On approximate solutions for two higher-order Caputo-Fabrizio fractional integro-differential equations

S Melike Aydogan,Asef Mousalou,Dumitru Baleanu,Shahram Rezapour

doi:10.1186/s13662-017-1258-3

Abstract

We investigate the existence of solutions for two high-order fractional differential equations including the Caputo-Fabrizio derivative. In this way, we introduce some new tools for obtaining solutions for the high-order equations. Also, we discuss two illustrative examples to confirm the reported results. In this way one gets the possibility of utilizing some continuous or discontinuous mappings as coefficients in the fractional differential equations of higher order.

Highlights

Up to now, there have been defined some fractional derivations of which most used are the Caputo and Riemann-Liouville operators
The applications of the fractional calculus with these two main derivatives can be observed within a huge range of real world phenomena
We discuss the existence of approximate solutions analytically corresponding to two CF fractional differential equations (FDE)

Summary

Introduction

There have been defined some fractional derivations of which most used are the Caputo and Riemann-Liouville operators. In order to increase the power and applicability of the fractional calculus some researchers suggested a new type of fractional derivatives possessing different kernels. Caputo and Fabrizio defined recently a new fractional derivative possessing a singular kernel [ ] and the properties of it were discussed in [ ]. Some researchers have used distinct methods for solving some different equations including the Caputo-Fabrizio (CF) fractional derivative (see [ – , , ] and the references therein) and multi-singular pointwise defined equations [ – ]. Despite these original results, still several issues regarding this new fractional derivative have to be developed.

Basic tools

Conclusions