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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:
65L99Numerical methods for ODE
34A08Fractional differential equations
26A33Fractional derivatives and integrals (real functions)
45J05Integro-ordinary differential equations
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)
[3]Bagley, R. L.; Torvik, P. J.: Fractional calculus in the transient analysis of viscoelastically damped structures, Aiaa j. 23, No. 6, 918-925 (1983) · Zbl 0562.73071 · doi:10.2514/3.9007
[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. 31, No. 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 29, No. 5, 441-444 (1984) · Zbl 0532.93025 · doi:10.1109/TAC.1984.1103551
[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 · doi:10.2307/2007768
[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 · doi:10.2307/2008136
[10]Lubich, C.: Discretized fractional calculus, SIAM J. Math. anal. 17, 704-719 (1986) · Zbl 0624.65015 · doi:10.1137/0517050
[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 · doi:emis:journals/ETNA/vol.5.1997/pp1-6.dir/pp1-6.html
[12]Diethelm, K.; Walz, G.: Numerical solution of fractional order differential equations by extrapolation, Numer. algorithms 16, 231-253 (1997) · Zbl 0926.65070 · doi:10.1023/A:1019147432240
[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 · doi:10.1023/A:1016592219341
[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 · doi:10.1016/j.cam.2003.09.028
[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 · doi:10.1016/j.amc.2006.08.162
[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 · doi:10.1016/S0377-0427(02)00558-7
[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, No. 3, 683-699 (2005) · Zbl 1062.65073 · doi:10.1016/j.amc.2003.11.026
[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 · doi:10.1016/j.cnsns.2007.02.006
[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 · doi:10.1016/j.physleta.2007.06.016
[22]He, J. H.: A new approach to linear partial differential equations, Commun. nonlinear sci. Numer. simul. 2, No. 4, 230-235 (1997) · Zbl 0923.35046 · doi:10.1016/S1007-5704(97)90029-0
[23]He, J. H.: Some applications of nonlinear fractional differential equations and their approximations, Bull. sci. Technol. 15, No. 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 · doi:10.1016/S0045-7825(98)00109-1
[25]Inokuti, M.; Sekine, H.; Mura, T.: General use of the Lagrange multiplier in non-linear mathematical physics, , 156-162 (1978)
[26]Batiha, B.; Noorani, M. S. M.; Hashim, I.: Application of variational iteration method to heat and wave-like equations, Phys. lett. A 369, No. 1–2, 55-61 (2007) · Zbl 1209.80040 · doi:10.1016/j.physleta.2007.04.069
[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 · doi:10.1088/0031-8949/76/4/018
[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 · doi:10.1016/j.camwa.2006.12.072
[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 · doi:10.1016/j.camwa.2008.03.030
[30]Draganescu, G. E.: Application of a variational iteration method to linear and nonlinear viscoelastic models with fractional derivatives, J. math. Phys. 47, No. 8, 082902 (2006) · Zbl 1112.74009 · doi:10.1063/1.2234273
[31]Odibat, Z.; Momani, S.: Application of variational iteration method to nonlinear differential equations of fractional order, Int. J. Nonlinear sci. Numer. simul. 1, No. 7, 15-27 (2006)
[32]Momani, S.; Odibat, Z.: Analytic approach to linear fractional partial differential equations arising in fluid mechanics, Phys. lett. A 355, 271-279 (2006)
[33]Momani, S.; Abuasad, S.: Application of he’s variational iteration meathod to helmhotz equation, Chaos solitons fractals 27, No. 5, 1119-1123 (2006) · Zbl 1086.65113 · doi:10.1016/j.chaos.2005.04.113
[34]Momani, S.; Odibat, Z.: Numerical comparison of methods for solving linear differential equations of fractional order, Chaos solitons fractals 31, No. 5, 1248-1255 (2007) · Zbl 1137.65450 · doi:10.1016/j.chaos.2005.10.068
[35]Momani, S.; Odibat, Z.: Numerical approach to differential equations of fractional order, J. comput. Appl. math. 207, 96-110 (2007) · Zbl 1119.65127 · doi:10.1016/j.cam.2006.07.015
[36]Odibat, Z.; Momani, S.: Numerical solution of Fokker–Planck equation with space- and time-fractional derivatives, Phys. lett. A 369, No. 5–6, 349-358 (2007) · Zbl 1209.65114 · doi:10.1016/j.physleta.2007.05.002
[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, No. 1, 262-271 (2007) · Zbl 1130.65132 · doi:10.1002/num.20247
[38]Molliq, R. Yulita; 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 · doi:10.1016/j.nonrwa.2008.02.026
[39]Soliman, A. A.: A numeric simulation an explicit solutions of KdV–Burgers and Lax’s seventh-order KdV equations, Chaos solitons fractals 29, No. 2, 294-302 (2006) · Zbl 1099.35521 · doi:10.1016/j.chaos.2005.08.054