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Numerical studies for a multi-order fractional differential equation. (English) Zbl 1209.65116

Summary: We implement the variational iteration method and the homotopy perturbation method, for solving the system of fraction differential equations (FDE) generated by a multi-order fraction differential equation. The fractional derivatives are described in the Caputo sense. In these schemes, the solution takes the form of a convergent series with easily computable components. Numerical results show that the two approaches are easy to implement and accurate when applied to partial differential equations of fractional order. An algorithm to convert a multi-order FDE has been suggested which is valid in the most general cases.

MSC:

65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
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[1] Abdou, M. A.; Soliman, A. A., Physica D, 211, 1 (2005)
[2] Abdou, M. A.; Soliman, A. A., J. Comput. Appl. Math., 1 (2004)
[3] Bagley, R. L.; Calico, R. A., J. Guidance Control Dynamics, 14, 2 (1999)
[4] Benson, D. A.; Wheatcraft, S. W.; Meerschaert, M. M., Water Resour. Res., 36, 6, 1403 (2000)
[5] Benson, D. A.; Wheatcraft, S. W.; Meerschaert, M. M., Water Resour. Res., 36, 6, 1413 (2000)
[6] Daftardar-Gejji, V.; Jafari, H., Appl. Math. Comput., 189, 1, 541 (2007)
[7] Diethelm, K.; Ford, N. J., Appl. Math. Comput., 154, 621 (2004)
[8] Djrbashian, M. M.; Nersesian, A. B., Izv. Acad. Nauk Armjanskoy SSR, 3, 1, 3 (1968), (in Russian)
[9] He, J. H., Int. J. Mod. Phys. B, 20, 10, 1141 (2006)
[10] He, J. H., Int. J. Non-Linear Mech., 34, 699 (1999)
[11] He, J. H.; Wu, X.-H., Chaos Solitons Fractals, 29, 108 (2006)
[12] Huang, F.; Liu, F., ANZIAM J., 46, 317 (2005)
[13] Ichise, M.; Nagayanagi, Y.; Kojima, T., J. Electroanal. Chem. Interfacial Electrochem., 33, 253 (1971)
[14] Kilbas, A. A.; Saigo, M.; Saxena, R. K., J. Integral Equations Appl., 14, 4, 377 (2002) · Zbl 1041.45011
[15] Koeller, R. C., Acta Mech., 58, 299 (1984)
[16] Koeller, R. C., J. Appl. Mech., 51, 299 (1984)
[17] Liu, F.; Anh, V.; Turner, I.; Zhuang, P., J. Appl. Math. Comput., 233 (2003)
[18] Mandelbrot, B., IEEE Trans. Inform. Theory, 13, 2, 289 (1967)
[19] Mark, R. J.; Hall, M. W., IEEE Trans. Acoust. Speech Signal Process., 29, 872 (1981)
[20] Miller, K. S.; Boss, B., An Introduction to the Fractional Calculus and Fractional Differential Equations (1993), Wiley: Wiley New York
[21] Oldham, K. B.; Spanier, J. C., The Fractional Calculus (1974), Academic Press: Academic Press San Diego, CA
[22] Podlubny, I., Fractional Differential Equations (1999), Academic Press: Academic Press San Diego, CA · Zbl 0918.34010
[23] Skaar, S. B.; Michel, A. N.; Miller, R. K., IEEE Trans. Automat. Control, 3, 4, 348 (1988)
[24] Sweilam, N. H.; Khader, M. M., Chaos Solitons Fractals, 32, 145 (2007)
[25] Sweilam, N. H., J. Comput. Appl. Math., 207, 64 (2007)
[26] Sun, H. H.; Onaral, N.; Tsao, Y., IEEE Trans. Ciomed. Eng., 31, 10, 664 (1984)
[27] Sun, H. H.; Abdelwahab, A. A.; Onaral, B., IEEE Trans. Automat. Control, 29, 5, 441 (1984)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.