Convergence of the variational iteration method for solving multi-order fractional differential equations

Computers and Mathematics with Applications

Volumn 60, Issue 10, 2010, Pages 2871-2879

  Yang, Shuiping ^a,b   Xiao, Aiguo ^a   Su, Hong ^a  

^a XIANGTAN UNIVERSITY   (China)

^b HUIZHOU UNIVERSITY   (China)

Author keywords

Convergence; Fractional calculus; Fractional differential equations; Variational iteration method

Indexed keywords

APPROXIMATE SOLUTION; CONVERGENCE; ERROR ESTIMATES; FRACTIONAL CALCULUS; FRACTIONAL DIFFERENTIAL EQUATIONS; NUMERICAL RESULTS; VARIATIONAL ITERATION METHOD;

CALCULATIONS; DIFFERENTIAL EQUATIONS; DIFFERENTIATION (CALCULUS);

ITERATIVE METHODS;

EID: 78049255335     PISSN: 08981221     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.camwa.2010.09.044     Document Type: Article

References (39)

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. I. Podlubny Fractional Differential Equations 1999 Academic Press New York
3. R.L. Bagley, and P.J. Torvik Fractional calculus in the transient analysis of viscoelastically damped structures AIAA J. 23 6 1983 918 925
4. M. Ichise, Y. Nagayanagi, and T. Kojima An analog simulation of non-integer order transfer functions for analysis of electrode processes J. Electroanal. Chem. 33 1971 253 265
5. H.H. Sun, B. Onaral, and Y. Tsao Application of positive reality principle to metal electrode linear polarization phenomena IEEE Trans. Biomed. Eng. BME-31 10 1984 664 674
6. H.H. Sun, A.A. Abdelwahab, and B. Onaral Linear approximation of transfer function with a pole of fractional order IEEE Trans. Autom. Control AC-29 5 1984 441 444
7. Chunguang Li, and Guanrong Chen Chaos and hyperchaos in the fractional-order Rssle equations Physica A 341 2004 55 61
8. C. Lubich RungeKutta theory for Volterra and Abel integral equations of the second kind Math. Comp. 41 1983 87 102
9. C. Lubich Fractional linear multistep methods for AbelVolterra integral equations of the second kind Math. Comp. 45 1985 463 469
10. C. Lubich Discretized fractional calculus SIAM J. Math. Anal. 17 1986 704 719
11. K. Diethelm An algorithm for the numerical solution of differential equations of fractional order Electron. Trans. Numer. Anal. 5 1997 1 6 (Pubitemid 38870919)
12. K. Diethelm, and G. Walz Numerical solution of fractional order differential equations by extrapolation Numer. Algorithms 16 1997 231 253 (Pubitemid 127429647)
13. K. Diethelm, N.J. Ford, and Alan D. Freed A predictorcorrector approach for the numerical solution of fractional differential equations Nonlinear Dynam. 29 2002 3 22
14. K. Diethelm, and N.J. Ford Numerical solution of the BagleyTorvik equation BIT 42 2002 490 507
15. F. Liu, V. Anh, and I. Turner Numerical solution of the space fractional FokkerPlanck equation J. Comput. Appl. Math. 166 2004 209 219
16. F. Liu, P. Zhuang, V. Anh, I. Turner, and K. Burrage Stability and convergence of the difference methods for the spacetime fractional advectiondiffusion equation Appl. Math. Comput. 191 2007 12 20
17. K. Diethelm, and Y. Luchko Numerical solution of linear multi-order differential equations of fractional order J. Comput. Anal. Appl. 6 2004 243 263
18. J.T. Edwards, N.J. Ford, and A.C. Simpson The numerical solution of linear multi-order fractional differential equations: systems of equations J. Comput. Appl. Math. 148 2002 401 418
19. A.E.M. El-Mesiry, A.M.A. El-Sayed, and H.A.A. El-Saka Numerical methods for multi-term fractional (arbitrary) orders differential equations Appl. Math. Comput. 160 3 2005 683 699
20. Vedat Suat Erturk, Shaher Momani, and Zaid Odibat Application of generalized differential transform method to multi-order fractional differential equations Commun. Nonlinear Sci. Numer. Simul. 13 2008 1642 1654 (Pubitemid 351250728)
21. N.H. Sweilam, M.M. Khader, and R.F. Al-Bar Numerical studies for a multi-order fractional differential equation Phys. Lett. A 371 2007 26 33
22. J.H. He A new approach to linear partial differential equations Commun. Nonlinear Sci. Numer. Simul. 2 4 1997 230 235
23. J.H. He Some applications of nonlinear fractional differential equations and their approximations Bull. Sci. Technol. 15 12 1999 86 90
24. J.H. He Approximate solution of non linear differential equation with convolution product nonlinearities Comput. Methods Appl. Mech. Engrg. 167 1998 69 73
25. M. Inokuti, H. Sekine, and T. Mura General use of the Lagrange multiplier in non-linear mathematical physics Variational Methods in the Mechanics of Solids 1978 Pergamon Press New York 156 162
26. B. Batiha, M.S.M. Noorani, and I. Hashim Application of variational iteration method to heat and wave-like equations Phys. Lett. A 369 12 2007 55 61
27. B. Batiha, M.S.M. Noorani, I. Hashim, and E.S. Ismail The multistage variational iteration method for class of nonlinear system of ODEs Phys. Scr. 76 2007 388 392
28. M.T. Darvishi, F. Khani, and A.A. Soliman The numerical simulation for stiff systems of ordinary differential equations Comput. Math. Appl. 54 2007 1055 1063
29. Davod Khojasteh Salkuyeh Convergence of the variational iteration method for solving linear systems of ODEs with constant coefficients Comput. Math. Appl. 56 2008 2027 2033
30. G.E. Draganescu Application of a variational iteration method to linear and nonlinear viscoelastic models with fractional derivatives J. Math. Phys. 47 8 2006 082902
31. Z. Odibat, and S. Momani Application of variational iteration method to nonlinear differential equations of fractional order Int. J. Nonlinear Sci. Numer. Simul. 1 7 2006 15 27
32. S. Momani, and Z. Odibat Analytic approach to linear fractional partial differential equations arising in fluid mechanics Phys. Lett. A 355 2006 271 279
33. S. Momani, and S. Abuasad Application of He's variational iteration meathod to Helmhotz equation Chaos Solitons Fractals 27 5 2006 1119 1123
34. S. Momani, and Z. Odibat Numerical comparison of methods for solving linear differential equations of fractional order Chaos Solitons Fractals 31 5 2007 1248 1255
35. S. Momani, and Z. Odibat Numerical approach to differential equations of fractional order J. Comput. Appl. Math. 207 2007 96 110
36. Z. Odibat, and S. Momani Numerical solution of FokkerPlanck equation with space- and time-fractional derivatives Phys. Lett. A 369 56 2007 349 358
37. S. Momani, Z. Odibat, and A. Alawneh Variational iteration method for solving the space- and time-fractional KdV equation Numer. Methods Partial Differential Equations 24 1 2007 262 271
38. R. Yulita Molliq, M.S.M. Noorani, and I. Hashim Variational iteration method for fractional heat- and wave-like equations Nonlinear Anal. RWA 10 2009 1854 1869
39. A.A. Soliman A numeric simulation an explicit solutions of KdVBurgers' and Lax's seventh-order KdV equations Chaos Solitons Fractals 29 2 2006 294 302