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Convergence of the variational iteration method for solving multi-order fractional differential equations. (English) Zbl 1207.65109
Summary: The variational iteration method (VIM) is applied to obtain approximate solutions of multi-order fractional differential equations (M-FDEs). We can easily obtain the satisfying solution just by using a few simple transformations and applying the VIM. A theorem for convergence and error estimates of the VIM for solving M-FDEs is given. Moreover, numerical results show that our theoretical analysis are accurate and the VIM is a powerful method for solving M-FDEs.

##### MSC:
 65L99 Numerical methods for ordinary differential equations 34A08 Fractional ordinary differential equations 26A33 Fractional derivatives and integrals 45J05 Integro-ordinary differential equations
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##### References:
 [1] A.Y. Luchko, R. Groreflo, 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. [2] Podlubny, I., Fractional differential equations, (1999), Academic Press New York · Zbl 0918.34010 [3] Bagley, R.L.; Torvik, P.J., Fractional calculus in the transient analysis of viscoelastically damped structures, Aiaa j., 23, 6, 918-925, (1983) · Zbl 0562.73071 [4] Ichise, M.; Nagayanagi, Y.; Kojima, T., An analog simulation of non-integer order transfer functions for analysis of electrode processes, J. electroanal. chem., 33, 253-265, (1971) [5] Sun, H.H.; Onaral, B.; Tsao, Y., Application of positive reality principle to metal electrode linear polarization phenomena, IEEE trans. biomed. eng., BME-31, 10, 664-674, (1984) [6] Sun, H.H.; Abdelwahab, A.A.; Onaral, B., Linear approximation of transfer function with a pole of fractional order, IEEE trans. autom. control, AC-29, 5, 441-444, (1984) · Zbl 0532.93025 [7] Li, Chunguang; Chen, Guanrong, Chaos and hyperchaos in the fractional-order Rössle equations, Physica A, 341, 55-61, (2004) [8] Lubich, C., Runge – kutta theory for Volterra and Abel integral equations of the second kind, Math. comp., 41, 87-102, (1983) · Zbl 0538.65091 [9] Lubich, C., Fractional linear multistep methods for abel – volterra integral equations of the second kind, Math. comp., 45, 463-469, (1985) · Zbl 0584.65090 [10] Lubich, C., Discretized fractional calculus, SIAM J. math. anal., 17, 704-719, (1986) · Zbl 0624.65015 [11] Diethelm, K., An algorithm for the numerical solution of differential equations of fractional order, Electron. trans. numer. anal., 5, 1-6, (1997) · Zbl 0890.65071 [12] Diethelm, K.; Walz, G., Numerical solution of fractional order differential equations by extrapolation, Numer. algorithms, 16, 231-253, (1997) · Zbl 0926.65070 [13] Diethelm, K.; Ford, N.J.; Freed, Alan D., 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.J., Numerical solution of the bagley – torvik equation, Bit, 42, 490-507, (2002) · Zbl 1035.65067 [15] Liu, F.; Anh, V.; Turner, I., Numerical solution of the space fractional fokker – planck equation, J. comput. appl. math., 166, 209-219, (2004) · Zbl 1036.82019 [16] Liu, F.; Zhuang, P.; Anh, V.; Turner, I.; Burrage, K., Stability and convergence of the difference methods for the space – time fractional advection – diffusion equation, Appl. math. comput., 191, 12-20, (2007) · Zbl 1193.76093 [17] Diethelm, K.; Luchko, Y., Numerical solution of linear multi-order differential equations of fractional order, J. comput. anal. appl., 6, 243-263, (2004) · Zbl 1083.65064 [18] Edwards, J.T.; Ford, N.J.; Simpson, A.C., The numerical solution of linear multi-order fractional differential equations: systems of equations, J. comput. appl. math., 148, 401-418, (2002) · Zbl 1019.65048 [19] El-Mesiry, A.E.M.; El-Sayed, A.M.A.; El-Saka, H.A.A., Numerical methods for multi-term fractional (arbitrary) orders differential equations, Appl. math. comput., 160, 3, 683-699, (2005) · Zbl 1062.65073 [20] Erturk, Vedat Suat; Momani, Shaher; Odibat, Zaid, Application of generalized differential transform method to multi-order fractional differential equations, Commun. nonlinear sci. numer. simul., 13, 1642-1654, (2008) · Zbl 1221.34022 [21] Sweilam, N.H.; Khader, M.M.; Al-Bar, R.F., Numerical studies for a multi-order fractional differential equation, Phys. lett. A, 371, 26-33, (2007) · Zbl 1209.65116 [22] He, J.H., A new approach to linear partial differential equations, Commun. nonlinear sci. numer. simul., 2, 4, 230-235, (1997) [23] He, J.H., Some applications of nonlinear fractional differential equations and their approximations, Bull. sci. technol., 15, 12, 86-90, (1999) [24] He, J.H., Approximate solution of non linear differential equation with convolution product nonlinearities, Comput. methods appl. mech. engrg., 167, 69-73, (1998) · Zbl 0932.65143 [25] Inokuti, M.; Sekine, H.; Mura, T., General use of the Lagrange multiplier in non-linear mathematical physics, (), 156-162 [26] Batiha, B.; Noorani, M.S.M.; Hashim, I., Application of variational iteration method to heat and wave-like equations, Phys. lett. A, 369, 1-2, 55-61, (2007) · Zbl 1209.80040 [27] Batiha, B.; Noorani, M.S.M.; Hashim, I.; Ismail, E.S., The multistage variational iteration method for class of nonlinear system of odes, Phys. scr., 76, 388-392, (2007) · Zbl 1132.34008 [28] Darvishi, M.T.; Khani, F.; Soliman, A.A., The numerical simulation for stiff systems of ordinary differential equations, Comput. math. appl., 54, 1055-1063, (2007) · Zbl 1141.65371 [29] Salkuyeh, Davod Khojasteh, Convergence of the variational iteration method for solving linear systems of ODEs with constant coefficients, Comput. math. appl., 56, 2027-2033, (2008) · Zbl 1165.65376 [30] Draganescu, G.E., Application of a variational iteration method to linear and nonlinear viscoelastic models with fractional derivatives, J. math. phys., 47, 8, 082902, (2006) · Zbl 1112.74009 [31] Odibat, Z.; Momani, S., Application of variational iteration method to nonlinear differential equations of fractional order, Int. J. nonlinear sci. numer. simul., 1, 7, 15-27, (2006) · Zbl 1401.65087 [32] Momani, S.; Odibat, Z., Analytic approach to linear fractional partial differential equations arising in fluid mechanics, Phys. lett. A, 355, 271-279, (2006) · Zbl 1378.76084 [33] Momani, S.; Abuasad, S., Application of he’s variational iteration meathod to helmhotz equation, Chaos solitons fractals, 27, 5, 1119-1123, (2006) · Zbl 1086.65113 [34] Momani, S.; Odibat, Z., Numerical comparison of methods for solving linear differential equations of fractional order, Chaos solitons fractals, 31, 5, 1248-1255, (2007) · Zbl 1137.65450 [35] Momani, S.; Odibat, Z., Numerical approach to differential equations of fractional order, J. comput. appl. math., 207, 96-110, (2007) · Zbl 1119.65127 [36] Odibat, Z.; Momani, S., Numerical solution of fokker – planck equation with space- and time-fractional derivatives, Phys. lett. A, 369, 5-6, 349-358, (2007) · Zbl 1209.65114 [37] Momani, S.; Odibat, Z.; Alawneh, A., Variational iteration method for solving the space- and time-fractional KdV equation, Numer. methods partial differential equations, 24, 1, 262-271, (2007) · Zbl 1130.65132 [38] Yulita Molliq, R.; Noorani, M.S.M.; Hashim, I., Variational iteration method for fractional heat- and wave-like equations, Nonlinear anal. RWA, 10, 1854-1869, (2009) · Zbl 1172.35302 [39] Soliman, A.A., A numeric simulation an explicit solutions of kdv – burgers’ and lax’s seventh-order KdV equations, Chaos solitons fractals, 29, 2, 294-302, (2006) · Zbl 1099.35521
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