Initialization of Fractional Differential Equations: Theory and Application

Abstract

It has been known that the initialization of fractional operators requires time-varying functions, a complicating factor. This paper simplifies the process of initialization of fractional differential equations by deriving Laplace transforms for the initialized fractional integral and derivative that generalize those for the integer-order operators. A companion paper in this conference determines the Laplace transforms for initialized fractional integrals of any order and fractional derivatives of order less than one. This paper extends the theory for the Laplace transform of the derivative to higher order and provides applications.

Fractional derivatives in spaces of generalized functions

“…[4], [11], and references therein) as well as a new topic (cf. [10], [7], [20], [5], [8], [6], [9], [17] among the others). The main disadvantage of this calculus up to now, is the lack of appropriate functional spaces for their investigations.…”

Fractional order iterative learning control for fractional order system with unknown initialization

“…where α ∈ (0, 1), and the initialization function Ψ(t) is expressed by the Lerch's phi function for constant history functions [22], [23], [28], [35].…”

“…It should be noted that the viscoelastic materials have become a main research object for both fractional calculus and ILC, where the fractional order viscoelasticity is the first successful application of fractional order theory [30], and the application of ILC to physiotherapy is a hot spot in recent years [31]. The fractional order system with non-zero history function belongs to the initialization issue of fractional order differential equations, which is an unmature but high-profile problem [22], [23]. The same history function is a fundamental requirement in FOILC so that the repeatability can be guaranteed.…”

“…Thus different history functions lead to different solutions [22], [23]. In current references of FOILC, the zero history function is assumed, which is imitated from many classical fractional order control strategies [27].…”

Multivariable fractional system approximation with initial conditions using integral state space representation

“…Several methods making it possible to take it into account were proposed. In [18] and earlier papers of the authors, Lorenzo and Hartley proposed to use the so called ''initialization functions'' which are differently defined when the Riemann-Liouville or Caputo definitions are considered. In a recent paper, Sabatier et al [19] proposed to use a new representation of fractional order system that involves a classical linear integer system and a system described by a parabolic equation.…”