307,433 research outputs found
Hopf algebras in dynamical systems theory
The theory of exact and of approximate solutions for non-autonomous linear
differential equations forms a wide field with strong ties to physics and
applied problems. This paper is meant as a stepping stone for an exploration of
this long-established theme, through the tinted glasses of a (Hopf and
Rota-Baxter) algebraic point of view. By reviewing, reformulating and
strengthening known results, we give evidence for the claim that the use of
Hopf algebra allows for a refined analysis of differential equations. We
revisit the renowned Campbell-Baker-Hausdorff-Dynkin formula by the modern
approach involving Lie idempotents. Approximate solutions to differential
equations involve, on the one hand, series of iterated integrals solving the
corresponding integral equations; on the other hand, exponential solutions.
Equating those solutions yields identities among products of iterated Riemann
integrals. Now, the Riemann integral satisfies the integration-by-parts rule
with the Leibniz rule for derivations as its partner; and skewderivations
generalize derivations. Thus we seek an algebraic theory of integration, with
the Rota-Baxter relation replacing the classical rule. The methods to deal with
noncommutativity are especially highlighted. We find new identities, allowing
for an extensive embedding of Dyson-Chen series of time- or path-ordered
products (of generalized integration operators); of the corresponding Magnus
expansion; and of their relations, into the unified algebraic setting of
Rota-Baxter maps and their inverse skewderivations. This picture clarifies the
approximate solutions to generalized integral equations corresponding to
non-autonomous linear (skew)differential equations.Comment: International Journal of Geometric Methods in Modern Physics, in
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Solution Properties for Pertubed Linear and Nonlinear Integrals Equations
In this study we consider perturbative series solution with respect to a
parameter {\epsilon} > 0. In this methodology the solution is considered as an
infinite sum of a series of functional terms which usually converges fast to
the exact desired solution. Then we investigate perturbative solutions for
kernel perturbed integral equations and prove the convergence in an appropriate
ranges of the perturbation series. Next we investigate perturbation series
solutions for nonlinear perturbations of integral equations of Hammerstein type
and formulate conditions for their convergence. Finally we prove the existence
of a maximal perturbation range for non linear integral equations
Heun Functions and Some of Their Applications in Physics
Most of the theoretical physics known today is described by using a small
number of differential equations. For linear systems, different forms of the
hypergeometric or the confluent hypergeometric equations often suffice to
describe the system studied. These equations have power series solutions with
simple relations between consecutive coefficients and/ or can be represented in
terms of simple integral transforms. If the problem is nonlinear, one often
uses one form of the Painlev\'{e} equations. There are important examples,
however, where one has to use higher order equations. Heun equation is one of
these examples, which recently is often encountered in problems in general
relativity and astrophysics. Its special and confluent forms take names as
Mathieu, Lam\'{e} and Coulomb spheroidal equations. For these equations
whenever a power series solution is written, instead of a two-way recursion
relation between the coefficients in the series, we find one between three or
four different ones. An integral transform solution using simpler functions
also is not obtainable. The use of this equation in physics and mathematical
literature exploded in the later years, more than doubling the number of papers
with these solutions in the last decade, compared to time period since this
equation was introduced in 1889 up to 2008. We use SCI data to conclude this
statement, which is not precise, but in the correct ballpark. Here this
equation will be introduced and examples for its use, especially in general
relativity literature will be given.Comment: 19 pages. Submitted version to journal Adv.High Energy Physics. An
earlier version of the paper was published in "Proceedings of the 13th
Regional Conference on Mathematical Physics, Antalya, Turkey, October 27-31,
2010", Edited by Ugur Camci and Ibrahim Semiz, pp. 23-39. World Scientific
(2013) (DOI: 10.1142/9789814417532_0002
-Module Techniques for Solving Differential Equations in the Context of Feynman Integrals
Feynman integrals are solutions to linear partial differential equations with
polynomial coefficients. Using a triangle integral with general exponents as a
case in point, we compare -module methods to dedicated methods developed for
solving differential equations appearing in the context of Feynman integrals,
and provide a dictionary of the relevant concepts. In particular, we implement
an algorithm due to Saito, Sturmfels, and Takayama to derive canonical series
solutions of regular holonomic -ideals, and compare them to asymptotic
series derived by the respective Fuchsian systems.Comment: 35 pages, 2 figures, 2 appendices; comments welcom
The Reproducing Kernel Hilbert Space Method for Solving System of Linear Weakly Singular Volterra Integral Equations
The exact solutions of a system of linear weakly singular Volterra integral equations (VIE) have been a difficult to find. The aim of this paper is to apply reproducing kernel Hilbert space (RKHS) method to find the approximate solutions to this type of systems. At first, we used Taylor's expansion to omit the singularity. From an expansion the given system of linear weakly singular VIE is transform into a system of linear ordinary differential equations (LODEs). The approximate solutions are represent in the form of series in the reproducing kernel space . By comparing with the exact solutions of two examples, we saw that RKHS is a powerful, easy to apply and full efficiency in scientific applications to build a solution without linearization and turbulence or discretization. 
On the procedure for the series solution of second-order homogeneous linear differential equation via the complex integration method
The theory of series solutions for second-order linear homogeneous ordinary differential equation is developed ab initio, using an elementary complex integral expression (based on Herrera’ work [3]) derived and applied in previous papers [8, 9]. As well as reproducing the usual expression for the recurrence relations for second-order equations, the general solution method is straight-forward to apply as an algorithm on its own, with the integral algorithm replacing the manipulation of power series by reducing the task of finding a series solution for second-order equations to the solution, instead, of a system of uncoupled simple equations in a single unknown. The integral algorithm also simplifies the construction of ‘logarithmic solutions’ to second-order Fuchs, equations. Examples, from the general science and mathematics literature, are presented throughout
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