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The variational iteration method: an efficient scheme for handling fractional partial differential equations in fluid mechanics. (English) Zbl 1189.65254

Summary: Variational iteration method has been used to handle linear and nonlinear differential equations. The main property of the method lies in its flexibility and ability to solve nonlinear equations accurately and conveniently. In this work, a general framework of the variational iteration method is presented for analytical treatment of fractional partial differential equations in fluid mechanics. The fractional derivatives are described in the Caputo sense. Numerical illustrations that include the fractional wave equation, fractional Burgers equation, fractional KdV equation, fractional Klein-Gordon equation and fractional Boussinesq-like equation are investigated to show the pertinent features of the technique. Comparison of the results obtained by the variational iteration method with those obtained by Adomian decomposition method reveals that the first method is very effective and convenient. The basic idea described in this paper is expected to be further employed to solve other similar linear and nonlinear problems in fractional calculus.

MSC:

65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
26A33 Fractional derivatives and integrals
76A02 Foundations of fluid mechanics
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[1] J.H. He, Nonlinear oscillation with fractional derivative and its applications, in: International Conference on Vibrating Engineering’98, Dalian, China, 1998, pp. 288-291; J.H. He, Nonlinear oscillation with fractional derivative and its applications, in: International Conference on Vibrating Engineering’98, Dalian, China, 1998, pp. 288-291
[2] He, J. H., Some applications of nonlinear fractional differential equations and their approximations, Bull. Sci. Technol., 15, 2, 86-90 (1999)
[3] He, J. H., Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput. Methods Appl. Mech. Engrg., 167, 57-68 (1998) · Zbl 0942.76077
[4] Podlubny, I., Fractional Differential Equations (1999), Academic Press: Academic Press New York · Zbl 0918.34010
[5] 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
[6] Mainardi, F.; Luchko, Y.; Pagnini, G., The fundamental solution of the space-time fractional diffusion equation, Frac. Calc. Appl. Anal., 4, 153-192 (2001) · Zbl 1054.35156
[7] Hanyga, A., Multidimensional solutions of time-fractional diffusion-wave equations, Proc. R. Soc. Lond. A, 458, 933-957 (2002) · Zbl 1153.35347
[8] Huang, F.; Liu, F., The time fractional diffusion and fractional advection-dispersion equation, ANZIAM, 46, 1-14 (2005) · Zbl 1072.35218
[9] Huang, F.; Liu, F., The fundamental solution of the space-time fractional advection-dispersion equation, J. Appl. Math. Comput., 18, 21-36 (2005) · Zbl 1086.35003
[10] Momani, S., Analytic and approximate solutions of the space- and time-fractional telegraph equations, Appl. Math. Comput., 170, 2, 1126-1134 (2005) · Zbl 1103.65335
[11] Momani, S., An explicit and numerical solutions of the fractional KdV equation, Math. Comput. Simul., 70, 110-118 (2005) · Zbl 1119.65394
[12] Debnath, L.; Bhatta, D., Solutions to few linear fractional inhomogeneous partial differential equations in fluid mechanics, Frac. Calc. Appl. Anal., 7, 21-36 (2004) · Zbl 1076.35096
[13] Adomian, G., A review of the decomposition method in applied mathematics, J. Math. Anal. Appl., 135, 501-544 (1988) · Zbl 0671.34053
[14] Adomian, G., Solving Frontier Problems of Physics: The Decomposition method (1994), Kluwer Academic Publishers: Kluwer Academic Publishers Boston · Zbl 0802.65122
[15] Wazwaz, A. M., A new algorithm for calculating Adomian polynomials for nonlinear operators, Appl. Math. Comput., 111, 53-69 (2000) · Zbl 1023.65108
[16] Wazwaz, A. M.; El-Sayed, S., A new modification of the Adomian decomposition method for linear and nonlinear operators, Appl. Math. Comput., 122, 393-405 (2001) · Zbl 1027.35008
[17] Öziş, T.; Yildirim, A., Comparison between Adomians method and He’s homotopy perturbation method, Comput. Math. Appl., 56, 5, 1216-1224 (2008) · Zbl 1155.65344
[18] Momani, S.; Abuasad, S., Application of He’s variational iteration method to Helmholtz equation, Chaos Solitons Fractals, 27, 5, 1119-1123 (2006) · Zbl 1086.65113
[19] He, J. H., Variational iteration method for delay differential equations, Commun. Nonlinear Sci. Numer. Simul., 2, 4, 235-236 (1997)
[20] He, J. H., Semi-inverse method of establishing generalized principles for fluid mechanics with emphasis on turbomachinery aerodynamics, Int. J. Turbo Jet-Engines, 14, 1, 23-28 (1997)
[21] He, J. H., Approximate solution of non linear differential equations with convolution product nonlinearities, Comput. Methods Appl. Mech. Engrg., 167, 69-73 (1998) · Zbl 0932.65143
[22] He, J. H., Variational iteration method- a kind of non-linear analytical technique: Some examples, Int. J. Nonlinear Mech., 34, 699-708 (1999) · Zbl 1342.34005
[23] He, J. H., Variational iteration method for autonomous ordinary differential systems, Appl. Math. Comput., 114, 115-123 (2000) · Zbl 1027.34009
[24] He, J. H., Variational theory for linear magneto-electro-elasticity, Int. J. Nonlinear Sci. Numer. Simul., 2, 4, 309-316 (2001) · Zbl 1083.74526
[25] He, J. H., Variational principle for Nano thin film lubrication, Int. J. Nonlinear Sci. Numer. Simul., 4, 3, 313-314 (2003)
[26] He, J. H., Variational principle for some nonlinear partial differential equations with variable coefficients, Chaos, Solitons and Fractals, 19, 4, 847-851 (2004) · Zbl 1135.35303
[27] He, J. H., Variational iteration method—Some recent results and new interpretations, J. Comput. Appl. Math., 207, 1, 3-17 (2007) · Zbl 1119.65049
[28] He, J. H.; Wu, X. H., Variational iteration method: New development and applications, Comput. Math. Appl., 54, 7-8, 881-894 (2007) · Zbl 1141.65372
[29] 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
[30] Wazwaz, A. M., The variational iteration method for solving linear and nonlinear systems of PDEs, Comput. Math. Appl., 54, 7-8, 895-902 (2007) · Zbl 1145.35312
[31] Wazwaz, A. M., The variational iteration method: A reliable analytic tool for solving linear and nonlinear wave equations, Comput. Math. Appl., 54, 7-8, 926-932 (2007) · Zbl 1141.65388
[32] Wazwaz, A. M., The variational iteration method: A powerful scheme for handling linear and nonlinear diffusion equations, Comput. Math. Appl., 54, 7-8, 933-939 (2007) · Zbl 1141.65077
[33] Odibat, Z., Reliable approaches of variational iteration method for nonlinear operators, Math. Comput. Model., 48, 1-2, 222-231 (2008) · Zbl 1145.65314
[34] Yusufoglu, E., Variational iteration method for construction of some compact and noncompact structures of Klein-Gordon equations, Int. J. Nonlinear Sci. Numer. Simul., 8, 2, 153-185 (2007)
[35] Biazar, J.; Ghazvini, H., He’s variational iteration method for solving hyperbolic differential equations, Int. J. Nonlinear Sci. Numer. Simul., 8, 3, 311-314 (2007) · Zbl 1193.65144
[36] Ozer, H., Application of the variational iteration method to the boundary value problems with jump discontinuities arising in solid mechanics, Int. J. Nonlinear Sci. Numer. Simul., 8, 4, 513-518 (2007)
[37] Mokhtari, R., Variational iteration method for solving nonlinear differential-difference equations, Int. J. Nonlinear Sci. Numer. Simul., 9, 1, 19-24 (2008) · Zbl 1401.65152
[38] Yildirim, A.; Öziş, T., Solutions of singular IVPs of LaneEmden type by the variational iteration method, Nonlinear Anal. TMA, 70, 6, 2480-2484 (2009) · Zbl 1162.34005
[39] Momani, S., Zaid Odibat Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method, Appl. Math. Comput., 177, 2, 484-492 (2006)
[40] Odibat, Z.; Momani, S., Approximate solutions for boundary value problems of time-fractional wave equation, Appl. Math. Comput., 181, 1, 767-774 (2006) · Zbl 1148.65100
[41] Momani, S., Numerical approach to differential equations of fractional order, J. Comput. Appl. Math., 207, 1, 96-110 (2007) · Zbl 1119.65127
[42] Momani, S.; Odibat, Z., Numerical comparison of methods for solving linear differential equations of fractional order, Chaos Solitons Fractals, 31, 5, 1248-1255 (2007) · Zbl 1137.65450
[43] Odibat, Z.; Momani, S., Numerical methods for nonlinear partial differential equations of fractional order, Appl. Math. Model., 32, 1, 28-39 (2008) · Zbl 1133.65116
[44] 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
[45] Odibat, Z.; Momani, S., Application of variational iteration method to nonlinear differential equations of fractional order, Int. J. Nonlinear Sci. Numer. Simul., 7, 1, 15-27 (2006) · Zbl 1401.65087
[46] Momani, S.; Odibat, Z., Comparison between the homotopy perturbation method and the variational iteration method for linear fractional partial differential equations, Comput. Math. Appl., 54, 7-8, 910-919 (2007) · Zbl 1141.65398
[47] Momani, S.; Odibat, Z.; Alawneh, A., Variational iteration method for solving the space- and time-fractional KdV equation, Numer. Methods Partial Differential Equations, 24, 1, 262-271 (2008) · Zbl 1130.65132
[48] Caputo, M., Linear models of dissipation whose \(Q\) is almost frequency independent. Part II, J. Roy. Astr. Soc., 13, 529-539 (1967)
[49] A.Y. Luchko, R. Groreflo, The initial value problem for some fractional differential equations with the Caputo derivative, Preprint series \(A 08 - 98\); A.Y. Luchko, R. Groreflo, The initial value problem for some fractional differential equations with the Caputo derivative, Preprint series \(A 08 - 98\)
[50] Miller, K. S.; Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations (1993), John Wiley and Sons Inc.: John Wiley and Sons Inc. New York · Zbl 0789.26002
[51] Oldham, K. B.; Spanier, J., The Fractional Calculus (1974), Academic Press: Academic Press New York · Zbl 0428.26004
[52] Podlubny, I., Geometric and physical interpretation of fractional integration and fractional differentiation, Frac. Calc. Appl. Anal., 5, 367-386 (2002) · Zbl 1042.26003
[53] Cherruault, Y., Convergence of Adomian’s method, Kybernetes, 18, 31-38 (1989) · Zbl 0697.65051
[54] Cherruault, Y.; Adomian, G., Decomposition methods: A new proof of convergence, Math. Comput. Model., 18, 103-106 (1993) · Zbl 0805.65057
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