66,176 research outputs found
A novel collocation method based on residual error analysis for solving integro-differential equations using hybrid Dickson and Taylor polynomials
In this study, a novel matrix method based on collocation points is proposed to solve some linear and nonlinear integro-differential equations with variable coefficients under the mixed conditions. The solutions are obtained by means of Dickson and Taylor polynomials. The presented method transforms the equation and its conditions into matrix equations which comply with a system of linear algebraic equations with unknown Dickson coefficients, via collocation points in a finite interval. While solving the matrix equation, the Dickson coefficients and the polynomial approximation are obtained. Besides, the residual error analysis for our method is presented and illustrative examples are given to demonstrate the validity and applicability of the method
Solutions modulo of Gauss-Manin differential equations for multidimensional hypergeometric integrals and associated Bethe ansatz
We consider the Gauss-Manin differential equations for hypergeometric
integrals associated with a family of weighted arrangements of hyperplanes
moving parallelly to themselves. We reduce these equations modulo a prime
integer and construct polynomial solutions of the new differential
equations as -analogs of the initial hypergeometric integrals.
In some cases we interpret the -analogs of the hypergeometric integrals as
sums over points of hypersurfaces defined over the finite field . That
interpretation is similar to the interpretation by Yu.I. Manin in [Ma] of the
number of point on an elliptic curve depending on a parameter as a solution of
a classical hypergeometric differential equation.
We discuss the associated Bethe ansatz.Comment: Latex, 19 pages, v2: misprints correcte
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
Taylor expansions and Castell estimates for solutions of stochastic differential equations driven by rough paths
We study the Taylor expansion for the solutions of differential equations
driven by -rough paths with . We prove a general theorem concerning the
convergence of the Taylor expansion on a nonempty interval provided that the
vector fields are analytic on a ball centered at the initial point. We also
derive criteria that enable us to study the rate of convergence of the Taylor
expansion. Finally and this is also the main and the most original part of this
paper, we prove Castell expansions and tail estimates with exponential decays
for the remainder terms of the solutions of the stochastic differential
equations driven by continuous centered Gaussian process with finite
variation and fractional Brownian motion with Hurst parameter
.Comment: Final version for publis
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