Existence of Iterative Cauchy Fractional Differential Equation

Rabha W Ibrahim

doi:10.1155/2013/838230

Abstract

Our main aim in this paper is to use the technique of nonexpansive operators in more general iterative and noniterative fractional differential equations (Cauchy type). The noninteger case is taken in sense of the Riemann-Liouville fractional operators. Applications are illustrated.

Highlights

Fractional calculus and its applications are important in several widely diverse areas of mathematical, physical, and engineering sciences. It generalized the ideas of integer order differentiation and n-fold integration
This is the main advantage of fractional derivatives in comparison with classical integer-order models, in which such effects are neglected
Fractional Cauchy problems are useful in physics

Summary

Introduction

Fractional calculus and its applications (that is the theory of derivatives and integrals of any arbitrary real or complex order) are important in several widely diverse areas of mathematical, physical, and engineering sciences. It generalized the ideas of integer order differentiation and n-fold integration. The Riemann-Liouville fractional derivative could hardly pose the physical interpretation of the initial conditions required for the initial value problems involving fractional differential equations This operator possesses advantages of fast convergence, higher stability, and higher accuracy to derive different types of numerical algorithms (see [2]).

Preliminaries

Existence Theorems and Approximation of Solutions