Generalized fractional derivatives and their applications to mechanical systems

摘要

New fractional derivatives, termed henceforth generalized fractional derivatives (GFDs), are introduced. Their definition is based on the concept that fractional derivatives (FDs) interpolate the integer- order derivatives. This idea generates infinite classes of FDs. The new FDs provide, beside the fractional order, any number of free parameters to better calibrate the response of a physical system or procedure. Their usefulness and consequences are subject of further investigation. Like the Caputo FD, the GFDs allow the application of initial conditions having direct physical significance. A numerical method is also developed for the solution of differential equations involving GFDs. Mechanical systems including fractional oscillators, viscoelastic plane bodies and plates described by such equations are analyzed.