The problem of the initial conditions and their physical meaning in linear differential equations of fractional order

摘要

The well-known approaches to fractional differential equations (such as those proposed by Grunwald–Post and Riemmann–Liouville) imply, as initial conditions, the evaluation of fractional derivatives of the unknown function. From an applicative point of view, an extensive use of the corresponding and well-defined theoretical solutions would require an appropriate interpretation of the physical meaning of such initial conditions. On the other hand some authors propose to overcome the problem by introducing a different definition of fractional derivative which only requires the evaluation of integer-order derivatives of the unknown function at the lower extreme of the definition interval.The problem we are concerned with is faced here in the light of an «intensive» concept of fractional derivative which has been elaborated in order to give differential bases to Emergy algebra. Such an approach, apart from some peculiar novelties, is potentially able to solve the above-mentioned problem by suggesting a clear and meaningful physical interpretation of the initial conditions.As far as the main novelties are concerned it is worth mentioning the following: (i) fractional differentiation generates a multiplicity of derivatives (instead of a unique derivative) whose number is strictly dependent on the order of derivation; (ii) the exponential function results as an «invariant» (in module) with respect to the order of differentiation; (iii) the exponential function is therefore able to play a «hinge» role in solving fractional differential equations similar to the role it plays in the case of ordinary differential equations.As a consequence of the above-mentioned properties it is possible to assert that: a time differential problem described by one fractional differential equation generates new “special” functions (the “binary”, “ternary”, “quaternary” functions and so on) which can be interpreted as being the mathematical description of the evolution of a unique system, made up of a prefixed number of parts, which are in turn so strictly related to each other that they form one sole entity. Consequently, the pertaining fractional initial conditions directly refer to and describe the physical interactions between the parts of the system at the given initial time.