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)
Adomian decomposition: a tool for solving a system of fractional differential equations. (English) Zbl 1061.34003
Summary: Adomian’s decomposition method is employed to obtain solutions of a system of fractional differential equations. The convergence of the method is discussed with some illustrative examples. In particular, for the initial value problem $$[D^{\alpha_1} y_1,\dots,D^{\alpha_n}y_n]^t =A(y_1, \dots,y_n)^t, \quad y_i(0)=c_i, \quad i=1,\dots,n,$$ where $A=[a_{ij}]$ is a real square matrix, the solution turns out to be $$\overline y(x)={\cal E}_{(\alpha_1,\dots, \alpha_n),1}(x^{\alpha_1}A_1,\dots, x^{\alpha_n}A_n) \overline y(0),$$ where ${\cal E}_{(\alpha_1,\dots,\alpha_n),1}$ denotes the multivariate Mittag-Leffler function defined for matrix arguments and $A_i$ is the matrix having $i$th row as $[a_i1\dots a_{in}]$, and all other entries are zero. Fractional oscillation and Bagley-Torvik equations are solved as illustrative examples.

34A12Initial value problems for ODE, existence, uniqueness, etc. of solutions
26A33Fractional derivatives and integrals (real functions)
Full Text: DOI
[1] Abboui, K.; Cherruault, Y.: New ideas for proving convergence of decomposition methods. Comput. appl. Math. 29, 103-105 (1995) · Zbl 0832.47051
[2] Adomian, G.: Solving frontier problems of physics: the decomposition method. (1994) · Zbl 0802.65122
[3] Atanackovic, T. M.; Stankovic, B.: On a system of differential equations with fractional derivatives arising in rod theory. J. phys. A 37, 1241-1250 (2004) · Zbl 1059.35011
[4] Babolian, E.; Biazar, J.; Islam, R.: Solution of the system of ordinary differential equations by Adomian decomposition method. Appl. math. Comput. 147, 713-719 (2004) · Zbl 1034.65053
[5] Chechkin, A. V.; Gorenflo, R.; Sokolov, I. M.; Gonchar, V. Yu.: Distributed order time fractional diffusion equation. Frac. calc. Appl. anal. 6, 259-279 (2003) · Zbl 1089.60046
[6] Daftardar-Gejji, V.; Babakhani, A.: Analysis of a system of fractional differential equations. J. math. Anal. appl. 293, 511-522 (2004) · Zbl 1058.34002
[7] 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
[8] 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
[9] Luchko, Yu.; Gorenflo, R.: An operational method for solving fractional differential equations with the Caputo derivatives. Acta math. Vietnam 24, 207-233 (1999) · Zbl 0931.44003
[10] Miller, K. S.; Ross, B.: An introduction to the fractional calculus and fractional differential equations. (1993) · Zbl 0789.26002
[11] Podlubny, I.: Fractional differential equations. (1999) · Zbl 0924.34008
[12] Samko, S. G.; Kilbas, A. A.; Marichev, O. I.: Fractional integrals and derivatives: theory and applications. (1993) · Zbl 0818.26003
[13] Shawagfeh, N. T.: Analytical approximate solutions for nonlinear fractional differential equations. Appl. math. Comput. 131, 517-529 (2002) · Zbl 1029.34003
[14] Torvik, P. J.; Bagley, R. L.: On the appearance of the fractional derivative in the behaviour of real materials. J. appl. Mech. 51, 294-298 (1984) · Zbl 1203.74022
[15] West, B. J.; Bologna, M.; Grigolini, P.: Physics of fractal operators. (2003)