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Fractional variational iteration method for fractional initial-boundary value problems arising in the application of nonlinear science. (English) Zbl 1231.35288
Summary: We suggest a fractional functional for the variational iteration method to solve the linear and nonlinear fractional order partial differential equations with fractional order initial and boundary conditions by using the modified Riemann-Liouville fractional derivative proposed by {\it G. Jumarie} [Appl. Math. Lett. 22, No. 3, 378--385 (2009; Zbl 1171.26305)]. A fractional order Lagrange multiplier is considered. The solution is plotted for different values of $\alpha $.

35R11Fractional partial differential equations
35K15Second order parabolic equations, initial value problems
35L15Second order hyperbolic equations, initial value problems
35L20Second order hyperbolic equations, boundary value problems
45K05Integro-partial differential equations
65M99Numerical methods for IVP of PDE
Full Text: DOI
[1] Binning, P.; Celia, M. A.: Practical implementation of the fractional flow approach to multi-phase flow simulation, Advan. watr. Resour. 22, 461-478 (1999)
[2] Shen, C.; Phanikumar, M. S.: An efficient space-fractional dispersion approximation for stream solute transport modeling, Advan. watr. Resour. 32, 1482-1494 (2009)
[3] Huang, Q.; Huang, G.; Zhan, H.: A finite element solution for the fractional advection--dispersion equation, Advan. watr. Resour. 31, 1578-1589 (2008)
[4] Wheatcraft, S. W.; Meerschaert, M. M.: Fractional conservation of mass, Advan. watr. Resour. 31, 1377-1381 (2008)
[5] Dozier, J.; Painter, T. H.; Rittger, K.; Frew, J. E.: Time-space continuity of daily maps of fractional snow cover and albedo from MODIS, Advan. watr. Resour. 31, 1515-1526 (2008)
[6] Kevorkian, J.; Cole, J. D.: Multiple scale and singular perturbation method, (1996) · Zbl 0846.34001
[7] He, J. H.: Homotopy perturbation technique, Comput. math. Appl. mech. Engy., 178-257 (1999) · Zbl 0956.70017
[8] Yildirim, A.; Kocak, H.: Homotopy perturbation method for solving the space--time fractional advection--dispersion equation, Advan. watr. Resour. 32, 1711-1716 (2009)
[9] Ganji, D. D.; Ganji, S. S.; Karimpour, S.; Ganji, Z. Z.: Numerical study of homotopy-perturbation method applied to Burgers equation in fluid, Numer. methods partial differential equations 26, 917-930 (2010) · Zbl 1267.76082
[10] Khan, Y.; Wu, Q.: Homotopy perturbation transform method for nonlinear equations using he’s polynomials, Comput. math. Appl. 61, 1963-1967 (2011) · Zbl 1219.65119 · doi:10.1016/j.camwa.2010.08.022
[11] Nadeem, S.; Akbar, N. S.: Peristaltic flow of a Jeffrey fluid with variable viscosity in an asymmetric channel, Z. naturforsch. 64a, 713-722 (2009)
[12] Nadeem, S.; Akbar, N. S.: Influence of heat transfer on a peristaltic transport of Herschel Bulkley fluid in a non-uniform inclined tube, Commun. nonlinear sci. Numer. simul. 14, 4100-4113 (2009) · Zbl 1221.76269 · doi:10.1016/j.cnsns.2009.02.032
[13] Nadeem, S.; Akbar, N. S.: Influence of heat transfer on a peristaltic flow of Johnson Segalman fluid in a non uniform tube, International communications in heat and mass transfer 36, 1050-1059 (2009)
[14] Nadeem, S.; Hayat, T.; Akbar, Noreen Sher; Malik, M. Y.: On the influence of heat transfer in peristalsis with variable viscosity, International journal of heat and mass transfer 52, 4722-4730 (2009) · Zbl 1176.80030 · doi:10.1016/j.ijheatmasstransfer.2009.04.037
[15] He, J. H.: Variational iteration method--a kind of non-linear analytical technique: some examples, Int. J. Non-linear mech. 34, 699-708 (1999) · Zbl 05137891
[16] He, J. H.; Wu, G. C.; Austin, F.: The variational iteration method which should be followed, Nonl. sci. Lett. A 1, 1-30 (2010)
[17] Faraz, N.; Khan, Y.; Austin, F.: An alternative approach to differential-difference equations using the variational iteration method, Z. naturforsch. 65a, 1055-1059 (2010)
[18] Al-Khaled, K.; Momani, S.: An approximate solution for a fractional diffusion-wave equation using the decomposition method, Appl. math. Comput. 165, 473-483 (2005) · Zbl 1071.65135 · doi:10.1016/j.amc.2004.06.026
[19] Khan, Y.; Faraz, N.: Modified fractional decomposition method having integral $(d{\xi}){\alpha}$, J. King saud. Uni. sci. 23, 157-161 (2011)
[20] Khan, Y.: An effective modification of the Laplace decomposition method for nonlinear equations, Int. J. Nonlinear sci. Numer. simul. 10, 1373-1376 (2009)
[21] Khan, Y.; Faraz, N.: Application of modified Laplace decomposition method for solving boundary layer equation, J. King saud. Uni. sci. 23, 115-119 (2011)
[22] Khan, Y.; Austin, F.: Application of the Laplace decomposition method to nonlinear homogeneous and non-homogenous advection equations, Z. naturforsch. 65a, 849-853 (2010)
[23] Nadeem, S.; Akbar, N. S.: Effects of heat transfer on the peristaltic transport of MHD Newtonian fluid with variable viscosity: application of Adomian decomposition method, Commun. nonlinear sci. Numer. simul. 14, 3844-3855 (2009)
[24] 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 · doi:10.1016/S0045-7825(98)00108-X
[25] Odibat, Z.; Momani, S.: The variational iteration method: an efficient scheme for handling fractional partial differential equations in fluid mechanics, Comput. math. Appl. 58, 2199-2208 (2009) · Zbl 1189.65254 · doi:10.1016/j.camwa.2009.03.009
[26] Das, S.: Analytical solution of a fractional diffusion equation by variational iteration method, Comput. math. Appl. 57, 483-487 (2009) · Zbl 1165.35398 · doi:10.1016/j.camwa.2008.09.045
[27] Momani, S.; Odibat, Z.: Comparison between the homotopy perturbation method and the variational iteration method for linear fractional partial differential equations, Comput. math. Appl. 58, 2199-2208 (2009) · Zbl 1189.65254
[28] Momani, S.; Odibat, Z.: Analytical approach to linear fractional partial differential equations arising in fluid mechanics, Phys. lett. A 355, 271-279 (2006) · Zbl 05675858
[29] Momani, S.; Odibat, Z.: Numerical comparison of the methods for solving linear differential equations of fractional order, Chaos solitons fractals 31, 1248-1255 (2007) · Zbl 1137.65450 · doi:10.1016/j.chaos.2005.10.068
[30] Faraz, N.; Khan, Y.; Yildirim, A.: Analytical approach to two-dimensional viscous flow with a shrinking sheet via variational iteration algorithm-II, J. King saud. Uni. sci. 23, 77-81 (2011)
[31] Inc, Mustafa: The approximate and exact solutions of the space and time-fractional Burgers equations with initial conditions by variational iteration method, J. math. Anal. appl. 345, 476-484 (2008) · Zbl 1146.35304 · doi:10.1016/j.jmaa.2008.04.007
[32] Inokuti, M.; Sekine, H.; Mura, T.: General use of the Lagrange multiplier in nonlinear mathematical physics, Variational methods in the mechanics of solids, 156-162 (1978)
[33] Jumarie, G.: Table of some basic fractional calculus formulae derived from a modified Riemann--Liouville derivative for non-differentiable functions, Appl. math. Lett. 22, 378-385 (2009) · Zbl 1171.26305 · doi:10.1016/j.aml.2008.06.003
[34] Podlubry, I.: Fractional differential equations, (1999)
[35] Diethelm, K.; Ford, N. J.: Analysis of fractional differential equations, J. math. Anal. appl. 265, 229-248 (2002) · Zbl 1014.34003 · doi:10.1006/jmaa.2001.7194
[36] Miller, K. S.; Ross, B.: An introduction to the fractional calculus and fractional differential equations, (2003) · Zbl 0789.26002
[37] Jumarie, G.: New stochastic fractional models for malthusian growth, the Poissonian birth process and optimal management of populations, Math. comput. Model. 44, 231-254 (2006) · Zbl 1130.92043 · doi:10.1016/j.mcm.2005.10.003
[38] Jumarie, G.: Laplace’s transform of fractional order via the mittage--Leffler funcation and modified Riemann--Liouville derivative, Appl. math. Lett. 22, 1659-1664 (2009) · Zbl 1181.44001 · doi:10.1016/j.aml.2009.05.011
[39] Wu, G. C.; He, J. H.: Fractional calculus of variations in fractal sapcetime, Nonlinear sci. Lett. A 1, No. 3, 281-287 (2010)
[40] Wu, G. C.; Lee, E. W. M.: Fractional variational iteration method and its application, Phys. lett. A 374, 2506-2509 (2010) · Zbl 1237.34007