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**Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable. functions. Further results.**
*(English)*
Zbl 1137.65001

Summary: The paper gives some results and improves the derivation of the fractional Taylor’s series of nondifferentiable functions obtained recently by the author [Appl. Math. Lett. 18, No. 7, 739–748 (2005; Zbl 1082.60029); ibid. 18, No. 7, 817–826 (2005; Zbl 1075.60068)] in the form \(f (x + h) = E_{\alpha } (h^{\alpha }D_{x }^{\alpha })f(x )\), \(0 < \alpha \leq 1\), where \(E_{\alpha }\) is the Mittag-Leffler function. Here, one defines fractional derivative as the limit of fractional difference, and by this way one can circumvent the problem which arises with the definition of the fractional derivative of constant using Riemann-Liouville definition. As a result, a modified Riemann-Liouville definition is proposed, which is fully consistent with the fractional difference definition and avoids any reference to the derivative of order greater than the considered one’s.

In order to support this F-Taylor series, one shows how its first term can be obtained directly in the form of a mean value formula. The fractional derivative of the Dirac delta function is obtained together with the fractional Taylor’s series of multivariate functions. The relation with irreversibility of time and symmetry breaking is exhibited, and to some extent, this F-Taylor’s series generalizes the fractional mean value formula obtained by K. M. Kolwankar and A. D. Gangal [Phys. Rev. Lett. 80, No. 2, 214–217 (1998; Zbl 0945.82005)].

In order to support this F-Taylor series, one shows how its first term can be obtained directly in the form of a mean value formula. The fractional derivative of the Dirac delta function is obtained together with the fractional Taylor’s series of multivariate functions. The relation with irreversibility of time and symmetry breaking is exhibited, and to some extent, this F-Taylor’s series generalizes the fractional mean value formula obtained by K. M. Kolwankar and A. D. Gangal [Phys. Rev. Lett. 80, No. 2, 214–217 (1998; Zbl 0945.82005)].

### MSC:

65B15 | Euler-Maclaurin formula in numerical analysis |

26A33 | Fractional derivatives and integrals |

40A05 | Convergence and divergence of series and sequences |

### Keywords:

fractional Taylor series; fractional Mac-Laurin series; fractional derivative; Mittag-Leffler function; nondifferentiable functions; Dirac delta function
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\textit{G. Jumarie}, Comput. Math. Appl. 51, No. 9--10, 1367--1376 (2006; Zbl 1137.65001)

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### References:

[1] | Decreusefond, L.; Ustunel, A.S., Stochastic analysis of the fractional Brownian motion, Potential anal., 10, 177-214, (1999) · Zbl 0924.60034 |

[2] | 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 |

[3] | Hu, Y.; Øksendal, B., Fractional white noise calculus and applications to finance, Infinite dim. anal. quantum probab. related topics, 6, 1-32, (2003) · Zbl 1045.60072 |

[4] | Jumarie, G., Stochastic differential equations with fractional Brownian motion input, Int. J. syst. sc., 6, 1113-1132, (1993) · Zbl 0771.60043 |

[5] | Jumarie, G., Maximum entropy, information without probability and complex fractals, (2000), Kluwer · Zbl 0982.94001 |

[6] | Jumarie, G., Further results on the modelling of complex fractals in finance, scaling observation and optimal portfolio selection, Systems analysis, modelling simulation, 45, 10, 1483-1499, (2002) · Zbl 1092.91025 |

[7] | Jumarie, G., Fractional Brownian motions via random walk in the complex plane and via fractional derivative, comparison and further results on their Fokker-Planck equations, Chaos, solitons and fractals, 4, 907-925, (2004) · Zbl 1068.60053 |

[8] | Mandelbrot, B.B.; van Ness, J.W., Fractional Brownian motions, fractional noises and applications, SIAM rev., 10, 422-437, (1968) · Zbl 0179.47801 |

[9] | Mandelbrot, B.B.; Cioczek-Georges, R., A class of micropulses and antipersistent fractional Brownian motions, Stochastic processes and their applications, 60, 1-18, (1995) · Zbl 0846.60055 |

[10] | Mandelbrot, B.B.; Cioczek-Georges, R., Alternative micropulses and fractional Brownian motion, Stochastic processes and their applications, 64, 143-152, (1996) · Zbl 0879.60076 |

[11] | Wyss, W., The fractional Black-Scholes equation, Fract. cale. appl. anal., 3, 1, 51-61, (2000) · Zbl 1058.91045 |

[12] | El Naschie, M.S., A review of E infinity theory and the mass spectrum of high energy particle physics, Chaos, solitons and fractals, 19, 209-236, (2004) · Zbl 1071.81501 |

[13] | Jumarie, G., A Fokker-Planck equation of fractional order with respect to time, Journal of math. physics, 33, 10, 3536-3542, (1992) · Zbl 0761.60071 |

[14] | Bakai, E., Fractional Fokker-Planck equation, solutions and applications, Physical review E, 63, 1-17, (2001) |

[15] | Jumarie, G., Schrödinger equation for quantum-fractal space-time of order n via the complex-valued fractional Brownian motion, Intern. J. of modern physics A, 16, 31, 5061-5084, (2001) · Zbl 1039.81008 |

[16] | Nelson, E., Quantum fluctuations, (1985), Princeton University Press Dordrecht · Zbl 0563.60001 |

[17] | Ord, G.N.; Mann, R.B., Entwined paths, difference equations and Dirac equations, Phys. rev A, 37, 0121XX3, (2003) |

[18] | Anh, V.V.; Leonenko, N.N., Scaling laws for fractional diffusion-wave equations with singular initial data, Statistics and probability letters, 48, 239-252, (2000) · Zbl 0970.35174 |

[19] | El-Sayed, A., Fractional order diffusion-wave equation, Int. J. theor. phys., 35, 311-322, (1996) · Zbl 0846.35001 |

[20] | Hanyga, A., Multidimensional solutions of time-fractional diffusion-wave equations, (), 933-957 · Zbl 1153.35347 |

[21] | Nottale, L., Fractal space-time and microphysics, (1993), World Scientific Princeton, NJ · Zbl 0789.58003 |

[22] | Nottale, L., Scale-relativity and quantization of the universe I. theoretical framework, Astronm astrophys, 327, 867-889, (1997) |

[23] | Nottale, L., The scale-relativity programme, Chaos, solitons and fractals, 10, 2-3, 459-468, (1999) · Zbl 0997.81526 |

[24] | Shawagfeh, N.T., Analytical approximate solutions for nonlinear fractional differential equations, Appl. math. and comp., 131, 517-529, (2002) · Zbl 1029.34003 |

[25] | Jumarie, G., On the representation of fractional Brownian motion as an integral with respect to (dt)α, Appl. math. lett., 18, 7, 739-748, (2005) · Zbl 1082.60029 |

[26] | Jumarie, G., On the solution of the stochastic differential equation of exponential growth driven by fractional Brownian motion, Appl. math. lett., 18, 7, 817-826, (2005) · Zbl 1075.60068 |

[27] | Kober, H., On fractional integrals and derivatives, Quart. J. math. Oxford, 11, 193-215, (1940) · Zbl 0025.18502 |

[28] | Letnivov, A.V., Theory of differentiation of fractional order, Math. sb., 3, 1-7, (1868) |

[29] | Liouville, J., Sur le calcul des differentielles ê indices quelconques (in French), J. ecole polytechnique, 13, 71, (1832) |

[30] | Caputo, M., Linear model of dissipation whose Q is almost frequency dependent II, Geophys. J. R. ast. soc., 13, 529-539, (1967) |

[31] | Djrbashian, M.M.; Nersesian, A.B., Fractional derivative and the Cauchy problem for differential equations of fractional order (in Russian), Izv. acad. nauk armjanskoi SSR, 3, 1, 3-29, (1968) |

[32] | Osler, T.J., Taylor’s series generalized for fractional derivatives and applications, SIAM. J. mathematical analysis, 2, 1, 37-47, (1971) · Zbl 0215.12101 |

[33] | Kolwankar, K.M.; Gangal, A.D., Holder exponents of irregular signals and local fractional derivatives, Pramana J. phys., 48, 49-68, (1997) |

[34] | Kolwankar, K.M.; Gangal, A.D., Local fractional Fokker-Planck equation, Phys. rev. lett., 80, 214-217, (1998) · Zbl 0945.82005 |

[35] | G. Jumarie, Fractional Hamilton-Jacobi equation for the optimal control of non-random fractional dynamics with fractional cost function, Journal of Applied Mathematics and Computing (to appear). · Zbl 1111.49014 |

[36] | G. Juniarie, Lagrange characteristics method for solving a class of nonlinear partial differential equations of fractional order, Appl. Math. Lett. (to appear). |

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