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)
A new fractional analytical approach via a modified Riemann-Liouville derivative. (English) Zbl 1251.65101
Summary: This work suggests a new analytical technique called the fractional homotopy perturbation method (FHPM) for solving fractional differential equations of any fractional order. This method is based on He’s homotopy perturbation method and the modified Riemann-Liouville derivative. The fractional differential equations are described in Jumarie’s sense. The results from introducing a modified Riemann-Liouville derivative in the cases studied show the high accuracy, simplicity and efficiency of the approach.
65L05Initial value problems for ODE (numerical methods)
34A08Fractional differential equations
34A34Nonlinear ODE and systems, general
[1]Oldham, K. B.; Spanier, J.: The fractional calculus: theory and applications of differentiation and integration to arbitrary order, (1974)
[2]Miller, K.; Ross, B.: An introduction to the fractional calculus and fractional differential equations, (1993)
[3]Podlubny, I.: Fractional differential equations, (1999)
[4]Momani, Shaher: Analytical approximate solution for fractional heat-like and wave-like equations with variable coefficients using the decomposition method, Appl. math. Comput. 165, 459-472 (2005) · Zbl 1070.65105 · doi:10.1016/j.amc.2004.06.025
[5]Odibat, Zaid; Momani, Shaher: Numerical solution of Fokker–Planck equation with space- and time-fractional derivatives, Phys. lett. A 369, 349-358 (2007) · Zbl 1209.65114 · doi:10.1016/j.physleta.2007.05.002
[6]Odibat, Zaid; Momani, Shaher: The variational iteration method: an efficient scheme for handling fractional partial differential equations in fluid mechanics, Comput. math. Appl. 58, 2199-2208 (2009) · Zbl 1189.65254 · doi:10.1016/j.camwa.2009.03.009
[7]Inc, Mustafa: The approximate and exact solutions of the space and time-fractional Burgers equations with initial conditions by variational iteration method, J. math. Appl. 345, 476-484 (2008) · Zbl 1146.35304 · doi:10.1016/j.jmaa.2008.04.007
[8]He, Ji-Huan: Approximate analytical solution for seepage flow with fractional derivative in porous media, Comput. methods appl. Mech. engrg. 167, 57-68 (1998) · Zbl 0942.76077 · doi:10.1016/S0045-7825(98)00108-X
[9]Odibat, Z.; Momani, S.: Application of variational iteration method to nonlinear differential equations of fractional order, Int. J. Nonlinear sci. Numer. simul. 7, No. 1, 27-34 (2006)
[10]Wu, G. C.; Lee, E. W. M.: Fractional variational iteration method and its application, Phys. lett. A 374, 2506-2509 (2010)
[11]Al-Rabtah, Adel; Ertürk, Vedat Suat; Momani, Shaher: Solutions of a fractional oscillator by using differential transform method, Comput. math. Appl. 59, 1356-1362 (2010) · Zbl 1189.34068 · doi:10.1016/j.camwa.2009.06.036
[12]Momani, Shaher; Odibat, Zaid: Homotopy perturbation method for nonlinear partial differential equations of fractional order, Phys. lett. A 365, 345-350 (2007) · Zbl 1203.65212 · doi:10.1016/j.physleta.2007.01.046
[13]Momani, Shaher; Odibat, Zaid: Numerical solutions of the space–time fractional advection–dispersion equation, Numer. methods partial differential equations 24, 1416-1429 (2008) · Zbl 1148.76044 · doi:10.1002/num.20324
[14]Wang, Qi: Homotopy perturbation method for fractional KdV–Burgers equation, Chaos solitons fractals 35, 843-850 (2008) · Zbl 1132.65118 · doi:10.1016/j.chaos.2006.05.074
[15]He, J. H.: Homotopy perturbation technique, Comput. methods appl. Mech. engrg. 178, No. 3–4, 257-262 (1999)
[16]He, J. H.: A coupling method of homotopy technique and perturbation technique for nonlinear problems, Internat. J. Non-linear mech. 35, No. 1, 37-43 (2000) · Zbl 1068.74618 · doi:10.1016/S0020-7462(98)00085-7
[17]Jumarie, G.: Modified Riemann–Liouville derivative and fractional Taylor series of nondifferentiable functions further results, Comput. math. Appl. 51, 1367-1376 (2006) · Zbl 1137.65001 · doi:10.1016/j.camwa.2006.02.001
[18]Caputo, M.: Linear model of dissipation whose Q is almost frequency dependent II, Geophys. J. R. astron. Soc. 13, 529-539 (1967)
[19]Djrbashian, M. M.; Nersesian, A. B.: Fractional derivative and the Cauchy problem for differential equations of fractional order, Izv. acad. Nauk arm. SSR 3, 3-29 (1968)
[20]Jumarie, G.: Fractional partial differential equations and modified Riemann–Liouville derivative new methods for solution, J. appl. Math. comput. 24, 31-48 (2007) · Zbl 1145.26302 · doi:10.1007/BF02832299
[21]Jumarie, G.: Stochastic differential equations with fractional Brownian motion input, Internat. J. Systems sci. 6, 1113-1132 (1993) · Zbl 0771.60043 · doi:10.1080/00207729308949547
[22]Jumarie, G.: New stochastic fractional models for malthusian growth, the Poissonian birth process and optimal management of populations, Math. comput. Modelling 44, 231-254 (2006) · Zbl 1130.92043 · doi:10.1016/j.mcm.2005.10.003
[23]Jumarie, G.: Laplace’s transform of fractional order via the Mittag-Leffler function and modified Riemann–Liouville derivative, Appl. math. Lett. 22, 1659-1664 (2009) · Zbl 1181.44001 · doi:10.1016/j.aml.2009.05.011
[24]Jumarie, G.: Fourier’s transform of fractional order via Mittag-Leffler function and modified Riemann–Liouville derivative, J. appl. Math. inform. 26, 1101-1121 (2008)
[25]Jumarie, G.: Table of some basic fractional calculus formulae derived from a modified Riemann–Liouville derivative for non-differentiable functions, Appl. math. Lett. 22, 378-385 (2009) · Zbl 1171.26305 · doi:10.1016/j.aml.2008.06.003
[26]J.H. He, Strongly nonlinear problems, Dissertation, de-Verlag im Internat GmbH, Berlin, 2006.
[27]He, J. H.: New interpretation of homotopy perturbation method, Internat. J. Modern phys. B 20, 1-7 (2006)
[28]Wu, G. C.; Shi, Y. G.; Wu, K. T.: Adomian decomposition method and non-analytical solutions of fractional differential equations, Romanian J. Phys. 56, 873-880 (2011)