Yıl 2015,
Cilt: 3 Sayı: 2, 0 - 0, 01.09.2015
Öz
Fractional calculus presents a new perspective for solution of scientific and engineering problems. There are still many fields where fractional calculus promise fresh understanding of real world problems. In this study, we present an investigation on fractional order motion, where we extend our comprehension from velocity and acceleration concepts to a fuzzy velocity and acceleration domains. Here, fundamentally, we suggest a debate on interpretation of fractional-order derivative system on the bases of integer-order derivative knowledge. We observed that the continuous fractional-order motion equation set can be considered to cover the discrete integer-order motion equation set and physical meaning of fractional-order motion systems can be better understood by using a fuzzy conceptualization of integer-order derivative systems according to the dissimilarity metric.
Kaynakça
- I. Podlubny, Fractional Differential Equations Vol. 198, Mathematics in Science and Engineering, Academic Press, New York and Tokyo, 1999.
- K. B. Oldham and J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, Dover Books on Mathematics, 2006.
- B. Ross, Fractional Calculus and its Applications, Springer - Verlag, Berlin, 1975.
- J. Sabatier, O. P. Agrawal and J. A. Machado, Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering Springer, 2007.
- A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, 2006.
- S. Das, Functional Fractional Calculus for System Identification and Controls, Springer, 2007.
- T. M. Saleh, M. Haeri, Rational approximations in the simulation and implementation of fractional-order dynamics: A descriptor system approach, Automatica,vol.46, no.1, pp.94-100, 2010.
- A. E. Matouk, Chaos, feedback control and synchronization of a fractional-order modified autonomous van der Pol–Duffing circuit, Communications in Nonlinear Science and Numerical Simulation, 16, 2 pp.975-986, 2011.
- A. S. Elwakil, Fractional-order circuits and systems: An emerging interdisciplinary research area, Circuits and Systems Magazine, IEEE, vol. 10, no.4, pp.40-50, 2010.
- S. Lu, F.J. Molz, G.J. Fix, Possible problems of scale dependency in applications of the three-dimensional fractional advection-dispersion equation to natural porous media, Water Resources Research, vol.38, pp. 41-47, 2002.
- L. Kexue, P. Jigen, Fractional resolvents and fractional evolution equations, Applied Mathematics Letters, vol.25, pp.808-812, 2012.
- A.M.A. El-Sayed, A.E.M. El-Mesiry, H.A.A. El-Saka, On the fractional-order logistic equation, Applied Mathematics Letters, vol.20, pp.817-823, 2007.
- E.M. El-Mesiry, A.M.A. El-Sayed, H.A.A. El-Saka, Numerical methods for multi-term fractional (arbitrary) orders differential equations Appl. Math. Comput. vol.160, pp.683-699, 2005.
- R.E. Gutiérrez, J.M. Rosário and J.A.T. Machado, Fractional Order Calculus: Basic Concepts and Engineering Applications, Hindawi Publishing Corporation Mathematical Problems in Engineering, vol.19, 2010.
- G.W. Leibnitz. Letter from hanover, germany, september 30, 1695 to g. a. l’hospital. Leibnizen Mathematische Schriften. Olms Verlag Hildesheim, Germany, 1962, First published in 1849.
- J. A. T. Machado, A probabilistic interpretation of the fractional-order differentiation Fractional Calculus and applied Analysis, vol. 6 pp.73-80, 2003.
- I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fractional Calculus & Applied Analysis, vol. 5 pp.367-386, 2002.
- M. Moshrefi-Torbati, J.K. Hammond, Physical and geometrical interpretation of fractional operators,Journal of the Franklin Institute, vol. 335, pp.1077-1086, 1998.
- F.J. Molz, G.J. Fix, S. Lu, A physical interpretation for the fractional derivative in Levy diffusion, Applied Mathematics Letters, vol.15, pp. 907-911, 2002.
- C. Giannantoni, The problem of the initial conditions and their physical meaning in linear differential equations of fractional order, Applied Mathematics and Computation, vol. 141, pp. 87-102, 2003.
- M. Dorota, D. F. M. Torres, Modified optimal energy and initial memory of fractional continuous-time linear systems, Signal Processing, vol.91, no.3, pp.379-385, 2011.
- R. E. Gutiérrez, J. M. Rosário, J. T. Machado, Fractional order calculus: basic concepts and engineering applications, Mathematical Problems in Engineering, vol.2010, 2010.
Yıl 2015,
Cilt: 3 Sayı: 2, 0 - 0, 01.09.2015
Öz
Kaynakça
- I. Podlubny, Fractional Differential Equations Vol. 198, Mathematics in Science and Engineering, Academic Press, New York and Tokyo, 1999.
- K. B. Oldham and J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, Dover Books on Mathematics, 2006.
- B. Ross, Fractional Calculus and its Applications, Springer - Verlag, Berlin, 1975.
- J. Sabatier, O. P. Agrawal and J. A. Machado, Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering Springer, 2007.
- A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, 2006.
- S. Das, Functional Fractional Calculus for System Identification and Controls, Springer, 2007.
- T. M. Saleh, M. Haeri, Rational approximations in the simulation and implementation of fractional-order dynamics: A descriptor system approach, Automatica,vol.46, no.1, pp.94-100, 2010.
- A. E. Matouk, Chaos, feedback control and synchronization of a fractional-order modified autonomous van der Pol–Duffing circuit, Communications in Nonlinear Science and Numerical Simulation, 16, 2 pp.975-986, 2011.
- A. S. Elwakil, Fractional-order circuits and systems: An emerging interdisciplinary research area, Circuits and Systems Magazine, IEEE, vol. 10, no.4, pp.40-50, 2010.
- S. Lu, F.J. Molz, G.J. Fix, Possible problems of scale dependency in applications of the three-dimensional fractional advection-dispersion equation to natural porous media, Water Resources Research, vol.38, pp. 41-47, 2002.
- L. Kexue, P. Jigen, Fractional resolvents and fractional evolution equations, Applied Mathematics Letters, vol.25, pp.808-812, 2012.
- A.M.A. El-Sayed, A.E.M. El-Mesiry, H.A.A. El-Saka, On the fractional-order logistic equation, Applied Mathematics Letters, vol.20, pp.817-823, 2007.
- E.M. El-Mesiry, A.M.A. El-Sayed, H.A.A. El-Saka, Numerical methods for multi-term fractional (arbitrary) orders differential equations Appl. Math. Comput. vol.160, pp.683-699, 2005.
- R.E. Gutiérrez, J.M. Rosário and J.A.T. Machado, Fractional Order Calculus: Basic Concepts and Engineering Applications, Hindawi Publishing Corporation Mathematical Problems in Engineering, vol.19, 2010.
- G.W. Leibnitz. Letter from hanover, germany, september 30, 1695 to g. a. l’hospital. Leibnizen Mathematische Schriften. Olms Verlag Hildesheim, Germany, 1962, First published in 1849.
- J. A. T. Machado, A probabilistic interpretation of the fractional-order differentiation Fractional Calculus and applied Analysis, vol. 6 pp.73-80, 2003.
- I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fractional Calculus & Applied Analysis, vol. 5 pp.367-386, 2002.
- M. Moshrefi-Torbati, J.K. Hammond, Physical and geometrical interpretation of fractional operators,Journal of the Franklin Institute, vol. 335, pp.1077-1086, 1998.
- F.J. Molz, G.J. Fix, S. Lu, A physical interpretation for the fractional derivative in Levy diffusion, Applied Mathematics Letters, vol.15, pp. 907-911, 2002.
- C. Giannantoni, The problem of the initial conditions and their physical meaning in linear differential equations of fractional order, Applied Mathematics and Computation, vol. 141, pp. 87-102, 2003.
- M. Dorota, D. F. M. Torres, Modified optimal energy and initial memory of fractional continuous-time linear systems, Signal Processing, vol.91, no.3, pp.379-385, 2011.
- R. E. Gutiérrez, J. M. Rosário, J. T. Machado, Fractional order calculus: basic concepts and engineering applications, Mathematical Problems in Engineering, vol.2010, 2010.
Toplam 22 adet kaynakça vardır.
Ayrıntılar
| Bölüm | Reviews |
|---|---|
| Yazarlar | |
| Yayımlanma Tarihi | 1 Eylül 2015 |
| Yayımlandığı Sayı | Yıl 2015 Cilt: 3 Sayı: 2 |
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