Non-perturbative analytical solutions of the space- and time-fractional Burgers equations. (English) Zbl 1099.35118

Summary: Non-perturbative analytical solutions for the generalized Burgers equation with time- and space-fractional derivatives of order \(\alpha\) and \(\beta\), \(0 < \alpha\), \(\beta \leq 1\), are derived using Adomian decomposition method. The fractional derivatives are considered in the Caputo sense. The solutions are given in the form of series with easily computable terms. Numerical solutions are calculated for the fractional Burgers equation to show the nature of solution as the fractional derivative parameter is changed.


35Q53 KdV equations (Korteweg-de Vries equations)
26A33 Fractional derivatives and integrals
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[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 · Zbl 0802.65122
[3] Ali, A. H.A.; Gardner, G. A.; Gardner, L. R.T., A collocation solution for Burgers equation using B-spline finite elements, Comput Math Appl Mech Eng, 100, 325-337 (1997) · Zbl 0762.65072
[4] Caputo, M., Linear models of dissipation whose \(Q\) is almost frequency independent. Part II, J Roy Astr Soc, 13, 529-539 (1967)
[5] El-Shahed, M., Adomian decomposition method for solving Burgers equation with fractional derivative, J Fac Cal, 24, 23-28 (2003) · Zbl 1057.35052
[7] Kaya, D.; Yokus, A., A numerical comparison of partial solutions in the decomposition method for linear and non-linear partial differential equations, Math Comput Simulat, 60, 507-512 (2002) · Zbl 1007.65078
[9] 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
[10] 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
[11] Oldham, K. B.; Spanier, J., The fractional calculus (1974), Academic Press: Academic Press New York · Zbl 0428.26004
[12] Biler, P.; Funaki, T.; Woyczynski, W. A., Fractal Burgers equation, J Different Equat, 148, 9-46 (1998) · Zbl 0911.35100
[13] Podlubny, I., Fractional differential equations (1999), Academic Press: Academic Press New York · Zbl 0918.34010
[14] Sugimoto, N., Burgers equation with a fractional derivative; Hereditary effects on non-linear acoustic waves, J Fluid Mech, 225, 631-653 (1991) · Zbl 0721.76011
[15] Wazwaz, A. M., Blow-up for solutions of some linear wave equations with mixed nonlinear boundary conditions, Appl Math Comput, 123, 133-140 (2001) · Zbl 1027.35016
[16] Wazwaz, A. M., A reliable modification of Adomian’s decomposition method, Appl Math Comput, 92, 1-7 (1998)
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