Solving the Oscillation Equation With Fractional Order Damping Term Using a New Fourier Transform Method
DOI:
https://doi.org/10.24297/jam.v13i5.6483Keywords:
Fourier transform, oscillation equation, damping term, the CTIT transformation, fractional derivativeAbstract
We propose an adapted Fourier transform method that gives the solution of an oscillation equation with a fractional damping term in ordinary domain. After we mention a transformation of cosmic time to individual time (CTIT), we explain how it can reduce the problem from fractional form to ordinary form when it is used with Fourier transformation, via an example for 1 < alpha < 2; where alpha is the order of fractional derivative. Then, we give an application of the results.
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2. Y. Zhang, H. M. Srivastava and M. C. Baleanu, Local fractional variational iteration algorithm II for nonhomogeneous
model associated with the non-differentiable heat flow, Advances in Mechanical Engineering,
2015, Vol. 7(10) 15.
3. A. Alsaedi, J. J. Nieto and V. Venktesh, Fractional electrical circuits, Advances in Mechanical Engineering, 2015,
Vol. 7(12) 17.
4. A. Atangana, J. J. Nieto, Numerical solution for the model of RLC circuit via the fractional derivative without
singular kernel, Advances in Mechanical Engineering, 2015, Vol. 7(10) 17.
5. A. Atangana, B. S. Alkahtani, Extension of the resistance, inductance, capacitance electrical circuit to fractional
derivative without singular kernel, Advances in Mechanical Engineering, 2015, Vol. 7(6) 16.
6. Q. Gao, Y. Hou, T. Deng, C. Wang and R. Hou, Extended state observer-based fractional order proportionalintegral-
derivative controller for a novel electro-hydraulic servo system with iso-actuation balancing and
positioning, Advances in Mechanical Engineering, 2015, Vol. 7(12) 111.
7. Q. Gao, K. Li, Y. Hou, R. Hou and C. Wang, Balancing and positioning for a gun control system based on fuzzy
fractional order proportional-integral-derivative strategy, Advances in Mechanical Engineering, 2016, Vol. 8(3) 19.
8. A. M. Lopes, J. T. Machado, Visualizing control systems performance: A fractional perspective, Advances in
Mechanical Engineering, 2015, Vol. 7(12) 18.
9. Y. Zhang, S. Wang, G. Liu and J.Yang, Computer aided diagnosis of abnormal breasts in mammogram images
by weighted type fractional Fourier transform, Advances in Mechanical Engineering, 2016, Vol. 8(2) 111.
10. S. Kempe, L. Gaul, Global solutions of fractional linear dierential equations, Proc. of ICIAM95, Zeitschrift Angew.
Math. Mech., vol76, suppl. 2, 1996, pp. 571â€572.
11. R. R. Nigmatullin, Y. E. Ryabov, ColeDavidson dielectric relaxation as a self-similar relaxation process, Phys.
Solid State, vol. 39, no.1, 1997, pp. 87â€90.
12. Y. F. Luchko, H. Matrinez, J. J. Trujillo, Fractional Fourier transform and some of its applications, Fractional
Calculus and Applied Analysis, Volume 11, Number 4 (2008).
13. G. Jumarie, Fouriers transform of fractional order via Mittag-Leffler function and modified Riemann-Liouville
derivative, J. Appl. Math. & Informatics, Vol. 26 (2008), No. 5â€6, pp. 1101â€1121.
14. L. Gaul, P.Klein, S. Kempe, Damping description involving fractional operators, Mech. Systems and Signal
Processings 5 (1991) 2, 81â€88.
15. H. Beyer, S. Kempe, Denition of Physically Consistent Damping Laws with Fractional Derivatives, Z. angew.
Math. Mech. 75 (1995) 8, 623â€635.
16. I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
17. I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fractional
Calculus and Applied Analysis 5 (4) (2002) 367386.
18. E. Demirci, N. Özalp, A method for solving differential equations of fractional order, J. Comput. Appl. Math. 236,
2754â€2762 (2012).
19. I. Koca, A method for solving differential equations of -fractional order, Applied Mathematics and Computation
266 (2015) 1â€5.
20. N. Ozalp, O. Ozturk Mizrak, Fractional Laplace transform method in the framework of the CTIT transformation,
Journal of Computational and Applied Mathematics 317 (2017), 90â€99.
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Published
2017-12-16
How to Cite
OZTURK MIZRAK, O. (2017). Solving the Oscillation Equation With Fractional Order Damping Term Using a New Fourier Transform Method. JOURNAL OF ADVANCES IN MATHEMATICS, 13(5), 7393–7397. https://doi.org/10.24297/jam.v13i5.6483
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