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Research Papers: Mathematical Theory

# Application of Incomplete Gamma Functions to the Initialization of Fractional-Order Systems

[+] Author and Article Information
Tom T. Hartley

University of Akron, Akron, OH 44325-3904thartley@uakron.edu

Carl F. Lorenzo

NASA Glenn Research Center, Cleveland, OH 44135carl.f.lorenzo@nasa.gov

J. Comput. Nonlinear Dynam 3(2), 021103 (Mar 11, 2008) (6 pages) doi:10.1115/1.2833480 History: Received June 08, 2007; Revised September 27, 2007; Published March 11, 2008

## Abstract

This paper reviews some properties of the gamma function, particularly the incomplete gamma function and its complement, as a function of the Laplace variable $s$. The utility of these functions in the solution of initialization problems in fractional-order system theory is demonstrated. Several specific differential equations are presented, and their initialization responses are found for a variety of initializations. Both the time-domain and Laplace-domain solutions are obtained and compared. The complementary incomplete gamma function is shown to be essential in finding the Laplace-domain solution of a fractional-order differential equation.

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## Figures

Figure 1

γ(a,x)∕Γ(a) versus x, incomplete gamma function scaled by Γ(a)

Figure 2

Γ(a,x)∕Γ(a) versus x, complementary incomplete gamma function scaled by Γ(a)

Figure 3

Time responses corresponding to Laplace transforms containing the gamma function, the incomplete gamma function, and the complementary incomplete gamma function

Figure 4

Log magnitude of γ(1∕2,2s)∕s1∕2

Figure 5

Log magnitude of Γ(1∕2,2s)∕s1∕2

Figure 6

Log magnitude of Γ(1∕2)∕s1∕2, the sum of the complex incomplete gamma functions of Figs.  56

Figure 7

Magnitude frequency content of the complementary incomplete gamma function and a fractional-order approximation

Figure 8

Initialization response of a semidifferential equation for various lengths of constant initialization, k=1

Figure 9

Initialization response of a semidifferential equation for impulse initializations occurring at several values negative time b

Figure 10

Initialization response of a qth-order differential equation for various values of q, and initialization period a=1.0, with k=1.0

Figure 11

Initialization response of a sesquidifferential equation for various times of constant initialization, k=2

Figure 12

Initialization response of a sesquidifferential equation for various times of ramp initialization, k=1

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