×

Analytical approach to linear fractional partial differential equations arising in fluid mechanics. (English) Zbl 1378.76084

Summary: In this letter, we implement relatively new analytical techniques, the variational iteration method and the Adomian decomposition method, for solving linear fractional partial differential equations arising in fluid mechanics. The fractional derivatives are described in the Caputo sense. The two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of fractional differential equations. In these methods, the solution takes the form of a convergent series with easily computable components. The corresponding solutions of the integer order equations are found to follow as special cases of those of fractional order equations. Some numerical examples are presented to illustrate the efficiency and reliability of the two methods.

MSC:

76M25 Other numerical methods (fluid mechanics) (MSC2010)
65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
35R11 Fractional partial differential equations
35Q35 PDEs in connection with fluid mechanics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] J.H. He, Nonlinear oscillation with fractional derivative and its applications, International Conference on Vibrating Engineering’98, Dalian, China, 1998, pp. 288-291; J.H. He, Nonlinear oscillation with fractional derivative and its applications, International Conference on Vibrating Engineering’98, Dalian, China, 1998, pp. 288-291
[2] He, J. H., Bull. Sci. Technol., 15, 2, 86 (1999)
[3] He, J. H., Comput. Methods Appl. Mech. Engrg., 167, 57 (1998) · Zbl 0942.76077
[4] Mainardi, F., Fractional calculus: Some basic problems in continuum and statistical mechanics, (Carpinteri, A.; Mainardi, F., Fractals and Fractional Calculus in Continuum Mechanics (1997), Springer-Verlag: Springer-Verlag New York), 291-348 · Zbl 0917.73004
[5] Podlubny, I., Fractional Differential Equations (1999), Academic Press: Academic Press New York · Zbl 0918.34010
[6] He, J. H., Commun. Nonlin. Sci. Numer. Simulation, 2, 4, 235 (1997)
[7] He, J. H., Comput. Methods Appl. Mech. Engrg., 167, 57 (1998) · Zbl 0942.76077
[8] He, J. H., Comput. Methods Appl. Mech. Engrg., 167, 69 (1998)
[9] He, J. H., Int. J. Nonlinear Mech., 34, 699 (1999) · Zbl 1342.34005
[10] He, J. H., Appl. Math. Comput., 114, 115 (2000)
[11] He, J. H.; Wan, Y. Q.; Guo, Q., Int. J. Circuit Theor. Appl., 32, 6, 629 (2004) · Zbl 1169.94352
[12] Adomian, G., J. Math. Anal. Appl., 135, 501 (1988) · Zbl 0671.34053
[13] Adomian, G., Solving Frontier Problems of Physics: The Decomposition Method (1994), Kluwer Academic Publishers: Kluwer Academic Publishers Boston · Zbl 0802.65122
[14] Shawagfeh, N.; Kaya, D., Appl. Math. Lett., 17, 323 (2004) · Zbl 1061.65062
[15] Momani, S., Math. Comput. Simulation, 70, 2, 110 (2005) · Zbl 1119.65394
[16] Momani, S., Chaos Solitons Fractals, 28, 4, 930 (2006) · Zbl 1099.35118
[17] S. Momani, Z. Odibat, Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method, Appl. Math. Comput., in press; S. Momani, Z. Odibat, Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method, Appl. Math. Comput., in press · Zbl 1096.65131
[18] S.S. Ray, R.K. Bera, Analytical solution of a fractional diffusion equation by Adomian decomposition method, Appl. Math. Comput. in press; S.S. Ray, R.K. Bera, Analytical solution of a fractional diffusion equation by Adomian decomposition method, Appl. Math. Comput. in press · Zbl 1089.65108
[19] Marinca, V., Int. J. Nonlin. Sci. Numer. Simulation, 3, 2, 107 (2002) · Zbl 1079.34028
[20] Drǎgǎnescu, G. E.; Cǎpǎlnǎsan, V., Int. J. Nonlin. Sci. Numer. Simulation, 4, 3, 219 (2003)
[21] Liu, H. M., Chaos Solitons Fractals, 23, 2, 573 (2005) · Zbl 1135.76597
[22] Hao, T. H., Int. J. Nonlin. Sci. Numer. Simulation, 6, 2, 209 (2005) · Zbl 1401.78004
[23] Momani, S.; Abuasad, S., Chaos Solitons Fractals, 27, 5, 1119 (2006) · Zbl 1086.65113
[24] Odibat, Z.; Momani, S., Int. J. Nonlin. Sci. Numer. Simulation, 7, 1, 27 (2006) · Zbl 1401.65087
[25] Debnath, L.; Bhatta, D., Fract. Calc. Appl. Anal., 7, 21 (2004) · Zbl 1076.35096
[26] Diethelm, K.; Ford, J. M.; Ford, N. J.; Weilbeer, M., J. Comput. Appl. Math., 186, 482 (2006) · Zbl 1078.65550
[27] Gorenflo, R., Afterthoughts on interpretation of fractional derivatives and integrals, (Rusev, P.; Dimovski, I.; Kiryakova, V., Transform Methods and Special Functions, Varna 96 (1998), Bulgarian Academy of Sciences, Institute of Mathematics ands Informatics: Bulgarian Academy of Sciences, Institute of Mathematics ands Informatics Sofia), 589-591 · Zbl 0914.00065
[28] A. Luchko, R. Gorenflo, The initial value problem for some fractional differential equations with the Caputo derivative, preprint series \(A\); A. Luchko, R. Gorenflo, The initial value problem for some fractional differential equations with the Caputo derivative, preprint series \(A\)
[29] Miller, K. S.; Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations (1993), John Wiley and Sons: John Wiley and Sons New York · Zbl 0789.26002
[30] Oldham, K. B.; Spanier, J., The Fractional Calculus (1974), Academic Press: Academic Press New York · Zbl 0428.26004
[31] Caputo, M., J. Roy. Astron. Soc., 13, 529 (1967)
[32] Podlubny, I., Fract. Calc. Appl. Anal., 5, 367 (2002) · Zbl 1042.26003
[33] Inokuti, M.; Sekine, H.; Mura, T., General use of the Lagrange multiplier in non-linear mathematical physics, (Nemat-Nasser, S., Variational Method in the Mechanics of Solids (1978), Pergamon Press: Pergamon Press Oxford), 156-162
[34] He, J. H., Int. J. Turbo Jet-Engines, 14, 1, 23 (1997)
[35] He, J. H., Int. J. Nonlin. Sci. Numer. Simulation, 2, 4, 309 (2001) · Zbl 1083.74526
[36] He, J. H., Generalized Variational Principles in Fluids (2003), Science and Culture Publishing House of China: Science and Culture Publishing House of China Hong Kong, pp. 222-230 (in Chinese) · Zbl 1054.76001
[37] He, J. H., Int. J. Nonlin. Sci. Numer. Simulation, 4, 3, 313 (2003)
[38] He, J. H., Chaos Solitons Fractals, 19, 4, 847 (2004)
[39] Liu, H. M., Int. J. Nonlin. Sci. Numer. Simulation, 5, 1, 95 (2004)
[40] Cherruault, Y., Kybernetes, 18, 31 (1989) · Zbl 0697.65051
[41] Cherruault, Y.; Adomian, G., Math. Comput. Modelling, 18, 103 (1993) · Zbl 0805.65057
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.