International Journal of Mathematics and Mathematical Sciences
Volume 13 (1990), Issue 3, Pages 481-496

On the solution of some simple fractional differential equations

L. M. B. C. Campos

Instituto Superior Tecnico, Lisboa Codex 1096, Portugal

Received 12 May 1988

Copyright © 1990 L. M. B. C. Campos. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The differintegration or fractional derivative of complex order ν, is a generalization of the ordinary concept of derivative of order n, from positive integer ν=n to complex values of ν, including also, for ν=n a negative integer, the ordinary n-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 generally a integro-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).