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Numerical approach to differential equations of fractional order. (English) Zbl 1119.65127
Summary: The variational iteration method and the Adomian decomposition method are implemented to give approximate solutions for systems of linear and nonlinear differential equations of fractional order. The two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of differential equations. In these schemes, the solution takes the form of a convergent series with easily computable components. This paper presents a numerical comparison between the two methods for solving systems of fractional differential equations. Numerical results show that the two approaches are easy to implement and accurate when applied to differential equations of fractional order.

65R20Integral equations (numerical methods)
45J05Integro-ordinary differential equations
45G10Nonsingular nonlinear integral equations
65L05Initial value problems for ODE (numerical methods)
34A34Nonlinear ODE and systems, general
26A33Fractional derivatives and integrals (real functions)
Full Text: DOI
[1] Abbaoui, K.; Cherruault, Y.: New ideas for proving convergence of decomposition methods. Comput. math. Appl. 29, No. 7, 103-108 (1996) · Zbl 0832.47051
[2] Abulwafa, E. M.; Abodu, M. A.; Mohmoud, A. A.: The solution of nonlinear coagulation problem with mass loss. Chaos solitons fractals 29, No. 2, 313-330 (2006)
[3] Adomian, G.: A review of the decomposition method in applied mathematics. J. math. Anal. appl. 135, 501-544 (1988) · Zbl 0671.34053
[4] Adomian, G.: Solving frontier problems of physics: the decomposition method. (1994) · Zbl 0802.65122
[5] Beyer, H.; Kempfle, S.: Definition of physically consistent damping laws with fractional derivatives. Z. angew. Math. mech. 75, 623-635 (1995) · Zbl 0865.70014
[6] Bildik, N.; Konuralp, A.: The use of variational iteration method, differential transform method and Adomian decomposition method for solving different types of nonlinear partial differential equations. Internat. J. Nonlinear sci. Numer. simul. 7, No. 1, 65-70 (2006) · Zbl 1115.65365
[7] L. Blank, Numerical treatment of differential equations of fractional order, Numerical Analysis Report 287, Manchester Center for Computational Mathematics, Manchester, 1996. · Zbl 0870.65137
[8] Caputo, M.: Linear models of dissipation whose Q is almost frequency independent, part II. J. roy. Astron. soc. 13, 529-539 (1967)
[9] Cherruault, Y.; Adomian, G.: Decomposition methods: a new proof of convergence. Math. comput. Modelling 18, 103-106 (1993) · Zbl 0805.65057
[10] Daftardar-Gejji, V.; Jafari, H.: An iterative method for solving nonlinear functional equations. J. math. Anal. appl. 316, 753-763 (2006) · Zbl 1087.65055
[11] Diethelm, K.: An algorithm for the numerical solution for differential equations of fractional order. Electron. trans. Numer. anal. 5, 1-6 (1997) · Zbl 0890.65071
[12] Diethelm, K.; Ford, N.: Analysis of fractional differential equations. J. math. Anal. appl. 265, 229-248 (2002) · Zbl 1014.34003
[13] Diethelm, K.; Ford, N.; Freed, A.: A predictor -- corrector approach for the numerical solution of fractional differential equations. Nonlinear dynam. 29, 3-22 (2002) · Zbl 1009.65049
[14] Diethelm, K.; Ford, N.; Freed, A.: Detailed error analysis for a fractional Adams method. Numer. algorithms 36, 31-52 (2004) · Zbl 1055.65098
[15] Diethelm, K.; Walz, G.: Numerical solution for fractional differential equations by extrapolation. Numer. algorithms 16, 231-253 (1997) · Zbl 0926.65070
[16] R. Gorenflo, Fractional calculus: some numerical methods, in: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus, New York, 1997.
[17] R. Gorenflo, F. Mainardi, Fractional calculus: integral and differential equations of fractional order, in: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus, New York, 1997. · Zbl 0934.35008
[18] Hao, T. H.: Search for variational principles in electrodynamics by Lagrange method. Internat. J. Nonlinear sci. Numer. simul. 6, No. 2, 209-210 (2005)
[19] He, J. H.: Variational iteration method for delay differential equations. Comm. nonlinear sci. Numer. simul. 2, No. 4, 235-236 (1997) · Zbl 0924.34063
[20] He, J. H.: Semi-inverse method of establishing generalized principles for fluid mechanics with emphasis on turbomachinery aerodynamics. Internat. J. Turbo jet-engines 14, No. 1, 23-28 (1997)
[21] He, J. H.: Approximate solution of nonlinear differential equations with convolution product nonlinearities. Comput. methods appl. Mech. eng. 167, 69-73 (1998) · Zbl 0932.65143
[22] He, J. H.: Approximate analytical solution for seepage flow with fractional derivatives in porous media. Comput. methods appl. Mech. eng. 167, 57-68 (1998) · Zbl 0942.76077
[23] He, J. H.: Variational iteration method --- a kind of non-linear analytical technique: some examples. Internat. J. Nonlinear mech. 34, 699-708 (1999) · Zbl 05137891
[24] He, J. H.: Variational iteration method for autonomous ordinary differential systems. Appl. math. Comput. 114, 115-123 (2000) · Zbl 1027.34009
[25] He, J. H.: Variational theory for linear magneto-electro-elasticity. Internat. J. Nonlinear sci. Numer. simul. 2, No. 4, 309-316 (2001) · Zbl 1083.74526
[26] He, J. H.: Variational principle for some nonlinear partial differential equations with variable coefficients. Chaos solitons fractals 19, No. 4, 847-851 (2004) · Zbl 1135.35303
[27] He, J. H.: Asymptotic methods for strongly nonlinear equations. Internat. J. Modern phys. B 20, No. 10, 1141-1199 (2006) · Zbl 1102.34039
[28] J.H. He, Non-perturbative methods for strongly nonlinear problems, dissertation.de-Verlag im Internet GmbH, Berlin, 2006.
[29] He, J. H.; Wu, X. H.: Construction of solitary solution and compaction-like solution by variational iteration method. Chaos solitons fractals 29, No. 1, 108-113 (2006) · Zbl 1147.35338
[30] Huang, F.; Liu, F.: The time-fractional diffusion equation and fractional advection -- dispersion equation. Anziam j. 46, 1-14 (2005)
[31] Huang, F.; Liu, F.: The fundamental solution of the space-time fractional advection -- dispersion equation. J. appl. Math. comput. 18, No. 2, 339-350 (2005) · Zbl 1086.35003
[32] Inokuti, M.; Sekine, H.; Mura, T.: General use of the Lagrange multiplier in non-linear mathematical physics. Variational method in the mechanics of solids, 156-162 (1978)
[33] Y. Luchko, R. Gorneflo, The initial value problem for some fractional differential equations with the Caputo derivative, Preprint series A08-98, Fachbreich Mathematik und Informatik, Freic Universitat Berlin, 1998.
[34] Mainardi, F.: Fractional relaxation -- oscillation and fractional diffusion-wave phenomena. Chaos solitons fractals 7, 1461-1477 (1996) · Zbl 1080.26505
[35] Miller, K. S.; Ross, B.: An introduction to the fractional calculus and fractional differential equations. (1993) · Zbl 0789.26002
[36] Momani, S.: An explicit and numerical solutions of the fractional KdV equation. Math. comput. Simul. 70, No. 2, 110-118 (2005) · Zbl 1119.65394
[37] Momani, S.: Non-perturbative analytical solutions of the space- and time-fractional Burgers equations. Chaos solitons fractals 28, No. 4, 930-937 (2006) · Zbl 1099.35118
[38] Momani, S.; Abuasad, S.: Application of he’s variational iteration method to Helmholtz equation. Chaos solitons fractals 27, No. 5, 1119-1123 (2006) · Zbl 1086.65113
[39] S. Momani, Z. Odibat, Analytical solution of a time-fractional Navier -- Stokes equation by Adomian decomposition method, Appl. Math. Comput., 177(2) (2006) 488 -- 494. · Zbl 1096.65131
[40] S. Momani, Z. Odibat, Numerical comparison of methods for solving linear differential equations of fractional order, Chaos Solitons Fractals, 355 (2006) 271 -- 279. · Zbl 05675858
[41] S. Momani, Z. Odibat, Analytical approach to linear fractional partial differential equations arising in fluid mechanics, Phys. Lett. A, in press. · Zbl 05675858
[42] S. Momani, R. Qaralleh, An efficient method for solving systems of fractional integro-differential equations, Comput. Math. Appl., accepted for publication. · Zbl 1137.65072
[43] Odibat, Z.; Momani, S.: Application of variational iteration method to nonlinear differential equations of fractional order. Internat. J. Nonlinear sci. Numer. simul. 1, No. 7, 15-27 (2006) · Zbl 05675858
[44] Z. Odibat, S. Momani, Approximate solutions for boundary value problems of time-fractional wave equation, Appl. Math. Comput., in press. · Zbl 1148.65100
[45] Oldham, K. B.; Spanier, J.: The fractional calculus. (1974) · Zbl 0292.26011
[46] Podlubny, I.: Numerical solution of ordinary fractional differential equations by the fractional difference method. Advances in difference equations (1997) · Zbl 0893.65051
[47] Podlubny, I.: Fractional differential equations. (1999) · Zbl 0924.34008
[48] Shawagfeh, N.: Analytical approximate solutions for nonlinear fractional differential equations. Appl. math. Comput. 131, No. 2, 517-529 (2002) · Zbl 1029.34003
[49] Shawagfeh, N.; Kaya, D.: Comparing numerical methods for the solutions of systems of ordinary differential equations. Appl. math. Lett. 17, 323-328 (2004) · Zbl 1061.65062
[50] Soliman, A. A.: A numerical simulation and explicit solutions of KdV -- Burgers and Lax’s seventh-order KdV equations. Chaos solitons fractals 29, No. 2, 294-302 (2006) · Zbl 1099.35521
[51] Torvik, P. T.; Bagley, R. L.: On the appearance of the fractional derivative in the behavior of real materials. J. appl. Mech. 51, 294-298 (1996) · Zbl 1203.74022
[52] Wazwaz, A.: A new algorithm for calculating Adomian polynomials of nonlinear operators. Appl. math. Comput. 111, 53-69 (2000) · Zbl 1023.65108
[53] Wazwaz, A.; El-Sayed, S.: A new modification of the Adomian decomposition method for linear and nonlinear operators. Appl. math. Comput. 122, 393-405 (2001) · Zbl 1027.35008