Particular Solutions of the Confluent Hypergeometric Differential Equation by Using the Nabla Fractional Calculus Operator

Resat Yilmazer,Mustafa Inc,Dumitru Baleanu,Fairouz Tchier

doi:10.3390/e18020049

Resat Yilmazer, Mustafa Inc + Show 2 more

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https://doi.org/10.3390/e18020049

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Journal: Entropy	Publication Date: Feb 5, 2016
Citations: 35	License type: CC BY 4.0

Affiliation: Fırat University, Çankaya University, King Saud University

Abstract

In this work; we present a method for solving the second-order linear ordinary differential equation of hypergeometric type. The solutions of this equation are given by the confluent hypergeometric functions (CHFs). Unlike previous studies, we obtain some different new solutions of the equation without using the CHFs. Therefore, we obtain new discrete fractional solutions of the homogeneous and non-homogeneous confluent hypergeometric differential equation (CHE) by using a discrete fractional Nabla calculus operator. Thus, we obtain four different new discrete complex fractional solutions for these equations.

Highlights

Due to their potential applications fractional and discrete fractional differential equations have attracted much attention in recent years [1,2,3,4,5]
Atici and Eloe introduced in [6] the discrete Laplace transform method for a family of finite fractional difference equations
Atici and Eloe [8] studied the properties of discrete fractional calculus (DFC) with the Nabla operator

Summary

Introduction

Due to their potential applications fractional and discrete fractional differential equations have attracted much attention in recent years [1,2,3,4,5]. Atici and Eloe [8] studied the properties of DFC with the Nabla operator They developed exponential laws and the product rule for the forward fractional calculus. Mohan [13] discussed the differentiability properties of solutions of Nabla fractional difference equations of non-integral order. Entropy 2016, 18, 49 where δ and η are real constants and r is an independent variable We recall that this equation was found by Kummer [15] and the confluent hypergeometric equation originates in physical problems. Ingo et al [22] applied entropy for the case of anomalous diffusion governed by the time and space fractional order diffusion equation They acquired a new perspective for fractional order models. Some conclusions and future perspectives are given in the last Section

Preliminaries

Main Results

Conclusions