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

A novel approach to fractional calculus: utilizing fractional integrals and derivatives of the Dirac delta function

While the definition of a fractional integral may be codified by Riemann and Liouville, an agreed-upon fractional derivative has eluded discovery for many years. This is likely a result of integral definitions including numerous constants of integration in their results. An elimination of constants of integration opens the door to an operator that reconciles all known fractional derivatives and shows surprising results in areas unobserved before, including the appearance of the Riemann Zeta Function and fractional Laplace and Fourier Transforms. A new class of functions, known as Zero Functions and closely related to the Dirac Delta Function, are necessary for one to perform elementary operations of functions without using constants. The operator also allows for a generalization of the Volterra integral equation, and provides a method of solving for Riemann's "complimentary" function introduced during his research on fractional derivatives

Conformable Derivative Operator in Modelling Neuronal Dynamics

This study presents two new numerical techniques for solving time-fractional one-dimensional cable differential equation (FCE) modeling neuronal dynamics. We have introduced new formulations for the approximate-analytical solution of the FCE by using modified homotopy perturbation method defined with conformable operator (MHPMC) and reduced differential transform method defined with conformable operator (RDTMC), which are derived the solutions for linear-nonlinear fractional PDEs. In order to show the efficiencies of these methods, we have compared the numerical and exact solutions of fractional neuronal dynamics problem. Moreover, we have declared that the proposed models are very accurate and illustrative techniques in determining to approximate-analytical solutions for the PDEs of fractional order in conformable sense

On the generalized Laplace transform

This article belongs to the Special Issue Discrete and Fractional Mathematics: Symmetry and Applications.In this paper we introduce a generalized Laplace transform in order to work with a very general fractional derivative, and we obtain the properties of this new transform. We also include the corresponding convolution and inverse formula. In particular, the definition of convolution for this generalized Laplace transform improves previous results. Additionally, we deal with the generalized harmonic oscillator equation, showing that this transform and its properties allow one to solve fractional differential equations.We would like to thank the referees for their comments, which have improved the paper. The research of José M. Rodríguez and José M. Sigarreta was supported by a grant from Agencia Estatal de Investigación (PID2019-106433GB-I00/AEI/10.13039/501100011033), Spain. The research of José M. Rodríguez is supported by the Madrid Government (Comunidad de Madrid-Spain) under the Multiannual Agreement with UC3M in the line of Excellence of University Professors (EPUC3M23), and in the context of the V PRICIT (Regional Programme of Research and Technological Innovation)

A New Approach in Solving Regular and Singular Conformable Fractional Coupled Burger's Equations

The conformable double ARA decomposition approach is presented in this current study to solve one-dimensional regular and singular conformable functional Burger's equations. We investigate the conformable double ARA transform's definition, existence requirements, and some basic properties. In this study, we introduce a novel interesting method that combines the double ARA transform with Adomian's decomposition method, in order to find the precise solutions of some nonlinear fractional problems. Moreover, we use the new approach to solve Burgers' equations for both regular and singular conformable fractional coupled systems. We also provide several instances to demonstrate the usefulness of the current study. Mathematica software has been used to get numerical results

Approximate and Exact Solutions in the Sense of Conformable Derivatives of Quantum Mechanics Models Using a Novel Algorithm

The entirety of the information regarding a subatomic particle is encoded in a wave function. Solving quantum mechanical models (QMMs) means finding the quantum mechanical wave function. Therefore, great attention has been paid to finding solutions for QMMs. In this study, a novel algorithm that combines the conformable Shehu transform and the Adomian decomposition method is presented that establishes approximate and exact solutions to QMMs in the sense of conformable derivatives with zero and nonzero trapping potentials. This solution algorithm is known as the conformable Shehu transform decomposition method (CSTDM). To evaluate the efficiency of this algorithm, the numerical results in terms of absolute and relative errors were compared with the reduced differential transform and the two-dimensional differential transform methods. The comparison showed excellent agreement with these methods, which means that the CSTDM is a suitable alternative tool to the methods based on the Caputo derivative for the solutions of time-fractional QMMs. The advantage of employing this approach is that, due to the use of the conformable Shehu transform, the pattern between the coefficients of the series solutions makes it simple to obtain the exact solution of both linear and nonlinear problems. Consequently, our approach is quick, accurate, and easy to implement. The convergence, uniqueness, and error analysis of the solution were examined using Banach’s fixed point theory

New solution of conformable Fornberg-Whitham differential equation via conformable sumudu decomposition method

In this work, a new analytical method called the conformable Sumudu decomposition method is introduced to obtain approximate solutions of fractional FornbergWhitham differential equation. The proposed method is a combination between conformable Sumudu integral transform and the Adomian decomposition method. The fractional derivatives are taken in terms of the conformable sense. In order to demonstrate the applicability, efficiency and simplicity of the presented method, we compare the behavior of the obtained approximate solutions with the exact solution given in the literature.Publisher's Versio

Solutions of systems with the Caputo-Fabrizio fractional delta derivative on time scales

Caputo-Fabrizio fractional delta derivatives on an arbitrary time scale are presented. When the time scale is chosen to be the set of real numbers, then the Caputo-Fabrizio fractional derivative is recovered. For isolated or partly continuous and partly discrete, i.e., hybrid time scales, one gets new fractional operators. We concentrate on the behavior of solutions to initial value problems with the Caputo-Fabrizio fractional delta derivative on an arbitrary time scale. In particular, the exponential stability of linear systems is studied. A necessary and sufficient condition for the exponential stability of linear systems with the Caputo-Fabrizio fractional delta derivative on time scales is presented. By considering a suitable fractional dynamic equation and the Laplace transform on time scales, we also propose a proper definition of Caputo-Fabrizio fractional integral on time scales. Finally, by using the Banach fixed point theorem, we prove existence and uniqueness of solution to a nonlinear initial value problem with the Caputo-Fabrizio fractional delta derivative on time scales.Comment: This is a preprint of a paper whose final and definite form is with 'Nonlinear Analysis: Hybrid Systems', ISSN: 1751-570X, available at [http://www.journals.elsevier.com/nonlinear-analysis-hybrid-systems]. Submitted 21/May/2018; Revised 07/Oct/2018; Accepted for publication 01-Dec-201