# zbMATH — the first resource for mathematics

Fractional integral representation of master equation. (English) Zbl 0988.82039
Summary: Using the definition of Liouville-Riemann fractional integral operator, the master equation can be represented in the domain of fractal time evolution with a critical exponent $$a$$ $$(0< a\leq 1)$$. The relation between the continuous time random walks and fractional master equation (FMF) has been achieved by obtaining the corresponding waiting time density $$\psi(t)$$. The latter is obtained in a closed form in terms of the generalized Mittag-Leffler (M-L) function. The asymptotic expansion of the (M-L) function show the same behavior considered in the theory of random walks. Applying the Fourier and Laplace-Mellin transforms to (FME), one obtains the solution, in closed form, in terms of the Fox function.

##### MSC:
 82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics 26A33 Fractional derivatives and integrals
Full Text:
##### References:
 [1] Halvin, S., Trus, B. and Weiss, G. H. J., J. Phys. A, 1985,\bf18, L1043. [2] Havlin, S. and Avraham, D. B., Adv. Phys., 1987,\bf36, 695. [3] Hilfer, R. and Anton, L., Phys. Rev. E, 1994,\bf51, R848. [4] Oldham, K. B. and Spanier, J., The Fractional Calculus. Academic Press, New York, 1974. · Zbl 0292.26011 [5] Schneider, W. and Wyss, W., J. Math. Phys., 1988,\bf25, 134. [6] Erdely, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G., Higher Transcendental Function. McGraw-Hill, New York, 1953. [7] March, N. H. and Tosi, M. P., Atomic Dynamics in Liquids. Macmillan, London, 1988. [8] Erdely, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G., Tables of Integral Transforms. McGraw-Hill, New York, 1953. [9] Levie, R. and Vogt, A., J. Electroanal. Chem., 1990,\bf278, 25.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.