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Research Papers: Mathematical Theory

Initialization of Fractional-Order Operators and Fractional Differential Equations

[+] Author and Article Information
Carl F. Lorenzo

Instrumentation and Controls Division, NASA Glenn Research Center, Cleveland, OH 44135

Tom T. Hartley

Department of Electrical and Computer Engineering, University of Akron, Akron, OH 44325-3904

J. Comput. Nonlinear Dynam 3(2), 021101 (Jan 25, 2008) (9 pages) doi:10.1115/1.2833585 History: Received June 07, 2007; Revised August 24, 2007; Published January 25, 2008

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. The new transforms unify the initialization of systems of fractional and ordinary differential equations. The paper provides background on past work in the area and determines the Laplace transforms for the initialized fractional integral and fractional derivatives of any (real) order. An application provides insight and demonstrates the theory.

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Copyright © 2008 by American Society of Mechanical Engineers
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Figures

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Figure 1

Semi-integrals of (t+2)U(t+2) versus t time, with initialization period −2<t<0 and integration of interest starting at t=0

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Figure 2

Block diagrams for the initialized ordinary order-one integral and the ordinary order-one derivative

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Figure 3

Block diagrams for the initialized fractional integral and the fractional derivative of order u=1−q, with 0⩽u⩽1

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Figure 4

Block diagrams for integer-order differentiation and uth-order fractional differentiation both based on “operational” integer-order differentiations. u=n-q. See Eqs. 18,20 for expressions for ψ(fi,−q,−a,0,t).

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Figure 5

Mechanical system

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