zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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.

35Q53KdV-like (Korteweg-de Vries) equations
26A33Fractional derivatives and integrals (real functions)
Full Text: DOI
[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) · 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
[6] Kaya D. An application of the decomposition method for the KdVB equation. Appl Math Comput, in press. · Zbl 1053.65087
[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
[8] Luchko Y, Gorneflo R. The initial value problem for some fractional differential equations with the Caputo derivative. Preprint series A08-98, Fachbreich Mathematik und Informatik, Freic Universitat Berlin, 1998.
[9] Mainardi, F.: Fractional calculus: ’some basic problems in continuum and statistical mechanics’. Fractals and fractional calculus in continuum mechanics, 291-348 (1997) · Zbl 0917.73004
[10] Miller, K. S.; Ross, B.: An introduction to the fractional calculus and fractional differential equations. (1993) · Zbl 0789.26002
[11] Oldham, K. B.; Spanier, J.: The fractional calculus. (1974) · Zbl 0292.26011
[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) · Zbl 0924.34008
[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)