zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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)
Analytical approximate solutions for nonlinear fractional differential equations. (English) Zbl 1029.34003
Summary: We consider a class of nonlinear fractional-differential equations bsed on the Caputo fractional derivative and, by extending the application of the Adomian decomposition method, we derive an analytical solution in the form of a series with easily computable terms. For linear equations, the method gives an exact solution, and, for nonlinear equations, it provides an approximate solution with good accuracy. Several examples are discussed.

MSC:
34A12Initial value problems for ODE, existence, uniqueness, etc. of solutions
34A45Theoretical approximation of solutions of ODE
WorldCat.org
Full Text: DOI
References:
[1] Podlubny, I.: Fractional differential equations. (1999) · Zbl 0924.34008
[2] Miller, K. S.; Ross, B.: An introduction to the fractional calculus and fractional differential equations. (1993) · Zbl 0789.26002
[3] Samko, G.; Kilbas, A. A.; Marichev, O. I.: Fractional integrals and derivatives: theory and applications. (1993) · Zbl 0818.26003
[4] Gorenflo, R.; Mainardi, F.: Fractional calculus: integral and differential equations of fractional order. Fractals & fractional calculus in continuum mechanics, 223-276 (1997)
[5] S. Kemple, H. Beyer, Global and causal solutions of fractional differential equations, in: Transform Methods and Special Functions: Varna96, Proceedings of 2nd International Workshop (SCTP), Singapore, 1997, pp. 210--216
[6] Luchko, Y.; Srivastava, H. M.: The exact solution of certain differential equations of fractional order by using operational calculus. Comput. math. Appl. 29, 73-85 (1995) · Zbl 0824.44011
[7] Y. Luchko, R. Gorenflo, The initial value problem for some fractional differential equations with the Caputo derivative, Preprint Series A08-98, Fachbereich Mathematik und Informatick, Freie Universitat Berlin, 1998 · Zbl 0940.45001
[8] Shawagfeh, N. T.: The decomposition method for fractional differential equations. J. frac. Calc. 16, 27-33 (1999) · Zbl 0956.34004
[9] Adomian, G.: Solving frontier problems of physics: the decomposition method. (1994) · Zbl 0802.65122
[10] K. Diethelm, A.D. Freed, On the solution of nonlinear fractional-order differential equations used in the modeling of viscoplasticity, in: Proceedings of the 2nd Conference on Scientific Computing in Chemical Engineering, Springer, Heidelberg, 1999
[11] Caputo, M.: Linear models of dissipation whose Q is almost frequency independent. Part II. J. roy. Astral. soc. 13, 529-539 (1967)
[12] Mainardi, F.: Fractional calculus: some basic problems in continuum and statistical mechanics. Fractals & fractional calculus in continuum mechanics, 291-348 (1997) · Zbl 0917.73004
[13] Wazwaz, A. M.: A new algorithm for calculating Adomian polynomials for nonlinear operators. Appl. math. Comput. 111, No. 1, 33-51 (2000) · Zbl 1023.65108
[14] Cherrualult, Y.; Adomian, G.: Decomposition method: a new proof of convergence. Math. comput. Modelling 18, 103-106 (1993) · Zbl 0805.65057
[15] Abboui, K.; Cherruault, Y.: New ideas for proving convergence of decomposition methods. Comput. math. Appl. 29, No. 7, 103-105 (1995) · Zbl 0832.47051
[16] A. Erdelyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher Transendental Functions, vol. VIII, McGraw-Hill, New York, 1955
[17] Oldham, K. B.; Spanier, J.: The fractional calculus. (1974) · Zbl 0292.26011
[18] 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 (1994) · Zbl 1203.74022
[19] K. Diethelm, N.J. Ford, Numerical solution of Bagley--Torvik equation, Berichte der Mathematischen Institute der Technischen Universitat Braunschweig, No. 00/14, 2000 · Zbl 1035.65067
[20] 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
[21] K. Diethelm, N.J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl. (to appear) · Zbl 1014.34003
[22] Wazwaz, A. M.: A reliable modification of Adomian’s decomposition method. Appl. math. Comput. 92, 1-7 (1998)