29 research outputs found
Inverse Nodal Problem for a Conformable Fractional Diffusion Operator
In this paper, a diffusion operator including conformable fractional
derivatives of order {\alpha} ({\alpha} in (0,1)) is considered. The
asymptotics of the eigenvalues, eigenfunctions and nodal points of the operator
are obtained. Furthermore, an effective procedure for solving the inverse nodal
problem is given
Trace Formulas for a Conformable Fractional Diffusion Operator
In this paper, the regularized trace formulas for a diffusion operator which
include conformable fractional derivatives of order {\alpha} (0<{\alpha \leq
1}) is obtained.Comment: 12 page
IMPULSIVE STURM-LIOUVILLE PROBLEMS ON TIME SCALES
In this paper, we consider an impulsive Sturm-Lioville problem on Sturmian time scales. We investigate the existence and uniqueness of the solution of this problem. We study some spectral properties and self-adjointness of the boundary-value problem. Later, we construct the Green function for this problem. Finally, an eigenfunction expansion is obtained
On generalized and fractional derivatives and their applications to classical mechanics
(Draft 3) A generalized differential operator on the real line is defined by
means of a limiting process. These generalized derivatives include, as a
special case, the classical derivative and current studies of fractional
differential operators. All such operators satisfy properties such as the sum,
product/quotient rules, chain rule, etc. We study a Sturm-Liouville eigenvalue
problem with generalized derivatives and show that the general case is actually
a consequence of standard Sturm-Liouville Theory. As an application of the
developments herein we find the general solution of a generalized harmonic
oscillator. We also consider the classical problem of a planar motion under a
central force and show that the general solution of this problem is still
generically an ellipse, and that this result is true independently of the
choice of the generalized derivatives being used modulo a time shift. The
previous result on the generic nature of phase plane orbits is extended to the
classical gravitational n-body problem of Newton to show that the global nature
of these orbits is independent of the choice of the generalized derivatives
being used in defining the force law (modulo a time shift). Finally,
restricting the generalized derivatives to a special class of fractional
derivatives, we consider the question of motion under gravity with and without
resistance and arrive at a new notion of time that depends on the fractional
parameter. The results herein are meant to clarify and extend many known
results in the literature and intended to show the limitations and use of
generalized derivatives and corresponding fractional derivatives.Comment: Final pre-publication draf
Solving conformable Gegenbauer differential equation and exploring its generating function
In this manuscript, we address the resolution of conformable Gegenbauer
differential equations. We demonstrate that our solution aligns precisely with
the results obtained through the power series approach. Furthermore, we delve
into the investigation and validation of various properties and recursive
relationships associated with Gegenbauer functions. Additionally, we introduce
and substantiate the conformable Rodriguez's formula and generating functio
Applied Mathematics and Fractional Calculus
In the last three decades, fractional calculus has broken into the field of mathematical analysis, both at the theoretical level and at the level of its applications. In essence, the fractional calculus theory is a mathematical analysis tool applied to the study of integrals and derivatives of arbitrary order, which unifies and generalizes the classical notions of differentiation and integration. These fractional and derivative integrals, which until not many years ago had been used in purely mathematical contexts, have been revealed as instruments with great potential to model problems in various scientific fields, such as: fluid mechanics, viscoelasticity, physics, biology, chemistry, dynamical systems, signal processing or entropy theory. Since the differential and integral operators of fractional order are nonlinear operators, fractional calculus theory provides a tool for modeling physical processes, which in many cases is more useful than classical formulations. This is why the application of fractional calculus theory has become a focus of international academic research. This Special Issue "Applied Mathematics and Fractional Calculus" has published excellent research studies in the field of applied mathematics and fractional calculus, authored by many well-known mathematicians and scientists from diverse countries worldwide such as China, USA, Canada, Germany, Mexico, Spain, Poland, Portugal, Iran, Tunisia, South Africa, Albania, Thailand, Iraq, Egypt, Italy, India, Russia, Pakistan, Taiwan, Korea, Turkey, and Saudi Arabia