Differential equations of fractional order:methods results and problem —I

Abstract

Thc paper deals with the so-called differential equations of fractional order in which an unknown function is contained under the operation of a derivative of fractional order. A survey of the methods and results in the theory of such ordinary fractional differential equations is given. In particular, the method based on the reduction of the Cauchy-type problem for the fractional differential equations to the Volterra integral equations is discussed, and the Laplace transform, operational calculas compositional methods for the solution of linear differential equations of fractional order are presented. Problems and new trends of research are discussed.

Keywords:

• Fractional calculus
• differential equations of fractional order
• Cauchy-type problem
• Volterra integral equation
• Laplace transform
• operational calculus and compositional methods
• Mittag-Leffler type functions

Keywords:

• 34A05
• 34A25
• 34A46
• 26A33
• 44A10
• 45D05
• 44A40
• 33E20