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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 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 \(\alpha \).

35R11 Fractional partial differential equations
35K15 Initial value problems for second-order parabolic equations
35L15 Initial value problems for second-order hyperbolic equations
35L20 Initial-boundary value problems for second-order hyperbolic equations
45K05 Integro-partial differential equations
65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
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[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), Springer-Verlag New York · 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
[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
[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.; Sher Akbar, Noreen; 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
[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 1342.34005
[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
[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
[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
[26] Das, S., Analytical solution of a fractional diffusion equation by variational iteration method, Comput. math. appl., 57, 483-487, (2009) · Zbl 1165.35398
[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 1378.76084
[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
[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] 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
[32] Inokuti, M.; Sekine, H.; Mura, T., General use of the Lagrange multiplier in nonlinear mathematical physics, (), 156-162
[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
[34] Podlubry, I., Fractional differential equations, (1999), Academic Press California, San Diego
[35] Diethelm, K.; Ford, N.J., Analysis of fractional differential equations, J. math. anal. appl., 265, 229-248, (2002) · Zbl 1014.34003
[36] Miller, K.S.; Ross, B., An introduction to the fractional calculus and fractional differential equations, (2003), John Willey and Sons, Inc. New York
[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
[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
[39] Wu, G.C.; He, J.H., Fractional calculus of variations in fractal sapcetime, Nonlinear sci. lett. A, 1, 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
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