×

Homotopy analysis method for fractional IVPs. (English) Zbl 1221.65277

Summary: The homotopy analysis method is applied to solve linear and nonlinear fractional initial-value problems (fIVPs). The fractional derivatives are described by Caputo’s sense. Exact and/or approximate analytical solutions of the fIVPs are obtained. The results of applying this procedure to the studied cases show the high accuracy and efficiency of the approach.

MSC:

65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
45A05 Linear integral equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Oldham, K. B.; Spanier, J., The fractional calculus (1974), Academic Press: Academic Press New York · Zbl 0428.26004
[2] Miller, K. S.; Ross, B., An introduction to the fractional calculus and fractional differential equations (1993), John Wiley and Sons: John Wiley and Sons New York · Zbl 0789.26002
[3] Luchko Y, Gorenflo R. The initial-value problem for some fractional differential equations with Caputo derivative. Preprint Series A08-98. Fachbereich Mathematik und Informatic, Berlin, Freie Universitat; 1998.; Luchko Y, Gorenflo R. The initial-value problem for some fractional differential equations with Caputo derivative. Preprint Series A08-98. Fachbereich Mathematik und Informatic, Berlin, Freie Universitat; 1998. · Zbl 0940.45001
[4] Podlubny, I., Fractional differential equations (1999), Academic Press: Academic Press New York · Zbl 0918.34010
[5] Shawagfeh, N. T., Analytical approximate solutions for nonlinear fractional differential equations, Appl Math Comput, 131, 517-529 (2002) · Zbl 1029.34003
[6] Ray, S. S.; Bera, R. K., Solution of an extraordinary differential equation by Adomian decomposition method, J Appl Math, 4, 331-338 (2004) · Zbl 1080.65069
[7] Ray, S. S.; Bera, R. K., An approximate solution of a nonlinear fractional differential equation by Adomian decomposition method, Appl Math Comput, 167, 561-571 (2005) · Zbl 1082.65562
[8] Abdulaziz, O.; Hashim, I.; Chowdhury, M. S.H.; Zulkifle, A. K., Assessment of decomposition method for linear and nonlinear fractional differential equations, Far East J Appl Math, 28, 1, 95-112 (2007) · Zbl 1134.26300
[9] Abdulaziz O, Hashim I, Ismail ES. Approximate analytical solutions to fractional modified KdV equations. Far East J Appl Math, [in press].; Abdulaziz O, Hashim I, Ismail ES. Approximate analytical solutions to fractional modified KdV equations. Far East J Appl Math, [in press]. · Zbl 1165.35441
[10] Momani, S.; Odibat, Z., Homotopy perturbation method for nonlinear partial differential equations of fractional order, Phys Lett A, 365, 345-350 (2007) · Zbl 1203.65212
[11] Odibat, Z.; Momani, S., Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order, Chaos Solitons Fractals, 36, 167-174 (2008) · Zbl 1152.34311
[12] Odibat, Z.; Momani, S., Application of variational iteration method to nonlinear differential equation of fractional order, Int J Nonlinear Sci Numer Simul, 1, 7, 271-279 (2006)
[13] Momani, S.; Odibat, Z., Numerical comparison of methods for solving linear differential equations of fractional order, Chaos Solitons Fractals, 31, 1248-1255 (2007) · Zbl 1137.65450
[14] Liao SJ. The proposed homotopy analysis technique for the solution of nonlinear problems. Ph.D thesis, Shanghai Jiao Tong University; 1992.; Liao SJ. The proposed homotopy analysis technique for the solution of nonlinear problems. Ph.D thesis, Shanghai Jiao Tong University; 1992.
[15] Liao, S. J., An approximate solution technique which does not depend upon small parameters: a special example, Int J Nonlinear Mech, 30, 371-380 (1995) · Zbl 0837.76073
[16] Liao, S. J., An approximate solution technique which does not depend upon small parameters (II): an application in fluid mechanics, Int J Nonlinear Mech, 32, 815-822 (1997) · Zbl 1031.76542
[17] Liao, S. J., An explicit, totally analytic approximation of Blasius viscous flow problems, Int J Nonlinear Mech, 34, 4, 759-778 (1999) · Zbl 1342.74180
[18] Liao, S. J., Beyond perturbation: introduction to the homotopy analysis method (2003), Chapman & Hall, CRC Press: Chapman & Hall, CRC Press Boca Raton
[19] Liao, S. J., On the homotopy analysis method for nonlinear problems, Appl Math Comput, 147, 499-513 (2004) · Zbl 1086.35005
[20] Liao, S. J.; Campo, A., Analytic solutions of the temperature distribution in Blasius viscous flow problems, J Fluid Mech, 453, 411-425 (2002) · Zbl 1007.76014
[21] Liao, S. J., On the analytic solution of magnetohydrodynamic flows of non-Newtonian fluids over a stretching sheet, J Fluid Mech, 488, 189-212 (2003) · Zbl 1063.76671
[22] Ayub, M.; Rasheed, A.; Hayat, T., Exact flow of a third grade fluid past a porous plate using homotopy analysis method, Int J Eng Sci, 41, 2091-2103 (2003) · Zbl 1211.76076
[23] Hayat, T.; Khan, M.; Asghar, S., Homotopy analysis of MHD flows of an Oldroyd 8-constant fluid, Acta Mech, 168, 213-232 (2004) · Zbl 1063.76108
[24] Hayat, T.; Khan, M.; Asghar, S., Magnetohydrodynamic flow of an Oldroyd 6-constant fluid, Appl Math Comput, 155, 417-425 (2004) · Zbl 1126.76388
[25] Abbasbandy, S., Homotopy analysis method for heat radiation equations, Int Comm Heat Mass Transfer, 34, 380-387 (2007)
[26] Abbasbandy, S., The application of homotopy analysis method to solve a generalized Hirota-Satsuma coupled KdV equation, Phys Lett A, 361, 478-483 (2007) · Zbl 1273.65156
[27] Abbasbandy S. Approximate solution for the nonlinear model of diffusion and reaction in porous catalysts by means of the homotopy analysis method. Chem Eng J 2007. doi:10.1016/j.cej.2007.03.022; Abbasbandy S. Approximate solution for the nonlinear model of diffusion and reaction in porous catalysts by means of the homotopy analysis method. Chem Eng J 2007. doi:10.1016/j.cej.2007.03.022
[28] Bataineh, A. S.; Noorani, M. S.M.; Hashim, I., Solving systems of ODEs by homotopy analysis method, Commun Nonlinear Sci Numer Sim, 13, 10, 2060-2070 (2008) · Zbl 1221.65194
[29] Bataineh AS, Noorani MSM, Hashim I. Application of homotopy analysis method to nonlinear heat transfer equation [submitted for publication].; Bataineh AS, Noorani MSM, Hashim I. Application of homotopy analysis method to nonlinear heat transfer equation [submitted for publication].
[30] Song L, Zhang H. Application of homotopy analysis method to fractional KdV-Burgers-Kuramoto equation. Phys Lett A, 2007. doi:10.1016/j.physleta.2007.02.083; Song L, Zhang H. Application of homotopy analysis method to fractional KdV-Burgers-Kuramoto equation. Phys Lett A, 2007. doi:10.1016/j.physleta.2007.02.083 · Zbl 1209.65115
[31] Gorenflo, R.; Mainardi, F., Fractional calculus: integral and differential equations of fractional order, (Fractals and fractional calculus in continuum mechanics (1997), Springer-Verlag: Springer-Verlag Wien and New York), 223-276 · Zbl 1438.26010
[32] Diethelm, K.; Ford, N. J.; Freed, A. D., A predictor-corrector approach for the numerical solution of fractional differential equation, Nonlinear Dyn, 29, 3-22 (2002) · Zbl 1009.65049
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.