On the solution of some simple fractional differential equations

Abstract

The differintegration or fractional derivative of complex orderν, is a generalization of the ordinary concept of derivative of ordern, from positive integerν=nto complex values ofν, including also, forν=−na negative integer, the ordinaryn-th primitive. Substituting, in an ordinary differential equation, derivatives of integer order by derivatives of non-integer order, leads to a fractional differential equation, which is generallyaintegro-differential equation. We present simple methods of solution of some classes of fractional differential equations, namely those with constant coefficients (standard I) and those with power type coefficients with exponents equal to the orders of differintegration (standard II). The fractional differential equations of standard I (II), both homogeneous, and inhomogeneous with exponential (power-type) forcing, can be solved in the ‘Liouville’ (‘Riemann’) systems of differintegration. The standard I (II) is linear with constant (non-constant) coefficients, and some results are also given for a class of non-linear fractional differential equations (standard III).