Research Article | Open Access

Ming Li, S. C. Lim, Carlo Cattani, Massimo Scalia, "Characteristic Roots of a Class of Fractional Oscillators", *Advances in High Energy Physics*, vol. 2013, Article ID 853925, 7 pages, 2013. https://doi.org/10.1155/2013/853925

# Characteristic Roots of a Class of Fractional Oscillators

**Academic Editor:**Gongnan Xie

#### Abstract

The fundamental theorem of algebra determines the number of characteristic roots of an ordinary differential equation of integer order. This may cease to be true for a differential equation of fractional order. The results given in this paper suggest that the number of the characteristic roots of a class of oscillators of fractional order may in general be infinitely great. Further, we infer that it may also be the case for the characteristic roots of a differential equation of fractional order greater than 1. The relationship between the range of the fractional order and the locations of characteristic roots of oscillators in the complex plane is considered.

#### 1. Introduction

Oscillators are an essential component in devices in electron positron collider systems (see, e.g., Zhao et al. [1], Ma et al. [2], Zang et al. [3], Ding et al. [4], Marder et al. [5], Barroso [6], Miller et al. [7], and Lemke [8], just citing a few). As a matter of fact, oscillations are phenomena widely observed in sciences and engineering relating to high energy physics (see, e.g., Akhmediev et al. [9], Bachas [10], Winter et al. [11], Dodonov [12], Tan [13], Diamandis et al. [14], Greenwald et al. [15], Mathews et al. [16], Faiman [17], Cocho et al. [18], Baldiotti et al. [19], Kyu Shin [20], Kirson [21], Clement [22], Sikström et al. [23], Asghari et al. [24], Um et al. [25], Bahar and Yasuk [26], Hassanabadi et al. [27], Bhattacharya and Roy [28], and Saad et al. [29], simply mentioning a few).

There are various structures of oscillators, such as Mathieu oscillator (Floris [30]), Liénard type oscillator (Yaşar [31]), relativistic oscillator (Osborne [32]), Schrödinger equation type oscillator (Cornwall and Tiktopoulos [33]), and Duffing oscillator (Baltanás et al. [34] and Erturk and Inman [35]). In fact, oscillators play a role in various fields, ranging from experimental physics to electronics engineering (see, e.g., Riley et al. [36], Soong and Grigoriu [37], Harris [38], Papoulis [39], Bendat and Piersol [40], Devasahayam [41], Karrenberg [42], Edson [43], and Balaban et al. [44]).

This research is in the domain of fractional oscillators that attract increasing interests of physicists and engineers. More specifically, we aim at revealing specific properties of characteristic roots of a class of fractional oscillators. In doing so, we first consider an ordinary differential equation of order given by where is a natural number and is any complex number. We always assume that at least one of the higher coefficients for . The characteristic equation of (1) is given by

The fundamental theorem of algebra says that the number of roots of (2) is (G. A. Korn and T. M. Korn [45]). This theorem is stated in the domain of complex variables (Krantz [46]).

Suppose that the roots of are . For each root of multiplicity of , either real or complex, we always consider the roots in what follows unless otherwise stated. Using the partial fraction expansion, can be expressed by

Now, we rewrite (3) by the following expression: where , , , and are constants. Without loss of generality, we can suppose that the only simple zero of is if is odd.

The factor in (4) corresponds to the oscillator equation in the form

The characteristic equation of (5) is in the form

There are two classes of fractional oscillators. One is in the form (Ryabov and Puzenko [47, Eq. (1)], Ahmad et al. [48], Radwan et al. [49], Drozdov [50, Eq. (9)], Tofighi and Pour [51], Tofighi [52, Eq. (2)], Blaszczyk et al. [53, Eq. (10)], and Narahari Achar et al. [54, 55])

The other is expressed with the form (Lim et al. [56–58], Muniandy and Lim [59], Eab and Lim [60], and Li et al. [61]) This research utilizes the form of (8).

According to the fundamental theorem of algebra, there are only two characteristic roots with respect to the oscillator equation (5). They are

One might be carelessly misled to consider that there exist only two characteristic roots regarding the fractional oscillator equation (8) because

However, we shall show that the number of the roots in the above expression dramatically differs from what in the following expression:

The contributions of this paper are twofold. One is to exhibit that the number of the characteristic roots of (8) is in general infinitely great. The other is to reveal the relationship between the range of and the locations of the characteristic roots of (8) in a complex plane. In addition, if all () are simple complex pair of roots, the ordinary differential equation of order (1) and its generalization given by may be taken as the product of oscillators of integer order and fractional order in series in the wide sense for being even, respectively.

The rest of the paper is organized as follows. We shall give the results in Section 2, including the proof that there are infinite characteristic roots regarding (8), and the explanation that (1) and (12) may be taken as oscillators in series in the wide sense. Discussions are given in Section 3, which is followed by Conclusions.

#### 2. Results

##### 2.1. Result 1

The number of the characteristic roots of (8) may be infinitely great.

Denote by **C** the set of complex numbers. Let . Suppose that a power function is given by

Then, the number of different values of relies on the value of for a given . More precisely, we express that by the following lemmas, which can be found in the literature, such as [45] or Yu [62].

Lemma 1. * If is a rational number expressed by the irreducible fraction , where , the number of values of is . *

Lemma 2. * If is an irrational number or imaginary number, the number of values of is infinitely great. *

The general expression of is in the form Therefore, from Lemma 2, we have the theorem below.

Theorem 3. * The number of the characteristic roots of the fractional oscillator (8) is infinitely great if .*

*Proof. *Let = 0. Then, the characteristic roots in (9) become the imaginary numbers expressed by

According to Lemma 2, the number of the roots of either or is infinitely great. Thus, Theorem 3 results.

##### 2.2. Result 2

Equations (1) and (12) may be taken as oscillators in series.

Denote in (4) by :

Without loss of generality, is assumed to be even. In addition, we suppose that all () are simple complex pair of roots. Then, we have the theorem below.

Theorem 4. *The ordinary differential equation (1) may be taken as an oscillator (i.e., product of oscillators) in the wide sense if is even and all () are simple complex pair of roots. By wide sense, one means that it is a system consisting of the product of a series of conventional 2-order oscillators.*

*Proof. * On the one hand, stands for the characteristic equation of the th oscillator of order 2 since is even and all () are simple complex pair of roots. On the other hand, the characteristic equation of (1) can be expressed by

Based on the theory of filter design (Mitra and Kaiser [63]), the system (1) in the case of being even may be expressed by Figure 1.

Therefore, the system (1) may be expressed by the product of a series of 2-order oscillators.

Denote by the characteristic equation of (12). Then, where

Thus, the system of fractional order expressed by (12) may be the product of a series of fractional oscillators of (8).

#### 3. Discussions

The previous section says that there are infinite roots in if . In the case of , reduces to the characteristic equation of the conventional oscillator (5) with two roots only. Thus, the fraction dramatically alters the behavior of characteristic roots of oscillators. For facilitating our discussions, we omit the subscript in what follows if not confused. More precisely, we specifically consider the fractional oscillator in the form

Figure 2 shows an RLC resonance circuit in series, where , , and represent resistor, inductor, and capacity, respectively. In Figure 2, is the electronic current and the power source. According to the Kirchhoff voltage law, one has

Let and . Denote by . Then, (21) becomes the form

Generalizing (22) to the fractional order yields

Below, we specifically study the circuit in Figure 2 with , as indicated in Figure 3.

In the case of Figure 3, (23) becomes the form

Denote by the impulse response function of (24). Then, using the techniques in fractional calculus and differential equations [64–81], we have (see [61] for details) where is the Bessel function of the first kind of order .

The following theorems reflect the particularity of roots of .

Theorem 5. * If , all roots of are located in the left side of the complex plane. *

*Proof. *Note that

From the above, we have the asymptotic expression in the form

Applying (27) to (25) produces

Denote by the Laplace transform of . Then, according to the final-value theorem, we have

The above implies that all poles of except the origin are strictly in the left side of plane. In the right of the plane, is analytic. This completes the proof.

Theorem 6. * If , at least, parts of roots of are located in the right side of the complex plane. *

*Proof. * Note that
From the above, we have the following:

Since implies , we immediately see that both the right side and the left one on the above expression are respectively unbounded when . Thus, for , the fractional oscillator (24) is nonstable according to the theory of systems (Gabel and Roberts [77], Dorf and Bishop [78]). Consequently, at least, some of poles of are in the right of the plane. Therefore, at least, parts of roots of are located in the right side of the complex plane.

Most of previous discussions take oscillators of fractional order (24) as a specific object. Note that the number of the characteristic roots of differential equation in general in the form of (12) may also be infinitely great. Hence, comes the following theorem in passing.

Theorem 7. * Fractional-order differential equation expressed by (12) has infinite characteristic roots if and if there is at least a pair of roots that are simple complex. *

*Proof. *The characteristic equation of (12) may be decomposed in the form of (18) due to . Because there is at least a pair of roots that are simple complex, the number of the characteristic roots of (19) is infinitely great. Thus, the number of characteristic roots of (12) is infinitely great. This completes the proof.

The previous discussions exhibit interesting phenomena of the characteristic roots of the oscillators of the fractional type of (24). In the future, we will work on exploring the answers of the questions described below.(i)Are all poles of with respect to (24) in the right of the plane when ?(ii)Might there be interesting oscillation behavior of (12) if all in (18) and if is even?

#### 4. Conclusions

We have explained that the number of the characteristic roots of fractional-order oscillators of (24) is usually infinitely great. This conclusion has been further inferred to the case of fractional-order differential equation of (12). We have exhibited that all characteristic roots of (24) are strictly located in the left side of the complex plane if and at least some of characteristic roots of (24) are in the right side of the complex plane if . In the case of , (24) reduce to an ordinary damping-free oscillator.

#### Acknowledgments

This work was partly supported by the National Natural Science Foundation of China (NSFC) under the project Grant nos. 61272402, 61070214, 60873264, and the 973 plan under the project no. 2011CB302800.

#### References

- L. Zhao, Y. Guo, B. Liu, H. Ding, and Q. Li, “A phase-locked oscillator in the high energy physics experiments,” in
*Proceedings of the 7th National Conferenc on Nuclear Electronics and Nuclear Detection Technology*, pp. 99–103, Jiangxi, China, October 1994, (Chinese). View at: Google Scholar - Q.-S. Ma, Q.-X. Liu, C. Su, Z.-K. Fan, A.-B. Chang, and H.-Y. Hu, “Experimental investigation of the X-band transit-time tube oscillator,”
*High Energy Physics and Nuclear Physics*, vol. 27, no. 6, pp. 542–545, 2003 (Chinese). View at: Google Scholar - J.-F. Zang, Q.-X. Liu, Y.-C. Lin, and J. Zhu, “High frequency characteristics of radial three-cavity transmit time oscillator,”
*High Power Laser and Particle Beams*, vol. 20, no. 12, pp. 2046–2050, 2008 (Chinese). View at: Google Scholar - H. Ding et al., “Beijing Spectrometer,”
*High Energy Physics and Nuclear Physics*, vol. 16, no. 9, pp. 769–789, 1992 (Chinese). View at: Google Scholar - B. M. Marder, M. C. Clark, L. D. Bacon, J. M. Hoffman, R. W. Lemke, and P. D. Coleman, “The split-cavity oscillator: a high-power E-beam modulator and microwave source,”
*IEEE Transactions on Plasma Science*, vol. 20, no. 3, pp. 312–331, 1992. View at: Publisher Site | Google Scholar - J. J. Barroso, “Split-cavity monotrons achieving 40 percent electronic efficiency,”
*IEEE Transactions on Plasma Science*, vol. 32, no. 3, pp. 1205–1211, 2004. View at: Publisher Site | Google Scholar - R. B. Miller, W. F. McCullough, K. T. Lancaster, and C. A. Muehlenweg, “Super-reltron theory and experiments,”
*IEEE Transactions on Plasma Science*, vol. 20, no. 3, pp. 332–343, 1992. View at: Publisher Site | Google Scholar - R. W. Lemke, “Dispersion analysis of symmetric transverse magnetic modes in a split cavity oscillator,”
*Journal of Applied Physics*, vol. 72, no. 9, pp. 4422–4428, 1992. View at: Publisher Site | Google Scholar - N. Akhmediev, J. M. Soto-Crespo, and P. Grelu, “Roadmap to ultra-short record high-energy pulses out of laser oscillators,”
*Physics Letters A*, vol. 372, no. 17, pp. 3124–3128, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH - C. Bachas, “A proof of exponential suppression of high-energy transitions in the anharmonic oscillator,”
*Nuclear Physics B*, vol. 377, no. 3, pp. 622–648, 1992. View at: Publisher Site | Google Scholar | MathSciNet - A. Winter, F. Ö. Ilday, O. D. Mücke et al., “Towards high-performance optical master oscillators for energy recovery linacs,”
*Nuclear Instruments and Methods in Physics Research A*, vol. 557, no. 1, pp. 299–304, 2006. View at: Publisher Site | Google Scholar - V. V. Dodonov, “Irregular behaviour of a quantum kicked oscillator,”
*Physics Letters A*, vol. 214, no. 1-2, pp. 27–32, 1996. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - C. Z. Tan, “Electric potential energy of the incident light and the Hamiltonian of the induced oscillators in non-absorbing isotropic dielectrics,”
*Physica B*, vol. 269, no. 3-4, pp. 373–378, 1999. View at: Publisher Site | Google Scholar - G. A. Diamandis, B. C. Georgalas, A. B. Lahanas, and E. Papantonopoulos, “On the high energy suppression of transition amplitudes in the driven anharmonic oscillator,”
*Physics Letters B*, vol. 306, no. 3-4, pp. 319–326, 1993. View at: Publisher Site | Google Scholar - Z. Greenwald, D. W. Woodard, A. R. Calawa, and L. F. Eastman, “The effect of a high energy injection on the performance of mm wave Gunn oscillators,”
*Solid State Electronics*, vol. 31, no. 7, pp. 1211–1214, 1988. View at: Publisher Site | Google Scholar - P. M. Mathews, M. Seetharaman, S. Raghavan, and V. T. A. Bhargava, “A simple accurate formula for the energy levels of oscillators with a quartic potential,”
*Physics Letters A*, vol. 83, no. 3, pp. 118–120, 1981. View at: Publisher Site | Google Scholar | MathSciNet - D. Faiman, “Harmonic oscillator analysis of the hyperon spectrum,”
*Nuclear Physics B*, vol. 32, no. 2, pp. 573–608, 1971. View at: Publisher Site | Google Scholar - G. Cocho, J. Flores, and A. Mondragón, “The relativistic harmonic oscillator and nuclear reactions,”
*Nuclear Physics A*, vol. 128, no. 1, pp. 110–128, 1969. View at: Publisher Site | Google Scholar - M. C. Baldiotti, R. Fresneda, and D. M. Gitman, “Quantization of the damped harmonic oscillator revisited,”
*Physics Letters A*, vol. 375, no. 15, pp. 1630–1636, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - H. Kyu Shin, “Probability of vibrational excitations in high energy collisions,”
*Chemical Physics Letters*, vol. 3, no. 3, pp. 125–127, 1969. View at: Publisher Site | Google Scholar - M. W. Kirson, “Oscillator parameters in nuclei,”
*Nuclear Physics A*, vol. 781, no. 3-4, pp. 350–362, 2007. View at: Publisher Site | Google Scholar - D. M. Clement, “Harmonic oscillator potential insertions in particle lines,”
*Nuclear Physics A*, vol. 174, no. 3, pp. 561–571, 1971. View at: Publisher Site | Google Scholar - C. M. Sikström, H. Nilsson, U. Litzén, A. Blom, and H. Lundberg, “Uncertainty of oscillator strengths derived from lifetimes and branching fractions,”
*Journal of Quantitative Spectroscopy and Radiative Transfer*, vol. 74, no. 3, pp. 355–368, 2002. View at: Publisher Site | Google Scholar - M. Asghari, P. Pedram, and K. Nozari, “Harmonic oscillator with minimal lengthminimal momentum, and maximal momentum uncertainties in SUSYQM framework,”
*Physics Letters B*, vol. 725, no. 4-5, pp. 451–455, 2013. View at: Publisher Site | Google Scholar - C.-I. Um, K.-H. Yeon, and T. F. George, “The quantum damped harmonic oscillator,”
*Physics Report*, vol. 362, no. 2-3, pp. 63–192, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - M. K. Bahar and F. Yasuk, “Exact solutions of the mass-dependent Klein-Gordon equation with the vector quark-antiquark interaction and harmonic oscillator potential,”
*Advances in High Energy Physics*, vol. 2013, Article ID 814985, 6 pages, 2013. View at: Publisher Site | Google Scholar | MathSciNet - H. Hassanabadi, Z. Molaee, M. Ghominejad, and S. Zarrinkamar, “Spin-one DKP equation in the presence of Coulomb and harmonic oscillator interactions in $(1+3)$-dimension,”
*Advances in High Energy Physics*, vol. 2012, Article ID 489641, 10 pages, 2012. View at: Publisher Site | Google Scholar | MathSciNet - S. Bhattacharya and S. Roy, “Dissipative effect and tunneling time,”
*Advances in Mathematical Physics*, vol. 2011, Article ID 138358, 13 pages, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - N. Saad, R. L. Hall, H. Çiftçi, and Ö. Yeşiltaş, “Study of the generalized quantum isotonic nonlinear oscillator potential,”
*Advances in Mathematical Physics*, vol. 2011, Article ID 750168, 20 pages, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - C. Floris, “Stochastic stability of damped Mathieu oscillator parametrically excited by a Gaussian noise,”
*Mathematical Problems in Engineering*, vol. 2012, Article ID 375913, 18 pages, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - E. Yaşar, “Integrating factors and first integrals for Liénard type and frequency-damped oscillators,”
*Mathematical Problems in Engineering*, vol. 2011, Article ID 916437, 10 pages, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - L. S. Osborne, “A relativistic oscillator model applied to lepton-nucleon reactions,”
*Physics Letters B*, vol. 63, no. 4, pp. 456–458, 1976. View at: Publisher Site | Google Scholar - J. M. Cornwall and G. Tiktopoulos, “Functional Schrödinger equation approach to high-energy multiparticle scattering,”
*Physics Letters B*, vol. 282, no. 1-2, pp. 195–200, 1992. View at: Publisher Site | Google Scholar - J. P. Baltanás, J. L. Trueba, and M. A. F. Sanjuán, “Energy dissipation in a nonlinearly damped Duffing oscillator,”
*Physica D*, vol. 159, no. 1-2, pp. 22–34, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH - A. Erturk and D. J. Inman, “Broadband piezoelectric power generation on high-energy orbits of the bistable Duffing oscillator with electromechanical coupling,”
*Journal of Sound and Vibration*, vol. 330, no. 10, pp. 2339–2353, 2011. View at: Publisher Site | Google Scholar - K. F. Riley, M. P. Hobson, and S. J. Bence,
*Mathematical Methods For Physics and Engineering*, Cambridge Press, Cambridge, UK, 2006. - T. T. Soong and M. Grigoriu,
*Random Vibration of Mechanical and Structural Systems*, Prentice Hall, New York, NY, USA, 1993. - C. M. Harris,
*Shock and Vibration Handbook*, McGraw-Hill, New York, NY, USA, 4th edition, 1995. - A. Papoulis,
*Circuits and Systems: A Modern Approach*, Oxford University Press, New York, NY, USA, 1995. - J. S. Bendat and A. G. Piersol,
*Random Data: Analysis and Measurement Procedure*, John Wiley & Sons, New York, NY, USA, 3rd edition, 2000. View at: MathSciNet - S. R. Devasahayam,
*Signals and Systems in Biomedical Engineering*, Springer, New York, NY, USA, 2013. - U. Karrenberg,
*Signals, Processes, and Systems: An Interactive Multimedia Introduction To Signal Processing*, Springer, New York, NY, USA, 2013. - W. A. Edson, “Intermittent behavior in oscillators,”
*Bell System Technical Journal*, vol. 24, no. l, pp. 1–22, 1945. View at: Publisher Site | Google Scholar - B. P. Balaban, K. B. Karafin, and S. D. Snyder, “Monte Carlo tolerance analysis of the integrated, single- substrate, RC, Touch-Tone oscillator,”
*Bell System Technical Journal*, vol. 50, no. 4, pp. 1263–1291, 1971. View at: Publisher Site | Google Scholar - G. A. Korn and T. M. Korn,
*Mathematical Handbook For Scientists and Engineers*, 1.6-3, McGraw-Hill, New York, NY, USA, 1961. - S. G. Krantz, “The fundamental theorem of algebra,” in
*Handbook of Complex Variables*, 1.1.7 and 3.1.4, pp. 7 and 32–33, 1999. View at: Google Scholar - Y. E. Ryabov and A. Puzenko, “Damped oscillations in view of the fractional oscillator equation,”
*Physical Review B*, vol. 66, no. 18, Article ID 184201, 8 pages, 2002. View at: Google Scholar - W. E. K. Ahmad, R. El-khazali, and A. S. R. Elwakil, “Fractional-order Wien-bridge oscillator,”
*Electronics Letters*, vol. 37, no. 18, pp. 1110–1112, 2001. View at: Publisher Site | Google Scholar - A. G. E. Radwan, A. S. Elwakil, and A. M. A. S. Soliman, “Fractional-order sinusoidal oscillators: design procedure and practical examples,”
*IEEE Transactions on Circuits and Systems I*, vol. 55, no. 7, pp. 2051–2063, 2008. View at: Publisher Site | Google Scholar | MathSciNet - A. D. Drozdov, “Fractional oscillator driven by a Gaussian noise,”
*Physica A*, vol. 376, no. 1-2, pp. 237–245, 2007. View at: Publisher Site | Google Scholar - A. Tofighi and H. N. Pour, “
*ε*-expansion and the fractional oscillator,”*Physica A*, vol. 374, no. 1, pp. 41–45, 2007. View at: Publisher Site | Google Scholar - A. Tofighi, “The intrinsic damping of the fractional oscillator,”
*Physica A*, vol. 329, no. 1-2, pp. 29–34, 2003. View at: Publisher Site | Google Scholar - T. Blaszczyk, M. Ciesielski, M. Klimek, and J. Leszczynski, “Numerical solution of fractional oscillator equation,”
*Applied Mathematics and Computation*, vol. 218, no. 6, pp. 2480–2488, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - B. N. Narahari Achar, J. W. Hanneken, and T. Clarke, “Response characteristics of a fractional oscillator,”
*Physica A*, vol. 309, no. 3-4, pp. 275–288, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH - B. N. Narahari Achar, J. W. Hanneken, and T. Clarke, “Damping characteristics of a fractional oscillator,”
*Physica A*, vol. 339, no. 3-4, pp. 311–319, 2004. View at: Publisher Site | Google Scholar | MathSciNet - S. C. Lim and L. P. Teo, “The fractional oscillator process with two indices,”
*Journal of Physics A*, vol. 42, no. 6, Article ID 065208, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. C. Lim, L. Ming, and L. P. Teo, “Locally self-similar fractional oscillator processes,”
*Fluctuation and Noise Letters*, vol. 7, no. 2, pp. L169–L179, 2007. View at: Publisher Site | Google Scholar - S. C. Lim, M. Li, and L. P. Teo, “Langevin equation with two fractional orders,”
*Physics Letters A*, vol. 372, no. 42, pp. 6309–6320, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. V. Muniandy and S. C. Lim, “Modeling of locally self-similar processes using multifractional Brownian motion of Riemann-Liouville type,”
*Physical Review E*, vol. 63, no. 4, Article ID 046104, 7 pages, 2001. View at: Google Scholar - C. H. Eab and S. C. Lim, “Path integral representation of fractional harmonic oscillator,”
*Physica A*, vol. 371, no. 2, pp. 303–316, 2006. View at: Publisher Site | Google Scholar | MathSciNet - M. Li, S. C. Lim, and S. Chen, “Exact solution of impulse response to a class of fractional oscillators and its stability,”
*Mathematical Problems in Engineering*, vol. 2011, Article ID 657839, 9 pages, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH - J. R. Yu,
*Complex Functions*, High Education Press, 1979, (Chinese). - S. K. Mitra and J. F. Kaiser,
*Handbook For Digital Signal Processing*, John Wiley & Sons, New York, NY, USA, 1993. - J. Klafter, S. C. Lim, and R. Metzler,
*Fractional Dynamics: Recent Advances*, World Scientific, Singapore, 2012. - R. Hilfer,
*Applications of Fractional Calculus in Physics*, World Scientific, Singapore, 2000. - C. Cattani, “Fractional calculus and Shannon wavelet,”
*Mathematical Problems in Engineering*, vol. 2012, Article ID 502812, 26 pages, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - M. H. Heydari, M. R. Hooshmandasl, F. Mohammadi, and C. Cattani, “Wavelets method for solving systems of nonlinear singular fractional Volterra integro-differential equations,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 19, no. 1, pp. 37–48, 2014. View at: Publisher Site | Google Scholar - Y. M. Wang, “Maximum-norm error estimates of ADI methods for a two-dimensional fractional sub-diffusion equation,”
*Advances in Mathematical Physics*, vol. 2013, Article ID 293706, 10 pages, 2013. View at: Publisher Site | Google Scholar - J. S. Duan, “The periodic solution of fractional oscillation equation with periodic input,”
*Advances in Mathematical Physics*, vol. 2013, Article ID 869484, 6 pages, 2013. View at: Google Scholar - M. Carlini, T. Honorati, and S. Castellucci, “Photovoltaic greenhouses: comparison of optical and thermal behaviour for energy savings,”
*Mathematical Problems in Engineering*, vol. 2012, Article ID 743764, 10 pages, 2012. View at: Publisher Site | Google Scholar - M. Carlini and S. Castellucci, “Modelling the vertical heat exchanger in thermal basin,” in
*Computational Science and Its Applications*, vol. 6785 of*Lecture Notes in Computer Science*, pp. 277–286, 2011. View at: Publisher Site | Google Scholar - C. Toma, “Advanced signal processing and command synthesis for memory-limited complex systems,”
*Mathematical Problems in Engineering*, vol. 2012, Article ID 927821, 13 pages, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - E. G. Bakhoum and C. Toma, “Specific mathematical aspects of dynamics generated by coherence functions,”
*Mathematical Problems in Engineering*, vol. 2011, Article ID 436198, 10 pages, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH - E. G. Bakhoum and C. Toma, “Transient aspects of wave propagation connected with spatial coherence,”
*Mathematical Problems in Engineering*, vol. 2013, Article ID 691257, 5 pages, 2013. View at: Publisher Site | Google Scholar - C. A. Monje, Y. Q. Chen, B. M. Vinagre, D. Xue, and V. Feliu,
*Fractional Order Systems and Controls—Fundamentals and Applications*, Springer, London, UK, 2010. View at: Publisher Site | MathSciNet - Y. Q. Chen, B. M. Vinagre, and I. Podlubny, “Fractional order disturbance observer for robust vibration suppression,”
*Nonlinear Dynamics*, vol. 38, no. 1–4, pp. 355–367, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH - R. A. Gabel and R. A. Roberts,
*Signals and Linear Systems*, John Wiley & Sons, New York, NY, USA, 1973. - R. C. Dorf and R. H. Bishop,
*Modern Control Systems*, Prentice Hall, New York, NY, USA, 9th edition, 2002. - O. M. Abuzeid, A. N. Al-Rabadi, and H. S. Alkhaldi, “Recent advancements in fractal geometric-based nonlinear time series solutions to the micro-quasistatic thermoviscoelastic creep for rough surfaces in contact,”
*Mathematical Problems in Engineering*, vol. 2011, Article ID 691270, 29 pages, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH - O. M. Abuzeid, A. N. Al-Rabadi, and H. S. Alkhaldi, “Fractal geometry-based hypergeometric time series solution to the hereditary thermal creep model for the contact of rough surfaces using the Kelvin-Voigt medium,”
*Mathematical Problems in Engineering*, vol. 2010, Article ID 652306, 22 pages, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH - S. X. Hu, Z. W. Liao, and W. F. Chen, “Sinogram restoration for low-dosed x-ray computed tomography using fractional-order perona-malik diffusion,”
*Mathematical Problems in Engineering*, vol. 2012, Article ID 391050, 13 pages, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH

#### Copyright

Copyright © 2013 Ming Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.