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Analytical treatment of differential equations with fractional coordinate derivatives. (English) Zbl 1228.65200
Summary: We outline a reliable strategy to use the homotopy perturbation method based on Jumarie’s derivative for solving fractional differential equations. In this framework, compact structures of fourth-order fractional diffusion-wave equations are considered as prototype examples. Moreover, convergence of the proposed approach for these types of equations is investigated. Results show that the response expressions are Mittag-Leffler stable.

65M99Numerical methods for IVP of PDE
35R11Fractional partial differential equations
26A33Fractional derivatives and integrals (real functions)
45K05Integro-partial differential equations
Full Text: DOI
[1] Podlubny, I.: Fractional differential equations, (1999) · Zbl 0924.34008
[2] Machado, J. Tenreiro; Kiryakova, V.; Mainardi, F.: Recent history of fractional calculus, Communications in nonlinear science and numerical simulation 16, 1140-1153 (2011) · Zbl 1221.26002 · doi:10.1016/j.cnsns.2010.05.027
[3] Hu, X. B.; Wu, Y. T.: Application of the Hirota bilinear formalism to a new integrable differential-difference equation, Physics letters A 246, 523-529 (1998)
[4] Adomian, G.: Solving frontier problems of physics: the decomposition method, (1994) · Zbl 0802.65122
[5] Fan, E.: Two applications of the homogeneous balance method, Physics letters A 256, 353-357 (2000) · Zbl 0947.35012 · doi:10.1016/S0375-9601(00)00010-4
[6] Vakhnenko, V. O.; Parkes, E. J.; Morrision, A. J.: A Bäcklund transformation and their inverse scatting transform method for the generalized Vakhnenko equation, Chaos, solitons and fractals 17, 683-692 (2003) · Zbl 1030.37047 · doi:10.1016/S0960-0779(02)00483-6
[7] Xu, H.; Liao, S. J.; You, X. C.: Analysis of nonlinear fractional partial differential equations with the homotopy analysis method, Communications in nonlinear science and numerical simulation 14, 1152-1156 (2009) · Zbl 1221.65286 · doi:10.1016/j.cnsns.2008.04.008
[8] He, J. H.: Variational iteration method--some recent results and new interpretations, Journal of computational and applied mathematics 207, No. 1, 3-17 (2007) · Zbl 1119.65049 · doi:10.1016/j.cam.2006.07.009
[9] He, J. H.; Shou, D. H.: Application of parameter-expanding method to strongly nonlinear oscillators, International journal of nonlinear sciences and numerical simulation 8, 121-124 (2007)
[10] Odibat, Z.; Momani, S.; Erturk, V. S.: Generalized differential transform method: application to differential equations of fractional order, Applied mathematics and computation 197, 467-477 (2008) · Zbl 1141.65092 · doi:10.1016/j.amc.2007.07.068
[11] He, J. H.: Homotopy perturbation technique, Computer methods in applied mechanics and engineering 178, 257-262 (1999) · Zbl 0956.70017
[12] Golbabai, A.; Sayevand, K.: A study on the multi-order time fractional differential equations with using homotopy perturbation method, Nonlinear science letters A 2, 141-147 (2010)
[13] Khan, N. A.; Ara, A.; Ali, S. A.: Analytical study of Navier--Stokes equationwith fractional order using he’s homotopy perturbation and variational iteration methods, International journal of nonlinear sciences and numerical simulation 10, 1127-1134 (2009)
[14] Yildirim, A.: An algorithm for solving the fractional nonlinear schördinger equation by means of the homotopy perturbation method, International journal of nonlinear sciences and numerical simulation 10, 445-450 (2009)
[15] Jafari, H.; Momani, S.: Solving fractional diffusion and wave equations by modified homotopy perturbation method, Physics letters A 370, 388-396 (2007) · Zbl 1209.65111 · doi:10.1016/j.physleta.2007.05.118
[16] Ganji, D. D.; Ganji, S. S.; Karimpur, S.; Ganji, Z. Z.: Numerical study of homotopy--perturbation method to Burgers equation in fluid, Numerical methods for partial differential equations 26, No. 4, 114-124 (2010) · Zbl 05668320
[17] Dehghan, M.; Shakeri, F.: The numerical solution of the second Painlevé equation, Numerical methods for partial differential equations 25, No. 5, 1238-1259 (2009) · Zbl 1172.65037 · doi:10.1002/num.20416
[18] Agrawal, O. P.: A general solution for a fourth-order fractional diffusion-wave equation defined in a bounded domain, Computers structures 79, 1497-1501 (2001)
[19] Jumarie, G.: Modified Riemann--Liouville derivative and fractional Taylor series of non-differentiable functions further results, Computers and mathematics with applications 51, 1367-1376 (2006) · Zbl 1137.65001 · doi:10.1016/j.camwa.2006.02.001
[20] Jumarie, G.: Table of some basic fractional calculus formulate derived from a modified Riemann--Liouville derivative for non-differentiable functions, Applied mathematics letters 22, 378-385 (2009) · Zbl 1171.26305 · doi:10.1016/j.aml.2008.06.003
[21] N. Faraz, Y. Khan, H. Jafari, M. Madani, Fractional variational iteration method via modified Riemann--Liouville derivative, in press (doi:10.1016/j.jksus.2010.07.025).
[22] Mainardi, F.; Gorenflo, R.: On Mittag-Leffler-type functions in fractional evolution processes, Journal of computational and applied mathematics 118, 283-299 (2000) · Zbl 0970.45005 · doi:10.1016/S0377-0427(00)00294-6
[23] Wazwaz, A.: A new algorithm for calculating Adomian polynomials for nonlinear operator, Applied mathematics and computation 111, 53-69 (2000) · Zbl 1023.65108 · doi:10.1016/S0096-3003(99)00047-8
[24] Linz, P.: Analytical and numrical methods for Volterra equations, (1985) · Zbl 0566.65094
[25] Agrawal, O. P.: Solution for a fractional diffusion-wave equation defined in a bounded domain, Nonlinear dynamics 29, 145-155 (2002) · Zbl 1009.65085 · doi:10.1023/A:1016539022492
[26] Mainardi, F.: The fundamental solutions for the fractional diffusion-wave equation, Applied mathematics letters 9, 23-28 (1996) · Zbl 0879.35036 · doi:10.1016/0893-9659(96)00089-4
[27] Agrawal, O. P.: A general solution for the fourth-order fractional diffusion-wave equation, Fractional calculus and applied analysis 3, 1-12 (2000) · Zbl 1111.45300
[28] Momani, S.; Hadid, S.: Lyapunov stability solutions of fractional integrodifferential equations, International journal of mathematics and mathematical sciences 47, 2503-2507 (2004) · Zbl 1074.45006 · doi:10.1155/S0161171204312366