1,068 research outputs found
A simple fractional-calculus approach to the solutions of the Bessel differential equation of general order and some of its applications
AbstractIn many recent works, several authors demonstrated the usefulness of fractional calculus operators in the derivation of (explicit) particular solutions of a significantly large number of linear ordinary and partial differential equations of the second and higher orders. The main object of the present paper is to show how this simple fractional-calculus approach to the solutions of the classical Bessel differential equation of general order would lead naturally to several interesting consequences which include (for example) an alternative investigation of the power-series solutions obtainable usually by the Frobenius method. The methodology presented here is based largely upon some of the general theorems on (explicit) particular solutions of a certain family of linear ordinary fractional differintegral equations
Numerical Approximations to Fractional Problems of the Calculus of Variations and Optimal Control
This chapter presents some numerical methods to solve problems in the
fractional calculus of variations and fractional optimal control. Although
there are plenty of methods available in the literature, we concentrate mainly
on approximating the fractional problem either by discretizing the fractional
term or expanding the fractional derivatives as a series involving integer
order derivatives. The former method, as a subclass of direct methods in the
theory of calculus of variations, uses finite differences, Grunwald-Letnikov
definition in this case, to discretize the fractional term. Any quadrature rule
for integration, regarding the desired accuracy, is then used to discretize the
whole problem including constraints. The final task in this method is to solve
a static optimization problem to reach approximated values of the unknown
functions on some mesh points.
The latter method, however, approximates fractional problems by classical
ones in which only derivatives of integer order are present. Precisely, two
continuous approximations for fractional derivatives by series involving
ordinary derivatives are introduced. Local upper bounds for truncation errors
are provided and, through some test functions, the accuracy of the
approximations are justified. Then we substitute the fractional term in the
original problem with these series and transform the fractional problem to an
ordinary one. Hereafter, we use indirect methods of classical theory, e.g.
Euler-Lagrange equations, to solve the approximated problem. The methods are
mainly developed through some concrete examples which either have obvious
solutions or the solution is computed using the fractional Euler-Lagrange
equation.Comment: This is a preprint of a paper whose final and definite form appeared
in: Chapter V, Fractional Calculus in Analysis, Dynamics and Optimal Control
(Editor: Jacky Cresson), Series: Mathematics Research Developments, Nova
Science Publishers, New York, 2014. (See
http://www.novapublishers.com/catalog/product_info.php?products_id=46851).
Consists of 39 page
The sixth Painleve transcendent and uniformization of algebraic curves
We exhibit a remarkable connection between sixth equation of Painleve list
and infinite families of explicitly uniformizable algebraic curves. Fuchsian
equations, congruences for group transformations, differential calculus of
functions and differentials on corresponding Riemann surfaces, Abelian
integrals, analytic connections (generalizations of Chazy's equations), and
other attributes of uniformization can be obtained for these curves. As
byproducts of the theory, we establish relations between Picard-Hitchin's
curves, hyperelliptic curves, punctured tori, Heun's equations, and the famous
differential equation which Apery used to prove the irrationality of Riemann's
zeta(3).Comment: Final version. Numerous improvements; English, 49 pages, 1 table, no
figures, LaTe
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