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)
Probability calculus of fractional order and fractional Taylor’s series application to Fokker-Planck equation and information of non-random functions. (English) Zbl 1197.60039
Summary: A probability distribution of fractional (or fractal) order is defined by the measure $\mu \{dx\} = p(x)(dx)\alpha$, $0 < \alpha < 1$. Combining this definition with the fractional Taylor’s series ${f(x+h)}=E_{\alpha}(D_x^\alpha h^\alpha)f(x)$ provided by the modified Riemann Liouville definition, one can expand a probability calculus parallel to the standard one. A Fourier’s transform of fractional order using the Mittag-Leffler function is introduced, together with its inversion formula; and it provides a suitable generalization of the characteristic function of fractal random variables. It appears that the state moments of fractional order are more especially relevant. The main properties of this fractional probability calculus are outlined, it is shown that it provides a sound approach to Fokker-Planck equation which are fractional in both space and time, and it provides new results in the information theory of non-random functions. Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.

MSC:
60G18Self-similar processes
35Q84Fokker-Planck equations
26A33Fractional derivatives and integrals (real functions)
60J60Diffusion processes
82C31Stochastic methods in time-dependent statistical mechanics
94A15General topics of information theory
WorldCat.org
Full Text: DOI
References:
[1] Al-Akaidi, M.: Fractal speech processing, (2004) · Zbl 1082.94003
[2] Anh, V. V.; Leonenko, N. N.: Spectral theory of renormalized fractional random fields, Teor imovirnost matem statyst 66, 3-14 (2002) · Zbl 1029.60040
[3] Barabasi, A. L.; Stanley, H. E.: Fractal concepts in surface growth, (1995)
[4] Campos, L. M. C.: On a concept of derivative of complex order with applications to special functions, IMA J appl math 33, 109-133 (1984) · Zbl 0565.30034 · doi:10.1093/imamat/33.2.109
[5] Campos, L. M. C.: Fractional calculus of analytic and branched functions, Recent advances in fractional calculus (1993) · Zbl 0789.30030
[6] Caputo, M.: Linear model of dissipation whose Q is almost frequency dependent II, Geophys JR ast soc 13, 529-539 (1967)
[7] Carpinteri R, Mainardi P, editors, Fractals and fractional calculus in continuum mechanics. CISM Lecture Notes, Vol. 378. 1997. · Zbl 0917.73004
[8] Decreusefond, L.; Ustunel, A. S.: Stochastic analysis of the fractional Brownian motion, Potential anal 10, 177-214 (1999) · Zbl 0924.60034 · doi:10.1023/A:1008634027843
[9] Djrbashian, M. M.; Nersesian, A. B.: Fractional derivative and the Cauchy problem for differential equations of fractional order, Izv acad nauk armjanskoi SSR 3, No. 1, 3-29 (1968)
[10] Duncan, T. E.; Hu, Y.; Pasik-Duncan, B.: Stochastic calculus for fractional Brownian motion, I. Theory, SIAM J control optim 38, 582-612 (2000) · Zbl 0947.60061 · doi:10.1137/S036301299834171X
[11] El Naschie, M. S.: Elementary prerequisites for E -- infinity (recommended background readings in nonlinear dynamics, geometry, topology), Chaos, solitons & fractals 30, 579-605 (2006)
[12] Falconer, K.: Techniques in fractal geometry, (1997) · Zbl 0869.28003
[13] Hilfer, R.: Fractional time evolution, Applications of fractional calculus in physics, 87-130 (2000) · Zbl 0994.34050
[14] Hu, Y.; øksendal, B.: Fractional white noise calculus and applications to finance, Infinite dim anal quantum probab relat topics 6, 1-32 (2003) · Zbl 1045.60072 · doi:10.1142/S0219025703001110
[15] , Fractal geometry in biological systems (1996)
[16] Itô, K.: On stochastic differential equations, Mem am soc 4 (1951) · Zbl 0054.05803
[17] Jumarie, G.: Stochastic differential equations with fractional Brownian motion input, Int J syst sci 24, No. 6, 1113-1132 (1993) · Zbl 0771.60043 · doi:10.1080/00207729308949547
[18] Jumarie, G.: Maximum entropy, information without probability and complex fractals, Fundamental theories of physics series (2000) · Zbl 0982.94001
[19] Jumarie, G.: Further results on Fokker -- Planck equation of fractional order, Chaos, solitons & fractals 12, 1873-1886 (2001) · Zbl 1046.82515
[20] Jumarie, G.: On the representation of fractional Brownian motion as an integral with respect to (dt)$\alpha $, Appl math lett 18, 739-748 (2005) · Zbl 1082.60029 · doi:10.1016/j.aml.2004.05.014
[21] Jumarie, G.: On the solution of the stochastic differential equation of exponential growth driven by fractional Brownian motion, Appl math lett 18, 817-826 (2005) · Zbl 1075.60068 · doi:10.1016/j.aml.2004.09.012
[22] Jumarie, G.: Modified Riemann -- Liouville derivative and fractional Taylor series of non-differentiable functions further results, Comput math appl 51, 1367-1376 (2006) · Zbl 1137.65001 · doi:10.1016/j.camwa.2006.02.001
[23] Jumarie, G.: New stochastic fractional models for malthusian growth, the Poissonian birth process and optimal management of populations, Math comput model 44, 231-254 (2006) · Zbl 1130.92043 · doi:10.1016/j.mcm.2005.10.003
[24] Jumarie, G.: Lagrangian mechanics of fractional order, Hamilton -- Jacobi fractional PDE and Taylor’s series of nondifferentiable functions, Chaos, solitons & fractals 32, No. 3, 969-987 (2007) · Zbl 1154.70011 · doi:10.1016/j.chaos.2006.07.053
[25] Kober, H.: On fractional integrals and derivatives, Quart J math Oxford 11, 193-215 (1940) · Zbl 0025.18502
[26] Kolwankar, K. M.; Gangal, A. D.: Holder exponents of irregular signals and local fractional derivatives, Pramana J phys 48, 49-68 (1997)
[27] Kolwankar, K. M.; Gangal, A. D.: Local fractional Fokker -- Planck equation, Phys rev lett 80, 214-217 (1998) · Zbl 0945.82005 · doi:10.1103/PhysRevLett.80.214
[28] Letnivov, A. V.: Theory of differentiation of fractional order, Math sb 3, 1-7 (1868)
[29] , Frontiers in mathematical biology (1994) · Zbl 0810.00005
[30] Levy Vehel J. Fractal, probability functions: an application to image analysis. In: Computer vision and pattern recognition, 1991 proceedings CVPR ’91’ IEEE. Computer Society Conference.
[31] Liouville, J.: Sur le calcul des differentielles à indices quelconques, J ecole polytech 13, 71 (1832)
[32] Mandelbrot, B. B.; Van Ness, J. W.: Fractional Brownian motions, fractional noises and applications, SIAM rev 10, 422-437 (1968) · Zbl 0179.47801 · doi:10.1137/1010093
[33] Mandelbrot, B. B.: The fractal geometry of nature, (1982) · Zbl 0504.28001
[34] Mandelbrot, B. B.; Cioczek-Georges, R.: A class of micropulses and antipersistent fractional Brownian motions, Stochast process appl 60, 1-18 (1995) · Zbl 0846.60055 · doi:10.1016/0304-4149(95)00046-1
[35] Mandelbrot, B. B.; Cioczek-Georges, R.: Alternative micropulses and fractional Brownian motion, Stochast process appl 64, 143-152 (1996) · Zbl 0879.60076 · doi:10.1016/S0304-4149(96)00089-0
[36] Mandelbrot, B.: Fractals and scaling in finance: discontinuity, concentration, risk, (1997) · Zbl 1005.91001
[37] Miller, K. S.; Ross, B.: An introduction to the fractional calculus and fractional differential equations, (1973) · Zbl 0789.26002
[38] Nishimoto, K.: Fractional calculus, (1989) · Zbl 0707.26009
[39] Oldham, K. B.; Spanier, J.: The fractional calculus. Theory and application of differentiation and integration to arbitrary order, (1974) · Zbl 0292.26011
[40] Ortigueira, M. D.: Introduction to fractional signal processing. Part I: Continuous time systems, IEE proc vision image signal process 1, 62-70 (2000)
[41] Osler, T. J.: Taylor’s series generalized for fractional derivatives and applications, SIAM J math anal 2, No. 1, 37-47 (1971) · Zbl 0215.12101 · doi:10.1137/0502004
[42] Oustaloup, A.: La derivation non entiere: theorie, synthese et applications (Non-integer derivation: theory, synthesis and applications), (1995) · Zbl 0864.93004
[43] Podlubny, I.: Fractional differential equations, (1999) · Zbl 0924.34008
[44] Ross, B.: Fractional calculus and its applications, Lecture notes in mathematics 457 (1974)
[45] Samko, S. G.; Kilbas, A. A.; Marichev, O. I.: Fractional integrals and derivatives. Theory and applications, (1987)
[46] Schlichter, J.; Friedrich, J.; Herenyi, L.; Fidy, J.: Protein dynamics a low temperatures, J chem phys 112, 3045-3050 (2000)
[47] , Lévy flights and related topics in physics 450 (1995)
[48] Stratonovich, R. L.: A new form of representing stochastic integrals and equations, J SIAM control 4, 362-371 (1966)
[49] West, B. J.: Fractal probability measures of learning, Methods 24, No. 4, 395-402 (2001)
[50] Zemanian, A. H.: Distribution theory and transform analysis, (1987) · Zbl 0643.46028