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On Sumudu transform method in discrete fractional calculus. (English) Zbl 1253.34014
Summary: Starting from the definition of the Sumudu transform on a general time scale, we define the generalized discrete Sumudu transform and present some of its basic properties. We obtain the discrete Sumudu transform of Taylor monomials, fractional sums, and fractional differences. We apply this transform to solve some fractional difference initial value problems.

MSC:
34A08Fractional differential equations
34N05Dynamic equations on time scales or measure chains
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Full Text: DOI
References:
[1] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science, Amsterdam, The Netherlands, 2006. · Zbl 1092.45003
[2] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives-Theory and Applications, Gordon and Breach Science Publishers, Linghorne, Pa, USA, 1993. · Zbl 0818.26003
[3] I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. · Zbl 0924.34008
[4] R. L. Magin, Fractional Calculus in Bioengineering, Begell House Publisher, Redding, Conn, USA, 2006.
[5] B. J. West, M. Bologna, and P. Grigolini, Physics of Fractal Operators, Institute for Nonlinear Science, Springer, New York, NY, USA, 2003. · doi:10.1007/978-0-387-21746-8
[6] N. Heymans and I. Podlubny, “Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives,” Rheologica Acta, vol. 45, pp. 765-771, 2006.
[7] K. S. Miller and B. Ross, “Fractional difference calculus,” in Proceedings of the Univalent Functions, Fractional Calculus, and Their Applications, pp. 139-152, Nihon University, 1989. · Zbl 0693.39002
[8] F. M. Atici and P. W. Eloe, “A transform method in discrete fractional calculus,” International Journal of Difference Equations, vol. 2, no. 2, pp. 165-176, 2007.
[9] F. M. Atici and P. W. Eloe, “Initial value problems in discrete fractional calculus,” Proceedings of the American Mathematical Society, vol. 137, no. 3, pp. 981-989, 2009. · Zbl 1166.39005 · doi:10.1090/S0002-9939-08-09626-3
[10] F. M. Atıcı and P. W. Eloe, “Discrete fractional calculus with the nabla operator,” Electronic Journal of Qualitative Theory of Differential Equations, no. 3, pp. 1-12, 2009. · Zbl 1189.39004 · emis:journals/EJQTDE/sped1/103.pdf · eudml:227223
[11] T. Abdeljawad and D. Baleanu, “Fractional differences and integration by parts,” Journal of Computational Analysis and Applications, vol. 13, no. 3, pp. 574-582, 2011. · Zbl 1225.39008
[12] T. Abdeljawad, “On Riemann and Caputo fractional differences,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1602-1611, 2011. · Zbl 1228.26008 · doi:10.1016/j.camwa.2011.03.036
[13] G. K. Watugala, “Sumudu transform: a new integral transform to solve differential equations and control engineering problems,” International Journal of Mathematical Education in Science and Technology, vol. 24, no. 1, pp. 35-43, 1993. · Zbl 0768.44003 · doi:10.1080/0020739930240105
[14] G. K. Watugala, “The Sumudu transform for functions of two variables,” Mathematical Engineering in Industry, vol. 8, no. 4, pp. 293-302, 2002. · Zbl 1025.44003
[15] M. A. Asiru, “Sumudu transform and the solution of integral equations of convolution type,” International Journal of Mathematical Education in Science and Technology, vol. 32, no. 6, pp. 906-910, 2001. · Zbl 1008.45003 · doi:10.1080/002073901317147870
[16] M. A. A\csiru, “Further properties of the Sumudu transform and its applications,” International Journal of Mathematical Education in Science and Technology, vol. 33, no. 3, pp. 441-449, 2002. · Zbl 1013.44001 · doi:10.1080/002073902760047940
[17] F. B. M. Belgacem, A. A. Karaballi, and S. L. Kalla, “Analytical investigations of the Sumudu transform and applications to integral production equations,” Mathematical Problems in Engineering, no. 3-4, pp. 103-118, 2003. · Zbl 1068.44001 · doi:10.1155/S1024123X03207018 · eudml:51518
[18] F. B. M. Belgacem and A. A. Karballi, “Sumudu transform fundemantal properties investigations and applications,” Journal of Applied Mathematics and Stochastic Analysis, vol. 2006, Article ID 91083, 23 pages, 2006. · Zbl 1115.44001 · doi:10.1155/JAMSA/2006/91083 · eudml:53202
[19] A. Kılı\ccman and H. Eltayeb, “On the applications of Laplace and Sumudu transforms,” Journal of the Franklin Institute, vol. 347, no. 5, pp. 848-862, 2010. · Zbl 1286.35185 · doi:10.1016/j.jfranklin.2010.03.008
[20] F. B. M. Belgacem, “Introducing and analysing deeper Sumudu properties,” Nonlinear Studies, vol. 13, no. 1, pp. 23-41, 2006. · Zbl 1102.44001
[21] F. Jarad and K. Tas, “Application of Sumudu and double Sumudu transforms to Caputo-Fractional dierential equations,” Journal of Computational Analysis and Applications, vol. 14, no. 3, pp. 475-483, 2012. · Zbl 1260.34013
[22] Q. D. Katatbeh and F. B. M. Belgacem, “Applications of the Sumudu transform to fractional differential equations,” Nonlinear Studies, vol. 18, no. 1, pp. 99-112, 2011. · Zbl 1223.44001
[23] M. Bohner and G. Sh. Guseinov, “The h-Laplace and q-Laplace transforms,” Journal of Mathematical Analysis and Applications, vol. 365, no. 1, pp. 75-92, 2010. · Zbl 1188.44008 · doi:10.1016/j.jmaa.2009.09.061
[24] S. Hilger, “Analysis on measure chains-a unified approach to continuous and discrete calculus,” Results in Mathematics, vol. 18, no. 1-2, pp. 18-56, 1990. · Zbl 0722.39001 · doi:10.1007/BF03323153
[25] M. Bohner and A. Peterson, Dynamic Equations on Time Scales, Birkhäuser, Boston, Mass, USA, 2001. · Zbl 0993.39010 · doi:10.1007/978-1-4612-0201-1
[26] M. Bohner and A. Peterson, Advances in Dynamic equations on Time Scales, Birkhäuser, Boston, Mass, USA, 2003. · Zbl 1025.34001 · doi:10.1007/978-0-8176-8230-9
[27] M. T. Holm, The theory of discrete fractional calculus: development and application [Ph.D. thesis], 2011. · Zbl 1248.39003
[28] F. Jarad, K. Bayram, T. Abdeljawad, and D. Baleanu, “On the discrete sumudu transform,” Romanian Reports in Physics. In press.