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

Kepler dynamics on a conformable Poisson manifold

The problem of Kepler dynamics on a conformable Poisson manifold is addressed. The Hamiltonian function is defined and the related Hamiltonian vector field governing the dynamics is derived, which leads to a modified Newton second law. Conformable momentum and Laplace-Runge-Lenz vectors are considered, generating $SO(3), SO(4),$ and $SO(1, 3)$ dynamical symmetry groups. The corresponding first Casimir operators of $SO(4)$ and $SO(1, 3)$ are, respectively, obtained. The recursion operators are constructed and used to compute the integrals of motion in action-angle coordinates. Main relevant properties are deducted and discussed

Certain Developments of Laguerre Equation and Laguerre Polynomials via Fractional Calculus

Recently, much interests have been paid in studying fractional calculus due to its effectiveness in modeling many of the natural phenomena. Motivated essentially by the success of the applications of the orthogonal polynomials, this paper is mainly devoted to developing Laguerre equation and Laguerre polynomials in the fractional calculus setting. We provide some type of generalizations of the classical Laguerre polynomials, via conformable fractional calculus. We start by solving the fractional Laguerre equation in the sense of conformable calculus about the fractional regular singular point. Next, we write the conformable fractional Laguerre polynomials (CFLPs), through various generating functions. Subsequently, Rodrigues’ type representation formula of fractional order is reported, besides certain types of recurrence relations are then developed. The conformable fractional integral and the fractional Laplace transform, and the orthogonal property of CFLPs, are established. As an application, we present a numerical technique to obtain solutions of interesting differential equations in the frame of conformable derivative. For this purpose, a new operational matrix of the fractional derivative of arbitrary order for CFLPs is derived. This operational matrix is applied together with the generalized Laguerre tau method for solving general linear multi-term fractional differential equations (FDEs). The method has the advantage of obtaining the solution in terms of the CFLPs’ parameters. Finally, some examples are given to illustrate the applicability and efficiency of the proposed method

NOVEL METHODS FOR SOLVING THE CONFORMABLE WAVE EQUATION

In this paper, a two-dimensional conformable fractional wave equation describing a circular membrane undergoing axisymmetric vibrations is formulated. It was found that the analytical solutions of the fractional wave equation using the conformable fractional formulation can be easily and efficiently obtained using separation of variables and double Laplace transform methods. These solutions are compared with the approximate solution obtained using the differential transform method for certain cases

New structure for exact solutions of nonlinear time fractional Sharma-Tasso-Olver equation via conformable fractional derivative

In this paper new fractional derivative and direct algebraic method are used to construct exact solutions of the nonlinear time fractional Sharma-Tasso-Olver equation. As a result, three families of exact analytical solutions are obtained. The results reveal that the proposed method is very effective and simple for obtaining approximate solutions of nonlinear fractional partial differential equations

Spectral expansion for singular conformable fractional Sturm-Liouville problem

With this study, the spectral function for singular conformable fractional Sturm-Lioville problem is demonstrated. Further, we establish a Parseval equality and spectral expansion formula by terms of the spectral function

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