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IJSTR >> Volume 9 - Issue 4, April 2020 Edition

International Journal of Scientific & Technology Research  
International Journal of Scientific & Technology Research

Website: http://www.ijstr.org

ISSN 2277-8616

Fractionalization Of Fourier Sine And Fourier Cosine Transforms And Their Applications

[Full Text]



Teekam Chand Mahor, Rajshree Mishra, Renu Jain



Mittag-Leffler function, Fractional Fourier transform, fractional derivative and fractional integral, fractional partial differential equation.



The fractional Fourier transform (FrFT) is a generalization of classical Fourier transform and received considerable attention of researchers since last four decades due to its wide ranging applicability in various fields such as, electrical engineering, optics, signal processing, signal analysis, optical communication and quantum mechanics. Fractional Fourier sine transform (FrFST) and fractional Fourier cosine transform (FrFCT) are closely related to the fractional Fourier transform (FrFT). In this paper, we introduce the new definition of FrFST and FrFCT of real order α using Mittag-Leffler function in fractional calculus environment. We have derived many algebraic properties including inversion formula, modulation theorem and Parseval’s identity of this new FrFST and FrFCT analytically. In addition, we have discussed FrFST and FrFCT of some standard functions and have applied these transforms to obtain analytical solutions of fractional heat-diffusion and fractional wave equations.



I.N. Sneddon, “Fourier transform”. Dover Publication. INC, New York, 1995.
J.F. James, “Fourier transform with applications in physics and engineering”. New York Cambridge University Press, 1995.
R. Bracewell, “The Fourier transform and its applications”, 3rded. New York, McGraw-Hill, 74-104, 1999.
N. Wiener, “Hermitian polynomials and Fourier analysis”, Studies in Applied Mathematics, 8(1-4), 70-73, 1929.
V. Namias, “The fractional order Fourier transform and its application to quantum mechanics”, J. Inst. Math. Appl., 25(3), 241-265, 1980.
A.C. McBride and F.H. Kerr, “Namias’s fractional Fourier transform”, IMA Journal of Applied Mathematics, 39(2), 159-175, 1987.
D.H. Bailey and P.N. Swarztrauber, “The fractional Fourier transform and applications”, SIAM Review, 33(3), 389-404, 1991.
A.W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform”, JOSA A, 10(10), 2181-2186, 1993.
L.B. Almeida, “The fractional Fourier transform and time frequency representation”, IEEE Transactions on Signal Processing, 42(11), 3084-3091, 1994.
H.M. Ozaktas, O. Arikan, M.A. Kutay and G. Bozdagt, “Digital computation of the fractional Fourier transform”, IEEE Transactions on Signal Processing, 44(9), 2141-2150, 1996.
R.Tao, Y. L. Li and Y. Wang, “Short-time fractional Fourier transform and its applications”, IEEE Transactions on Signal Processing, 58(5), 2568-2580, 2010.
H.M Ozaktas, M.A. Kutay and Z. Zalevsky, “The fractional Fourier transform with applications in optics and signal processing”, New York Wiley. 2000.
B. West, M. Bologna and P. Grigolini, “Physics of fractal operators”, International Edition Springer, 2003.
G. Jumarie, “Fourier’s transform of fractional Order via Mittag-Leffler function and modified Riemann-Liouville derivative”, Journal Applied & Informatics, 26(5_6), 1101-1121, 2008.
A.W. Lohmann, D. Mendlovic, Z. Zalevsky and R.G. Dorsch, "Some Important Fractional Transformation for Signal Processing”, Opt. commun. 125, 18-20, 1996.
S.C. Pei and M.H. Yeh, “The discrete fractional cosine and sine transforms”, IEEE Transaction on Signal Processing, 49(6), 1198-1207, 2001.
G. Cariolaro, T. Erseghe and P. Kraniuaskas, “The fractional discrete cosine transform”, IEEE Transaction on Signal Processing, 50(4), 902-911, 2002.
T. Alieva and M.J. Bastiaans, “Fractional cosine and sine transform in relation to the fractional Fourier and Hartley transforms”, 7th International Symposium on signal processing and its Applications, Malaysia, 1, 561-564, 2003.
C. Vijaya and J.S. Bhat, Signal Compression using Discrete Fractional Fourier Transform and Set Partitioning in hierarchical tree, Signal Processing, 86(8), 1976-1983, 2006.
G. Jumarie, “Modified Riemann Liouville derivative and fractional Taylor series of non-differentiable functions further result”, Computer and Mathematics with Applications, 51(9-10), 1367-1376, 2006.
G.M. Mittag-Leffler, “Sur la nouvelle function E_α (x)”, CR Acad. Sci. Paris, 137(2), 554-558, 1903.
G. Jumarie, “Table of some basic fractional calculus formulae derived from a modified Riemann-Liouville derivative for non- differenyiable functions”, Applied Mathematics Letters, 22, 378-385, 2009.
B. Gua, X. Pu and F. Haung, “Fractional partial differential equations and their numerical solutions”, World Scientific, 2015.