A general iteration formula of VIM for fractional heat- and wave-like equations

Journal of Applied Mathematics

Volumn 2013, Issue , 2013, Pages

  Yin, Fukang ^a   Song, Junqiang ^a   Cao, Xiaoqun ^a  

^a NATIONAL UNIVERSITY OF DEFENSE TECHNOLOGY   (China)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84876526778     PISSN: 1110757X     EISSN: 16870042     Source Type: Journal    
DOI: 10.1155/2013/428079     Document Type: Article

References (41)

1. Odibat Z., Momani S., Numerical methods for nonlinear partial differential equations of fractional order. Applied Mathematical Modelling 2008 32 1 28 39 2-s2.0-34548384362 10.1016/j.apm.2006.10.025 (Pubitemid 47355246)
2. Momani S., Odibat Z., Analytical approach to linear fractional partial differential equations arising in fluid mechanics. Physics Letters A 2006 355 4-5 271 279 2-s2.0-33646878106 10.1016/j.physleta.2006.02.048 (Pubitemid 43783194)
3. Momani S., Analytical approximate solution for fractional heat-like and wave-like equations with variable coefficients using the decomposition method. Applied Mathematics and Computation 2005 165 2 459 472 10.1016/j.amc.2004.06.025 MR2137804 ZBL1070.65105 (Pubitemid 40567744)
4. Momani S., Odibat Z., Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method. Applied Mathematics and Computation 2006 177 2 488 494 10.1016/j.amc.2005.11.025 MR2291973 ZBL1096.65131 (Pubitemid 43866876)
5. Wu G.-C., He J.-H., Fractional adomian decomposition method. http://arxiv.org/abs/1006.5264
6. Abbasbandy S., Numerical solutions of the integral equations: homotopy perturbation method and Adomian's decomposition method. Applied Mathematics and Computation 2006 173 1 493 500 10.1016/j.amc.2005.04.077 MR2203403 ZBL1090.65143 (Pubitemid 43208792)
7. Liu I.-C., Megahed A. M., Homotopy perturbation method for thin film flow and heat transfer over an unsteady stretching sheet with internal heating and variable heat flux. Journal of Applied Mathematics 2012 2012 12 10.1155/2012/418527 418527 MR2965707 ZBL1251.80002
8. Hosseinnia S. H., Ranjbar A., Momani S., Using an enhanced homotopy perturbation method in fractional differential equations via deforming the linear part. Computers & Mathematics with Applications 2008 56 12 3138 3149 10.1016/j.camwa.2008.07.002 MR2474569 ZBL1165.65375
9. Abbasbandy S., Homotopy perturbation method for quadratic Riccati differential equation and comparison with Adomian's decomposition method. Applied Mathematics and Computation 2006 172 1 485 490 10.1016/j.amc.2005.02.014 MR2197917 ZBL1088.65063 (Pubitemid 43132856)
10. Abbasbandy S., Modified homotopy perturbation method for nonlinear equations and comparison with Adomian decomposition method. Applied Mathematics and Computation 2006 172 1 431 438 10.1016/j.amc.2005.02.015 MR2197913 ZBL1088.65043 (Pubitemid 43132852)
11. He J. H., Variational iteration method-a kind of non-linear analytical technique: some examples. International Journal of Non-Linear Mechanics 1999 34 4 699 708 2-s2.0-0000092673
12. He J.-H., Variational iteration method for autonomous ordinary differential systems. Applied Mathematics and Computation 2000 114 2-3 115 123 10.1016/S0096-3003(99)00104-6 MR1778658 ZBL1027.34009
13. He J.-H., Approximate analytical solution for seepage flow with fractional derivatives in porous media. Computer Methods in Applied Mechanics and Engineering 1998 167 1-2 57 68 10.1016/S0045-7825(98)00108-X MR1665221 ZBL0942.76077
14. He J. H., Asymptotic methods for solitary solutions and compactons. Abstract and Applied Analysis 2012 2012 130 916793 10.1155/2012/916793
15. He J. H., Some applications of nonlinear fractional differential equations and their approximations. Bulletin of Science and Technology 1999 15 2 86 90
16. Nofal T. A., Approximate solutions for nonlinear initial value problems using the modified variational iteration method. Journal of Applied Mathematics 2012 2012 19 10.1155/2012/370843 370843 MR2948160 ZBL1251.65148
17. R Y. M., Noorani M. S. M., Hashim I., Variational iteration method for fractional heat- and wave-like equations. Nonlinear Analysis: Real World Applications 2009 10 3 1854 1869 10.1016/j.nonrwa.2008.02.026 MR2502991 ZBL1172.35302
18. Odibat Z. M., Momani S., Application of variational iteration method to nonlinear differential equations of fractional order. International Journal of Nonlinear Sciences and Numerical Simulation 2006 7 1 27 34 2-s2.0-30344464250 (Pubitemid 43246579)
19. Drǎgǎnescu G. E., Application of a variational iteration method to linear and nonlinear viscoelastic models with fractional derivatives. Journal of Mathematical Physics 2006 47 8 9 10.1063/1.2234273 082902 MR2258596 ZBL1112.74009 (Pubitemid 44327387)
20. Ganji D. D., Hashemi Kachapi S. H., Analysis of nonlinear equations in fluids. Progress in Nonlinear Science 2011 2 1 293
21. Ganji D. D., Hashemi Kachapi S. H., Analytical and numerical methods in engineering and applied sciences. Progress in Nonlinear Science 2011 3 1 579
22. Wu G.-C., Variational iteration method for q -difference equations of second order. Journal of Applied Mathematics 2012 2012 5 10.1155/2012/102850 102850 MR2948133
23. Faraz N., Khan Y., Jafari H., Yildirim A., Madani M., Fractional variational iteration method via modified Riemann-Liouville derivative. Journal of King Saud University-Science 2011 23 4 413 417
24. Ali E. J., Modified treatment of initial boundary value problems for one dimensional heat-like and wave-like equations using variational iteration method. Applied Mathematical Sciences 2012 6 33-36 1613 1626 MR2905110 ZBL1246.65232
25. Wu G.-C., Lee E. W. M., Fractional variational iteration method and its application. Physics Letters A 2010 374 25 2506 2509 10.1016/j.physleta.2010.04. 034 MR2640023 ZBL1237.34007
26. Wu G.-C., A fractional variational iteration method for solving fractional nonlinear differential equations. Computers & Mathematics with Applications 2011 61 8 2186 2190 10.1016/j.camwa.2010.09.010 MR2785581 ZBL1219.65085
27. Odibat Z., Momani S., The variational iteration method: an efficient scheme for handling fractional partial differential equations in fluid mechanics. Computers & Mathematics with Applications 2009 58 11-12 2199 2208 10.1016/j.camwa.2009.03.009 MR2557350 ZBL1189.65254
28. Liao S., On the homotopy analysis method for nonlinear problems. Applied Mathematics and Computation 2004 147 2 499 513 10.1016/S0096-3003(02)00790-7 MR2012589 ZBL1086.35005
29. Abbasbandy S., Shirzadi A., Homotopy analysis method for multiple solutions of the fractional Sturm-Liouville problems. Numerical Algorithms 2010 54 4 521 532 10.1007/s11075-009-9351-7 MR2670667 ZBL1197.65096
30. Podlubny I., Fractional Differential Equations: An Introduction To Fractional Derivatives, Fractional Differential Equations, To Methods of Their Solution and Some of Their Applications 1998 198 San Diego, Calif, USA Academic Press Mathematics in Science and Engineering
31. Jumarie G., Stochastic differential equations with fractional Brownian motion input. International Journal of Systems Science 1993 24 6 1113 1131 10.1080/00207729308949547 MR1222322 ZBL0771.60043
32. Jumarie G., On the representation of fractional Brownian motion as an integral with respect to (d t) a. Applied Mathematics Letters 2005 18 7 739 748 10.1016/j.aml.2004.05.014 MR2144721 ZBL1082.60029 (Pubitemid 40612588)
33. Jumarie G., On the solution of the stochastic differential equation of exponential growth driven by fractional Brownian motion. Applied Mathematics Letters 2005 18 7 817 826 10.1016/j.aml.2004.09.012 MR2144731 ZBL1075.60068 (Pubitemid 40612599)
34. Jumarie G., Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results. Computers & Mathematics with Applications 2006 51 9-10 1367 1376 10.1016/j.camwa.2006.02.001 MR2237634 ZBL1137.65001
35. Jumarie G., New stochastic fractional models for Malthusian growth, the Poissonian birth process and optimal management of populations. Mathematical and Computer Modelling 2006 44 3-4 231 254 10.1016/j.mcm.2005.10.003 MR2239054 ZBL1130.92043 (Pubitemid 43880369)
36. Jumarie G., Stock exchange fractional dynamics defined as fractional exponential growth driven by (usual) Gaussian white noise. Application to fractional Black-Scholes equations. Insurance Mathematics & Economics 2008 42 1 271 287 10.1016/j.insmatheco.2007.03.001 MR2392089 ZBL1141.91455
37. Jumarie G., Table of some basic fractional calculus formulae derived from a modified Riemann-Liouville derivative for non-differentiable functions. Applied Mathematics Letters 2009 22 3 378 385 10.1016/j.aml.2008.06.003 MR2483503 ZBL1171.26305
38. Jumarie G., Laplace's transform of fractional order via the Mittag-Leffler function and modified Riemann-Liouville derivative. Applied Mathematics Letters 2009 22 11 1659 1664 10.1016/j.aml.2009.05.011 MR2569059 ZBL1181.44001
39. Kexue L., Jigen P., Laplace transform and fractional differential equations. Applied Mathematics Letters 2011 24 12 2019 2023 10.1016/j.aml.2011. 05.035 MR2826118 ZBL1238.34013
40. Qing-Li W., He J. H., Zheng-Biao L., Fractional model for heat conduction in polar bear hairs. Thermal Science 2012 16 2 339 342
41. Shou D.-H., He J.-H., Beyond Adomian method: the variational iteration method for solving heat-like and wave-like equations with variable coefficients. Physics Letters A 2008 372 3 233 237 10.1016/j.physleta.2007.07.011 MR2380083 ZBL1217.35091