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Advances in Engineering Software 41 (2010) 70–74

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Advances in Engineering Software
journal homepage: www.elsevier.com/locate/advengsoft

A solution to the fundamental linear complex-order differential equation
Jay L. Adams *, Tom T. Hartley, Lynn I. Adams
The University of Akron, 34 Hawthrone Avenue, #4, Akron, OH 44325-3904, United States

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This paper provides the solution to the complex-order differential equation, 0 dt xðtÞ ¼ kxðtÞ þ buðtÞ, where both q and k are complex. The time-response solution is shown to be a series that is complex-valued. Combining this system with its complex conjugate-order system yields the following generalized differ  q 2ReðqÞ q  q    q xðtÞ À k0 dt xðtÞÀ k0 dt xðtÞ þ kkxðtÞ ¼ p0 dt uðtÞ þ p0 dt uðtÞ À ðk þ kÞuðtÞ.   transfer The ential equation, 0 dt  P pkn    function of this system is pðsq À kÞÀ1 þ pðsq À kÞÀ1 , having a time-response 2 1 t ðnþ1ÞuÀ1 Re Cððnþ1ÞqÞ n¼0    pkn cosððn þ 1Þv ln tÞ À Im Cððnþ1ÞqÞ sinððn þ 1Þv ln tÞ . The transfer function has an infinite number of complex–conjugate pole pairs. Bounds on the parameters u ¼ReðqÞ; v ¼ ImðqÞ, and k are determined for system stability. Ó 2009 Civil-Comp Ltd. and Elsevier Ltd. All rights reserved.
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Article history: Received 1 September 2007 Accepted 2 December 2008 Available online 5 March 2009 Keywords: Fractional calculus Fractional-order systems Fractional-order differential equations Complex-order derivatives Complex-order systems Complex-order differential equations1. Introduction The idea of an integer-order differintegral operator has previously been extended to the differintegral operator of non-integer but real order. In depth discussions can be found in [12,9]. Further generalizations have been made to the complex-order operator [7,6,10]. Hartley et al. showed that when a complex-order operator is paired with its conjugate-order operator, realtime-responses are created. This is analogous to a complex pole of an integer order system. A single complex pole has a complex time-response, but when encountered with its complex–conjugate pole, the resulting time-response is real-valued. Complex poles always occur in complex conjugates. Similarly, for a real time-response complex-order operators always occur in complex–conjugate pairs. [4]. Systemswith complex-order differintegrals can arise from a variety of situations. The CRONE controller makes use of conjugated-order differintegrals in a limited manner [11]. Such a system can also be artificially constructed and implemented using the techniques of Jiang et al. [5]. Such a system results from identification, as proposed by Adams et al. [1]. This paper develops the fundamentalcomplex–conjugated order differential equation. Because the meaning of complex time-responses is not understood, the purpose of this paper is to develop complex-order systems whose time-response is purely real. The behavior of these

systems is explored in the time domain, the Laplace domain and the frequency domain. Examples are presented. 2. Behavior 2.1. Time-response Hartley and Lorenzo [3] developed theF-function as the impulse response of the real-order system,

GðsÞ ¼

1 : sq À k
1 X n¼0

The F-function is given by

F q ðk; tÞ ¼

k t ðnþ1Þq ; Cððn þ 1ÞqÞ

n

ð1Þ

where k and q are real. The F-function is used rather than the more commonly used Mittag–Leffler function, because the F-function is more convenient. Noting that the derivation of the F-function in [3] does not dependon q; k, or p being real, it is easily seen that for HðsÞ given by

HðsÞ ¼

sq

     p psq þ psq À ðpk þ pkÞ p ; þ  ¼   À k sq À k ðsq À kÞðsq À kÞ

ð2Þ

the F-function, as given in Eq. (1), can be used to find the inverse Laplace transform of each term in the system, where p; q, and k can n each be complex. It is noted that k ¼ kn when n is an integer and that CðzÞ ¼ CðzÞ....