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)
Numerical comparison of methods for solving linear differential equations of fractional order. (English) Zbl 1137.65450
Summary: We implement relatively new analytical techniques, the variational iteration method and the Adomian decomposition method, for solving linear differential equations of fractional order. The two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of fractional differential equations. In these schemes, the solution takes the form of a convergent series with easily computable components. This paper will present a numerical comparison between the two methods and a conventional method such as the fractional difference method for solving linear differential equations of fractional order. The numerical results demonstrate that the new methods are quite accurate and readily implemented.

MSC:
65R20Integral equations (numerical methods)
45J05Integro-ordinary differential equations
26A33Fractional derivatives and integrals (real functions)
65L05Initial value problems for ODE (numerical methods)
34A30Linear ODE and systems, general
WorldCat.org
Full Text: DOI
References:
[1] Basset, A. B.: On the descent of a sphere in a viscous liquid. Quart J math 42, 369-381 (1910) · Zbl 41.0826.01
[2] 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
[3] Shawagfeh, N. T.: Analytical approximate solutions for nonlinear fractional differential equations. Appl math comput 131, 517-529 (2002) · Zbl 1029.34003
[4] Podlubny, I.: Fractional differential equations. (1999) · Zbl 0924.34008
[5] Gao, X.; Yu, J.: Synchronization of two coupled fractional-order chaotic oscillators. Chaos, solitons & fractals 26, No. 1, 141-145 (2005) · Zbl 1077.70013
[6] Lu, J. G.: Chaotic dynamics and synchronization of fractional-order arneodo’s systems. Chaos, solitons & fractals 26, No. 4, 1125-1133 (2005) · Zbl 1074.65146
[7] Lu, J. G.; Chen, G.: A note on the fractional-order Chen system. Chaos, solitons & fractals 27, No. 3, 685-688 (2006) · Zbl 1101.37307
[8] He, J. H.: Variational iteration method for delay differential equations. Commun nonlinear sci numer simulat 2, No. 4, 235-236 (1997) · Zbl 0924.34063
[9] He, J. H.: Approximate analytical solution for seepage flow with fractional derivatives in porous media. Comput meth appl mech eng 167, 57-68 (1998) · Zbl 0942.76077
[10] He, J. H.: Approximate solution of non linear differential equations with convolution product nonlinearities. Comput meth appl mech eng 167, 69-73 (1998) · Zbl 0932.65143
[11] He, J. H.: Variational iteration method --- a kind of non-linear analytical technique: some examples. Int J nonlinear mech 34, 699-708 (1999) · Zbl 05137891
[12] He, J. H.: Variational iteration method for autonomous ordinary differential systems. Appl math comput 114, 115-123 (2000) · Zbl 1027.34009
[13] He, J. H.; Wan, Y. Q.; Guo, Q.: An iteration formulation for normalized diode characteristics. Int J circ theory appl 32, No. 6, 629-632 (2004) · Zbl 1169.94352
[14] Adomian, G.: A review of the decomposition method in applied mathematics. J math anal appl 135, 501-544 (1988) · Zbl 0671.34053
[15] Adomian, G.: Solving frontier problems of physics: the decomposition method. (1994) · Zbl 0802.65122
[16] Shawagfeh, N.; Kaya, D.: Comparing numerical methods for the solutions of systems of ordinary differential equations. Appl math lett 17, 323-328 (2004) · Zbl 1061.65062
[17] Momani, S.; Al-Khaled, K.: Numerical solutions for systems of fractional differential equations by the decomposition method. Appl math comput 162, No. 3, 1351-1365 (2005) · Zbl 1063.65055
[18] Momani S. Numerical simulation of a dynamic system containing fractional derivatives. Accepted for presentation at the International Symposium on Nonlinear Dynamics, Shanghai, China, December 20 -- 21; 2005. · Zbl 1119.65394
[19] Marinca, V.: An approximate solution for one-dimensional weakly nonlinear oscillations. Int J nonlinear sci numer simulat 3, No. 2, 107-110 (2002) · Zbl 1079.34028
[20] Dra&caron, G. E.; Gaˇ Nescu; Ca&caron, V.; Paˇ Lnaˇ San: Nonlinear relaxation phenomena in polycrystalline solids. Int J nonlinear sci numer simulat 4, No. 3, 219-226 (2003)
[21] Liu, H. M.: Generalized variational principles for ion acoustic plasma waves by he’s semi-inverse method. Chaos, solitons & fractals 23, No. 2, 573-576 (2005) · Zbl 1135.76597
[22] Hao, T. H.: Search for variational principles in electrodynamics by Lagrange method. Int J nonlinear sci numer simulat 6, No. 2, 209-210 (2005)
[23] Momani, S.; Abuasad, S.: Application of he’s variational iteration method to Helmholtz equation. Chaos, solitons & fractals 27, No. 5, 1119-1123 (2006) · Zbl 1086.65113
[24] Odibat, Z.; Momani, S.: Application of variational iteration method to nonlinear differential equations of fractional order. Int J nonlinear sci numer simulat 6, No. 1, 27-34 (2005)
[25] Luchko Y, Gorneflo R. The initial value problem for some fractional differential equations with the Caputo derivative. Preprint series A08 -- 98, Fachbereich Mathematik und Informatik, Freie Universitat Berlin, 1998.
[26] Miller, K. S.; Ross, B.: An introduction to the fractional calculus and fractional differential equations. (1993) · Zbl 0789.26002
[27] Oldham, K. B.; Spanier, J.: The fractional calculus. (1974) · Zbl 0292.26011
[28] Caputo, M.: Linear models of dissipation whose Q is almost frequency independent. Part II. J roy austral soc 13, 529-539 (1967)
[29] Inokuti, M.; Sekine, H.; Mura, T.: General use of the Lagrange multiplier in non-linear mathematical physics. Variational method in the mechanics of solids, 156-162 (1978)
[30] He, J. H.: Semi-inverse method of establishing generalized principles for fluid mechanics with emphasis on turbomachinery aerodynamics. Int J turbo jet-engines 14, No. 1, 23-28 (1997)
[31] He, J. H.: Variational theory for linear magneto-electro-elasticity. Int J nonlinear sci numer simulat 2, No. 4, 309-316 (2001) · Zbl 1083.74526
[32] He, J. H.: Generalized variational principles in fluids (in chinese). (2003) · Zbl 1054.76001
[33] He, J. H.: Variational principle for nano thin film lubrication. Int J nonlinear sci numer simulat 4, No. 3, 313-314 (2003)
[34] He, J. H.: Variational principle for some nonlinear partial differential equations with variable coefficients. Chaos, solitons & fractals 19, No. 4, 847-851 (2004) · Zbl 1135.35303
[35] Liu, H. M.: Variational approach to nonlinear electrochemical system. Int J nonlinear sci numer simulat 5, No. 1, 95-96 (2004)