Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method. (English) Zbl 1096.65131

Momani, Shaher; Odibat, Zaid

Appl. Math. Comput. 177, No. 2, 488-494 (2006).

Summary: The aim of the present analysis is to apply the Adomian decomposition method for the solution of a time-fractional Navier-Stokes equation in a tube. By using an initial value, the explicit solution of the equation is presented in closed form and then its numerical solution is represented graphically. The present method performs extremely well in terms of efficiency and simplicity.

MSC:

65R20	Numerical methods for integral equations
35Q30	Navier-Stokes equations
76D05	Navier-Stokes equations for incompressible viscous fluids
45K05	Integro-partial differential equations
45G10	Other nonlinear integral equations
26A33	Fractional derivatives and integrals
76M25	Other numerical methods (fluid mechanics) (MSC2010)

Keywords:

Caputo fractional derivative; numerical examples; Adomian decomposition method; time-fractional Navier-Stokes equation

References:

[1]	Adomian, G., A review of the decomposition method in applied mathematics, J. Math. Anal. Appl., 135, 501-544 (1988) · Zbl 0671.34053
[2]	Adomian, G., Solving Frontier Problems of Physics: The Decomposition Method (1994), Kluwer Academic Publishers: Kluwer Academic Publishers Boston, MA · Zbl 0802.65122
[3]	Shawagfeh, N. T., Analytical approximate solutions for nonlinear fractional differential equations, Appl. Math. Comput., 131, 517-529 (2002) · Zbl 1029.34003
[4]	Momani, S.; Al-Khaled, K., Numerical solutions for systems of fractional differential equations by the decomposition method, Appl. Math. Comput., 162, 3, 1351-1365 (2005) · Zbl 1063.65055
[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]	Momani, S., Analytic and approximate solutions of the space- and time-fractional telegraph equations, Appl. Math. Comput., 170, 1126-1134 (2005) · Zbl 1103.65335
[8]	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
[9]	Podlubny, I., Fractional Differential Equations (1999), Academic Press: Academic Press New York · Zbl 0918.34010
[10]	El-Shahed, M.; Salem, A., On the generalized Navier-Stokes equations, Appl. Math. Comput., 156, 1, 287-293 (2004) · Zbl 1134.76323
[12]	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
[13]	Oldham, K. B.; Spanier, J., The Fractional Calculus (1974), Academic Press: Academic Press New York · Zbl 0428.26004
[14]	Caputo, M., Linear models of dissipation whose \(Q\) is almost frequency independent. Part II, J. Roy. Astral. Soc., 13, 529-539 (1967)
[15]	Cherruault, Y., Convergence of Adomian’s method, Kybernetes, 18, 31-38 (1989) · Zbl 0697.65051
[16]	Cherruault, Y.; Adomian, G., Decomposition methods: a new proof of convergence, Math. Comput. Modell., 18, 103-106 (1993) · Zbl 0805.65057
[17]	Biazar, J.; Babolian, E.; Kember, G.; Nouri, A.; Islam, R., An alternate algorithm for computing Adomian polynomials in special cases, Appl. Math. Comput., 138, 523-529 (2002) · Zbl 1027.65076