New Result on the Critical Exponent for Solution of an Ordinary Fractional Differential Problem

Xinguang Zhang,Yonghong Wu,Yujun Cui,Lishan Liu

doi:10.1155/2017/3976469

Abstract

We give some background of fractional Cauchy problems and correct an existing result of Cauchy problem for the fractional differential inequality, and we also give an example to illustrate our statement.

Highlights

It is well known that many engineering and industrial systems involve the convection-diffusion process such as in seepage flows [1], oil flow through porous media in underground reservoirs [2, 3], and flow of chemical and metal solutions through porous media in heap leaching [4, 5]
It has been established that the convection diffusion equation based on Ficks law fails to model the anomalous feature of diffusion in porous media, and fractional order convectiondiffusion equations have the potential to accurately model the convection-diffusion process [6,7,8,9,10]
This implies that fractional order models possess much more advantage than integer order models, especially in viscoelasticity, electrochemistry, and porous media [11,12,13,14,15], and many new applications for fractional models in various fields have been reported recently [16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34]

Summary

Introduction

It is well known that many engineering and industrial systems involve the convection-diffusion process such as in seepage flows [1], oil flow through porous media in underground reservoirs [2, 3], and flow of chemical and metal solutions through porous media in heap leaching [4, 5]. Fractional order integral and derivative operators are nonlocal which can describe the behaviour of many complex processes and systems with long-memory in time This implies that fractional order models possess much more advantage than integer order models, especially in viscoelasticity, electrochemistry, and porous media [11,12,13,14,15], and many new applications for fractional models in various fields have been reported recently [16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34]. Under the above assumptions, problem (2) admits global nontrivial solutions when b ≥ 0, as shown in the following example.

Conclusion

Correction of Theorem 1