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The problem of the initial conditions and their physical meaning in linear differential equations of fractional order. (English) Zbl 1037.34004
This paper considers the practically important question of how to set up the initial conditions when solving a fractional differential equation involving fractional derivatives of Riemann-Liouville or Grunwald-Post type. Some authors have chosen to use practically simpler Caputo-type fractional derivative because here the inital conditions would be specified in terms of integer order derivatives, which have some clear physical meaning and are relatively simply measured. However, this paper considers what might be the physical meaning of initial conditions (or indeed of the evaluation of a derivative) of fractional order. The discussion leads to a new formulation of fractional derivative which emphasises the multiplicity of definitions of fractional derivative in the existing literature.
MSC:
34A25Analytical theory of ODE (series, transformations, transforms, operational calculus, etc.)
34A12Initial value problems for ODE, existence, uniqueness, etc. of solutions