Recent Developments and Applications on Discrete Fractional Equations and Related TopicsView this Special Issue
Research Article | Open Access
Rabha W. Ibrahim, Hamid A. Jalab, "The Fractional Complex Step Method", Discrete Dynamics in Nature and Society, vol. 2013, Article ID 515973, 8 pages, 2013. https://doi.org/10.1155/2013/515973
The Fractional Complex Step Method
It is well known that the complex step method is a tool that calculates derivatives by imposing a complex step in a strict sense. We extended the method by employing the fractional calculus differential operator in this paper. The fractional calculus can be taken in the sense of the Caputo operator, Riemann-Liouville operator, and so forth. Furthermore, we derived several approximations for computing the fractional order derivatives. Stability of the generalized fractional complex step approximations is demonstrated for an analytic test function.
The concept of derivative is one of the most important concepts in science and engineering. It can be described from two equally valid points of view: the geometrical point of view and the physical one. From the geometrical point of view, the derivative can be seen as the tangent line to a function in a certain evaluation point. From the physical point of view, the derivative can be seen as a measure of the rate of change of the function in this point. Further method of computing the derivative of a function comes from its expansion in a Taylor series. Most naturally, derivatives of real functions are evaluated using real numbers, but the less intuitive idea of using an imaginary number in real functions differentiation has been shown capable of overcoming the term cancellation inherent to the ordinary FD method, as well as reducing the associated approximation error. The utilize of complex variables in numerical differentiation was imposed by Lyness and Moler , describing a method for calculating the derivatives of any analytic function. Lai and Crassidis  used the complex representation of the Taylor series to avoid using the real part for computing the second derivative. Cerviño and Bewley  extended the method with an application to pseudospectral simulation codes. Kim et al. [4, 5] employed the complex step perturbation in nonlinear robust performance analysis. Recently, the complex set method has been applied by many authors [6–10].
Fractional calculus (real and complex) is a rapidly growing subject of interest for physicists and mathematicians. The reason for this is that problems may be discussed in a much more stringent and elegant way than using traditional methods. Fractional differential equations have emerged as a new branch of applied mathematics which has been used for many mathematical models in science and engineering. In fact, fractional differential equations are considered as an alternative model to nonlinear differential equations. Varieties of them play important roles and tools not only in mathematics but also in physics, dynamical systems, control systems, and engineering to create the mathematical modeling of many physical phenomena. Furthermore, they are employed in social science such as food supplement, climate, and economics. Several different derivatives were introduced: Riemann Liouville, Hadamard, Grunwald Letnikov, Riesz, and Erdelyi-Kober operators and Caputo [11–17]. It is well known that the physical interpretation of the fractional derivative is an open problem today. There is no formal interpretation of the physical meaning of the fractional derivative. Since the appearance of the idea of differentiation and integration of arbitrary order, there was not any acceptable geometric and physical interpretation of these operations for more than 300 year. Recently, the physical interpretation is an open problem. In , it is shown that geometric interpretation of fractional integration is “Shadows on the walls” and its Physical interpretation is “Shadows of the past.”
In this work, we extended the complex step method by employing the fractional calculus differential operator. Furthermore, we derived several approximations for computing the fractional order derivatives. Stability of the generalized fractional complex step approximations is demonstrated for an analytic test function. Moreover, examples are illustrated.
2. Fractional Calculus
The concept of the fractional calculus (i.e., calculus of integrals and derivatives of any arbitrary real or complex order) was performed over 300 years ago. Abel in 1823 studied the generalized tautochrone problem and for the first time applied fractional calculus techniques in a physical problem. Later Liouville considered fractional calculus to problems in potential theory. Since that time, the fractional calculus has haggard the attention of many researchers in all areas of sciences.
This section concerns some basic preliminaries and notations regarding the fractional calculus.
Definition 1. The fractional (arbitrary) order integral of the function of order is defined by When , we write , where denoted the convolution product (see ), , and , and as where is the delta function.
Definition 2. The fractional (arbitrary) order derivative of the function of order is defined by
Remark 3 (see ). Consider the function , we have The Leibniz rule is
3. The Fractional Complex Step Method
The fractional complex step method (FCSM) can be expressed from the Taylor series expansion of , as follows:
Taking the imaginary part on both sides and reordering (6), we obtain the FCSM expression for the fractional derivative where is the error. In the same way, we can consider the second fractional derivative using the real term of (6) as follows:
As a generalization of the above approximate method, we let , the corresponding fractional Taylor series expansion, become where and are real numbers related to the real and imaginary differential steps. It is clear that when , (9) reduses to (7). The first approximation that can be found by using (9) is When , (10) reduces to the results obtained by Abreu et al. . Moreover, we have the following approximate fractional derivatives, which can be considered as applications of the work in :
4. Numerical Tests
In this section, we illustrate examples to examine our abstract results. We compute the fractional derivative of the function for , by applying (7) and (10)–(20). Moreover, the mean square error (MSE) is determined for , , and .
Numerical approximations for the fractional derivative, of order , based on the imaginary part of the function are computed and compared with the exact value. Table 1 shows the approximate method of the fractional derivative using (7). The mean square error is determined for the cases , and , where the exact values are , and , respectively, for . Figure 1 shows the decreasing of this error with respect to . Tables 2, 3, 4, 5, and 6 indicate the fractional derivative , using (10)–(20), where . In addition, Figures 2, 3, 4, 5, and 6 view the decreasing of MSE with respect to for (10), (13), and (17)–(20), while for (15). Note that (17)–(20) are equivalence. The results are computed with the help of MATLAB 2010 .
We extended the complex step method by employing the fractional calculus differential operator (the fractional complex step). The approximation is provided for , and in the same manner, we can consider the approximation for all . This derivative concept imposes improving many different approximations for the fractional derivatives of any complex valued analytic function utilizing its real and imaginary parts. We provided different approximations for the operator . Moreover, is approximated in (8). This work can be applied in physics and computer sciences such as image processing and signal processing.
The authors are thankful to Professor Rafael Abreu for helpful suggestions for the improvement of this paper. This research has been funded by the University of Malaya, under Grant no. UMRG 104-12ICT.
- J. Lyness and C. Moler, “Numerical differentiation of analytic functions,” Journal of Numerical Analysis, vol. 4, no. 4, pp. 202–210, 1967.
- K. L. Lai and J. L. Crassidis, “Extensions of the first and second complex-step derivative approximations,” Journal of Computational and Applied Mathematics, vol. 219, no. 1, pp. 276–293, 2008.
- L. I. Cerviño and T. R. Bewley, “On the extension of the complex-step derivative technique to pseudospectral algorithms,” Journal of Computational Physics, vol. 187, no. 2, pp. 544–549, 2003.
- J. Kim, D. G. Bates, and I. Postlethwaite, “Complex-step gradient approximation for robustness analysis of nonlinear systems,” in Proceedings of the 16th Triennial World Congress of International Federation of Automatic Control (IFAC '05), pp. 1–6, Czech Republic, July 2005.
- J. Kim, D. G. Bates, and I. Postlethwaite, “Nonlinear robust performance analysis using complex-step gradient approximation,” Automatica, vol. 42, no. 1, pp. 177–182, 2006.
- J. Weiya, B. H. Dennis, and B. P. Wang, “Improved sensitivity analysis using a complex variable semi-analytical method,” Structural and Multidisciplinary Optimization, vol. 41, no. 3, pp. 433–439, 2010.
- A. Voorhees, H. Millwater, and R. Bagley, “Complex variable methods for shape sensitivity of finite element models,” Finite Elements in Analysis and Design, vol. 47, no. 10, pp. 1146–1156, 2011.
- A. H. Al-Mohy and N. J. Higham, “The complex step approximation to the fréchet derivative of a matrix function,” Numerical Algorithms, vol. 53, no. 1, pp. 133–148, 2010.
- Y. Cao, “Complex Step Hessian,” 2008, http://www.mathworks.com/matlabcentral/fileexchange/18177-complex-step-hessian/content/hessiancsd.m.
- R. Abreu, D. Stich, and J. Morales, “On the generalization of the complex step method,” Journal of Computational and Applied Mathematics, vol. 241, pp. 84–102, 2013.
- K. S. Miller and B. Ross, An Introduction to the Fractional Integrals and Derivatives-Theory and Application, John Wiley & Sons, New York, NY, USA, 1993.
- I. Podlubny, Fractional Differential Equations, Academic Press, New York, NY, USA, 1999.
- R. Hilfer, Application of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
- B. J. West, M. Bologna, and P. Grigolini, Physics of Fractal Operators, Springer, New York, NY, USA, 2003.
- A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, Elsevier, New York, NY, USA, 2006.
- J. Sabatier, O. P. Agrawal, and J. A. Tenreiro Machado, Advance in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, New York, NY, USA, 2007.
- V. Lakshmikantham, S. Leela, and J. Vasundhara Devi, Theory of Fractional Dynamic Systems, Cambridge Scientific Publications, Cambridge, UK, 2009.
- I. Podlubny, “Geometric and physical interpretation of fractional integration andfractional differentiation,” Fractional Calculus and Applied Analysis, vol. 5, no. 4, 2002.
- K. M. Kolwankar and A. D. Gangal, “Fractional differentiability of nowhere differentiable functions and dimensions,” Chaos, vol. 6, no. 4, pp. 505–513, 1996.
- F. Ben Adda and J. Cresson, “About non-differentiable functions,” Journal of Mathematical Analysis and Applications, vol. 263, no. 2, pp. 721–737, 2001.
- Z. M. Odibat and N. T. Shawagfeh, “Generalized Taylor's formula,” Applied Mathematics and Computation, vol. 186, no. 1, pp. 286–293, 2007.
- X. Yang and F. Gao, “Fundamentals of local fractional iteration of the continuously nondifferentiable functions derived from local fractional calculus,” Communications in Computer and Information Science, vol. 153, no. 2, pp. 398–404, 2011.
- W. Squire and G. Trapp, “Using complex variables to estimate derivatives of real functions,” SIAM Review, vol. 40, no. 1, pp. 110–112, 1998.
- Matlab, The MathWorks, Natick, Massachusetts, USA, 2010, http://www.mathworks.com/.
Copyright © 2013 Rabha W. Ibrahim and Hamid A. Jalab. 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.