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Differential equations of fractional order: Methods, results and problems. I. (English) Zbl 1031.34002
Summary: The authors study 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 on the theory of such ordinary fractional differential equations is given. In particular, the method based on the reduction of the Cauchy-type problem for fractional differential equations to the Volterra integral equations is discussed, and the Laplace transform, operational calculus and compositional methods for the solution of linear differential equations of fractional order are presented. Problems and new trends of research are discussed.

MSC:
34A25Analytical theory of ODE (series, transformations, transforms, operational calculus, etc.)
26A33Fractional derivatives and integrals (real functions)
45D05Volterra integral equations
34A12Initial value problems for ODE, existence, uniqueness, etc. of solutions