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)
Numerical solutions for systems of fractional differential equations by the decomposition method. (English) Zbl 1063.65055
Summary: We use the Adomian decomposition method to solve systems of nonlinear fractional differential equations and a linear multi-term fractional differential equation by reducing it to a system of fractional equations each of order at most unity. We begin by showing how the decomposition method applies to a class of nonlinear fractional differential equations and give two examples to illustrate the efficiency of the method. Moreover, we show how the method can be applied to a general linear multi-term equation and solve several applied problems.

65L05Initial value problems for ODE (numerical methods)
34A34Nonlinear ODE and systems, general
26A33Fractional derivatives and integrals (real functions)
65L60Finite elements, Rayleigh-Ritz, Galerkin and collocation methods for ODE
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] Baker, C. T. H.; Derakhshan, M. S.: Stability barriers to the construction of ${\rho},{\sigma}$-reducible and fractional quadrature rules. Ser. numer. Math. 85, 1-15 (1988) · Zbl 0652.65014
[4] Bagley, R. L.; Torvik, P. J.: On the appearance of the fractional derivative in the behavior of real materials. J. appl. Mech. 51, 294-298 (1984) · Zbl 1203.74022
[5] Basset, A. B.: On the descent of a sphere in a viscous liquid. Quart. J. Math. 42, 369-381 (1910) · Zbl 41.0826.01
[6] L. Blank, Numerical treatment of differential equations of fractional order, MCCM Numerical Analysis Report No. 287, The University of Manchester, 1996 · Zbl 0870.65137
[7] Caputo, M.: Linear models of dissipation whose Q is almost frequency independent. Part II. J. roy. Astr. soc. 13, 529-539 (1967)
[8] Diethelm, K.: An algorithm for the numerical solution of differential equations of fractional order. Electron. trans. Numer. anal. 5, 1-6 (1997) · Zbl 0890.65071
[9] Diethelm, K.; Ford, N. J.: Numerical solution of the bagley--torvik equation. Bit 42, 490-507 (2002) · Zbl 1035.65067
[10] K. Diethelm, Y. Luchko, Numerical solution of linear multi-term differential equations of fractional order, J. Comput. Anal. Appl., in press · Zbl 1083.65064
[11] Edwards, J. T.; Ford, N. J.; Simpson, A. C.: The numerical solution of linear multi-term fractional differential equations: systems of equations. J. comput. Appl. math. 148, 401-418 (2002) · Zbl 1019.65048
[12] Hull, T. E.; Enright, W. H.; Fellen, B. M.; Sedgwick, A. E.: Comparing numerical methods for ordinary differential equations. SIAM J. Numer. anal. 9, 603 (1972) · Zbl 0221.65115
[13] D. Kaya, A reliable method for the numerical solution of the kinetics problems, Appl. Math. Comput, in press · Zbl 1057.65042
[14] AY. Luchko, R. Groreflo, 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
[15] 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
[16] Metzler, R.; ; Klafter, J.: Boundary value problems fractional diffusion equations. Physica A 278, 107-125 (2000) · Zbl 0984.82032
[17] Miller, K. S.; Ross, B.: An introduction to the fractional calculus and fractional differential equations. (1993) · Zbl 0789.26002
[18] Momani, S. M.: On the existence of solutions of a system of ordinary differential equations of fractional order. Far east J. Math. sci. 1, No. 2, 265-270 (1999) · Zbl 0933.34003
[19] Oldham, K. B.; Spanier, J.: The fractional calculus. (1974) · Zbl 0292.26011
[20] Podlubny, I.: Fractional differential equations. (1999) · Zbl 0924.34008
[21] Shawagfeh, N. T.: Analytical approximate solutions for nonlinear fractional differential equations. Appl. math. Comput. 131, No. 2--3, 517-529 (2002) · Zbl 1029.34003
[22] 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
[23] Wazwaz, A. M.: A reliable modification of Adomian’s decomposition method. Appl. math. Comput. 92, 1-7 (1998)