefficients. We
characterise convergent, -summable and multisummable formal power series
solutions in terms of analytic continuation properties and growth estimates of
the Cauchy data. We also introduce and characterise sequences preserving
summability, which make a very useful tool, especially in the context of moment
differential equations.Comment: 19 page
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