Fractional Derivative and Integral

ABSTRACT: In this thesis we studied fractional order derivative and integral. In Chapter1, a brief history on the foundation of fractional derivative and integration has been given. In the second chapter, some definitions and theorems have been provided. Also some needed special functions such as Gamma, Beta, Mittag-Leffler and Wright function have taken place in this chapter. Properties of fractional derivative and integral are discussed in Chapter 3. We started to this chapter by the discussion of the Abel integral equation and it’s application. In the first section of Chapter 3, fractional integral in the space of integrable functions and related properties has been given. The second section is devoted to basic definitions and properties of fractional derivative and integral. Definition of fractional integral and derivative of complex order take place in the third section together with some related theorems. Fourth section contains fractional integrals of some elementary functions. In the last section of Chapter 3, we discussed fractional differentiation and integration as reciprocal operations. Keywords: Fractional Equation, Fractional Derivative, Fractional Integral. …………………………………………………………………………………………………………………………