The Mittag–Leffler Functions for a Class of First-Order Fractional Initial Value Problems: Dual Solution via Riemann–Liouville Fractional Derivative

Abdelhalim Ebaid,Hind K Al-Jeaid

doi:10.3390/fractalfract6020085

Abstract

In this paper, a new approach is developed to solve a class of first-order fractional initial value problems. The present class is of practical interest in engineering science. The results are based on the Riemann–Liouville fractional derivative. It is shown that the dual solution can be determined for the considered class. The first solution is obtained by means of the Laplace transform and expressed in terms of the Mittag–Leffler functions. The second solution was determined through a newly developed approach and given in terms of exponential and trigonometric functions. Moreover, the results reduce to the ordinary version as the fractional-order tends to unity. Characteristics of the dual solution are discussed in detail. Furthermore, the advantages of the second solution over the first one is declared. It is revealed that the second solution is real at certain values of the fractional-order. Such values are derived theoretically and accordingly, and the behavior of the real solution is shown through several plots. The present analysis may be introduced for obtaining the solution in a straightforward manner for the first time. The developed approach can be further extended to include higher-order fractional initial value problems of oscillatory types.

Highlights

The fractional calculus (FC) is a growing field of research that is usually utilized to investigate the physical phenomena of the memory effect [1,2,3]
The fractional physical model of the projectile motion was discussed by Ebaid [15] and Ebaid et al [16] utilizing the Caputo fractional derivative (CFD), and their results have been compared with experimental data
The above models have been formulated in the form of secondorder fractional initial value problems (2nd-order FIVPs)

Summary

Introduction

The fractional calculus (FC) is a growing field of research that is usually utilized to investigate the physical phenomena of the memory effect [1,2,3]. Many scientific models have been analyzed via the FC approach [4,5,6,7,8]. The fractional physical model of the projectile motion was discussed by Ebaid [15] and Ebaid et al [16] utilizing the Caputo fractional derivative (CFD), and their results have been compared with experimental data. Ahmed et al [17] implemented the Riemann–Liouville fractional derivative (RLFD) to analyze the same problem. The above models have been formulated in the form of secondorder fractional initial value problems (2nd-order FIVPs)

Objectives

Results

Conclusion