×

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 \(\alpha \).

MSC:

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

Citations:

Zbl 1171.26305
PDFBibTeX XMLCite
Full Text: DOI

References:

[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: 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, (Nemat-Nasser, S., Variational Methods in the Mechanics of Solids (1978), Pergamon Press: Pergamon Press New York), 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: 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.: 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.