×

Fractional Euler limits and their applications. (English) Zbl 1371.35304

The Euler limit of the Mittag-Leffler function is conjectured by discretizing the Caputo fractional derivative. Then the Cauchy integral formula for the same Mittag-Leffler function is derived by using the Laplace transform. By making a deformation of the contour to the real axis, further integral formulas are obtained. Several analytic properties of these integrals are discussed. It is shown that both the negative exponential and the negative Mittag-Leffler function are completely monotone in the case of a negative argument. Finally, probabilistic interpretations and matrix arguments are discussed, as well as applications to Schlögl reactions. Instead of theorems, several graphs are shown to illustrate the reasoning.

MSC:

35Q92 PDEs in connection with biology, chemistry and other natural sciences
35R60 PDEs with randomness, stochastic partial differential equations
35R11 Fractional partial differential equations
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] L. Abadias and P. J. Miana, {\it A subordination principle on Wright functions and regularized resolvent families}, J. Function Spaces (2015), 158145. · Zbl 1354.47028
[2] L. Altenberg, {\it Resolvent positive linear operators exhibit the reduction phenomenon}, Proc. Nat. Acad. Sci., 109 (2012), pp. 3705-3710, . · Zbl 1287.47034
[3] C. N. Angstmann, I. C. Donnelly, and B. I. Henry, {\it Pattern formation on networks with reactions: A continuous-time random-walk approach}, Phys. Rev. E, 87 (2013), 032804. · Zbl 1320.60135
[4] C. N. Angstmann, I. C. Donnelly, B. I. Henry, T. A. M. Langlands, and P. Straka, {\it Generalized continuous time random walks, master equations, and fractional Fokker-Planck equations}, SIAM J. Appl. Math., 75 (2015), pp. 1445-1468. · Zbl 1327.35370
[5] W. Arendt, C. Batty, M. Hieber, and F. Neubrander, {\it Vector-Valued Laplace Transforms and Cauchy Problems}, Birkhauser, Basel, 2011. · Zbl 1226.34002
[6] B. Baeumer and M. M. Meerschaert, {\it Stochastic solutions for fractional Cauchy problems}, Fract. Calc. Appl. Anal., 4 (2001), pp. 481-500. · Zbl 1057.35102
[7] E. G. Bajlekova, {\it Fractional Evolution Equations in Banach Spaces}, Ph.D. thesis, Technische Universiteit Eindhoven, 2001. · Zbl 0989.34002
[8] B. P. Belinskiy and T. J. Kozubowski, {\it Exponential mixture representation of geometric stable densities}, J. Math. Anal. Appl., 246 (2000), pp. 465-479. · Zbl 0963.60005
[9] Y. Berkowitz, Y. Edery, H. Scher, and B. Berkowitz, {\it Fickian and non-Fickian diffusion with bimolecular reactions}, Phys. Rev. E, 87 (2013), 032812.
[10] A. Berman and R. J. Plemmons, {\it Nonnegative Matrices in the Mathematical Sciences}, Classics in Appl. Math. 9, SIAM, Philadelphia, 1994. · Zbl 0815.15016
[11] E. Blanc, S. Engblom, A. Hellander, and P. Lötstedt, {\it Mesoscopic modeling of stochastic reaction-diffusion kinetics in the subdiffusive regime}, Multiscale Model. Simul., 14 (2016), pp. 668-707, . · Zbl 1381.35086
[12] S. Bochner, {\it Harmonic Analysis and the Theory of Probability}, Dover, New York, 2005. · Zbl 0068.11702
[13] K. Burrage and G. Lythe, {\it Accurate stationary densities with partitioned numerical methods for stochastic differential equations}, SIAM J. Numer. Anal., 47 (2009), pp. 1601-1618. · Zbl 1197.60068
[14] A. V. Chechkin, R. Gorenflo, and I. M. Sokolov, {\it Fractional diffusion in inhomogeneous media}, J. Phys. A, 38 (2005), p. L679. · Zbl 1082.76097
[15] B. Drawert, M. Trogdon, S. Toor, L. Petzold, and A. Hellander, {\it Molns: A cloud platform for interactive, reproducible, and scalable spatial stochastic computational experiments in systems biology using pyurdme}, SIAM J. Sci. Comput., 38 (2016), pp. C179-C202, .
[16] S. Fedotov, {\it Non-Markovian random walks and nonlinear reactions: Subdiffusion and propagating fronts}, Phys. Rev. E, 81 (2010), 011117, .
[17] P. J. Forrester and C. J. Thompson, {\it The Golden-Thompson inequality: Historical aspects and random matrix applications}, J. Math. Phys., 55 (2014). · Zbl 1308.15018
[18] D. Fulger, E. Scalas, and G. Germano, {\it Monte Carlo simulation of uncoupled continuous-time random walks yielding a stochastic solution of the space-time fractional diffusion equation}, Phys. Rev. E, 77 (2008), 021122.
[19] R. Garrappa, {\it Numerical evaluation of two and three parameter Mittag-Leffler functions}, SIAM J. Numer. Anal., 53 (2015), pp. 1350-1369, . · Zbl 1331.33043
[20] D. Gillespie, {\it Markov Processes: An Introduction for Physical Scientists}, Academic Press, New York, 1992. · Zbl 0743.60001
[21] R. Gorenflo, F. Mainardi, and A. Vivoli, {\it Subordination in fractional diffusion processes via continuous time random walk}, in More Progresses in Analysis, World Scientific, River Edge, NJ, 2009. · Zbl 1181.26015
[22] H. J. Haubold, A. M. Mathai, and R. K. Saxena, {\it Mittag-Leffler functions and their applications}, J. Appl. Math., 2011 (2011), 298628. · Zbl 1218.33021
[23] B. I. Henry, T. A. M. Langlands, and S. L. Wearne, {\it Anomalous diffusion with linear reaction dynamics: From continuous time random walks to fractional reaction-diffusion equations}, Phys. Rev. E, 74 (2006), 031116.
[24] N. J. Higham, {\it Functions of Matrices: Theory and Computation}, SIAM, Philadelphia, 2008. · Zbl 1167.15001
[25] R. Hilfer and L. Anton, {\it Fractional master equations and fractal time random walks}, Phys. Rev. E, 51 (1995), R848.
[26] A. Iserles, {\it A First Course in the Numerical Analysis of Differential Equations}, Cambridge University Press, Cambridge, UK, 2009. · Zbl 1171.65060
[27] T. Kato, {\it Perturbation Theory for Linear Operators}, Springer-Verlag, Berlin, 1976. · Zbl 0342.47009
[28] V. M. Kenkre, E. W. Montroll, and M. F. Shlesinger, {\it Generalized master equations for continuous-time random walks}, J. Statist. Phys., 9 (1973), p. 45.
[29] J. Klafter, A. Blumen, and M. F. Shlesinger, {\it Stochastic pathway to anomalous diffusion}, Phys. Rev. A, 35 (1987), pp. 3081-3085.
[30] J. Klafter and I. M. Sokolov, {\it First Steps in Random Walks: From Tools to Applications}, Oxford University Press, New York, 2011. · Zbl 1242.60046
[31] S. Kotz, T. Kozubowski, and K. Podgorski, {\it The Laplace Distribution and Generalizations: A Revisit with Applications to Communications, Economics, Engineering and Finance}, Birkhauser, Basel, 2001. · Zbl 0977.62003
[32] T. J. Kozubowski, {\it Computer simulation of geometric stable distributions}, J. Comput. Appl. Math., 116 (2000), pp. 221-229. · Zbl 0952.65007
[33] T. J. Kozubowski and S. T. Rachev, {\it Univariate geometric stable laws}, J. Comput. Anal. Appl., 1 (1999), pp. 177-217. · Zbl 1055.60503
[34] T. Kurtz, {\it Representations of Markov processes as multiparameter time changes}, Ann. Probab., 8 (1980), pp. 682-715. · Zbl 0442.60072
[35] P. Lax and L. Zalcman, {\it Complex Proofs of Real Theorems}, AMS, Providence, RI, 2012. · Zbl 1243.30001
[36] P. D. Lax and R. D. Richtmyer, {\it Survey of the stability of linear finite difference equations}, Comm. Pure Appl. Math, 9 (1956), pp. 267-293. · Zbl 0072.08903
[37] S. Macnamara, {\it Cauchy integrals for computational solutions of master equations}, ANZIAM J., 56 (2015), pp. 32-51, .
[38] S. MacNamara, K. Burrage, and R. Sidje, {\it Multiscale Modeling of Chemical Kinetics via the Master Equation}, SIAM Multiscale Model. Simul., 6 (2008), pp. 1146-1168. · Zbl 1153.60370
[39] M. Magdziarz, A. Weron, and K. Weron, {\it Fractional Fokker-Planck dynamics: Stochastic representation and computer simulation}, Phys. Rev. E, 75 (2007), 016708. · Zbl 1123.60045
[40] F. Mainardi, {\it Fractional Calculus and Waves in Linear Viscoelasticity}, Imperial College Press, London, 2010. · Zbl 1210.26004
[41] F. Mainardi, M. Raberto, R. Gorenflo, and E. Scalas, {\it Fractional calculus and continuous-time finance II: The waiting-time distribution}, Phys. A, 287 (2000), pp. 468-481. · Zbl 1138.91444
[42] A. Mathai, R. K. Saxena, and H. J. Haubold, {\it The H-Function: Theory and Applications}, Springer, Berlin, 2009. · Zbl 1218.33021
[43] W. McLean, {\it Regularity of solutions to a time-fractional diffusion equation}, ANZIAM J., 52 (2010), pp. 123-138. · Zbl 1228.35266
[44] W. McLean and V. Thomée, {\it Time discretization of an evolution equation via Laplace transforms}, IMA J. Numer. Anal., 24 (2004), pp. 439-463. · Zbl 1068.65146
[45] R. Metzler and J. Klafter, {\it The random walk’s guide to anomalous diffusion: A fractional dynamics approach}, Phys. Rep., 339 (2000), pp. 1-77. · Zbl 0984.82032
[46] G. M. Mittag-Leffler, {\it Sur la nouvelle fonction \({E}_α(x)\)}, C. R. Acad. Sci. Paris, 137 (1903), pp. 554-558. · JFM 34.0435.01
[47] G. M. Mittag-Leffler, {\it Une generalisation de l’integrale de Laplace-Abel}, C. R. Acad. Sci. Paris, 137 (1903), pp. 537-539. · JFM 34.0434.02
[48] E. W. Montroll and G. H. Weiss, {\it Random Walks on Lattices. II}, J. Math. Phys., 6 (1965), pp. 167-181. · Zbl 1342.60067
[49] K. Mustapha and W. McLean, {\it Superconvergence of a discontinuous Galerkin method for fractional diffusion and wave equations}, SIAM J. Numer. Anal., 51 (2013), pp. 516-525. · Zbl 1267.26005
[50] I. Podlubny, {\it Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications}, Academic Press, San Diego, 1999. · Zbl 0924.34008
[51] J. Prüss, {\it Evolutionary Integral Equations and Applications}, Birkhauser, Basel, 2012. · Zbl 1258.45008
[52] M. Raberto, F. Rapallo, and E. Scalas, {\it Semi-Markov Graph Dynamics}, PLoS ONE, 6 (2011), e23370.
[53] I. M. Sokolov, M. G. W. Schmidt, and F. Sagués, {\it Reaction-subdiffusion equations}, Phys. Rev. E, 73 (2006), 031102, .
[54] R. Speth, W. Green, S. MacNamara, and G. Strang, {\it Balanced splitting and rebalanced splitting}, SIAM J. Numer. Anal., 51 (2013), pp. 3084-3105. · Zbl 1284.65121
[55] G. Strang, {\it Introduction to Linear Algebra}, Wellesley-Cambridge Press, Wellesley, MA, 2009. · Zbl 1067.15500
[56] G. Strang and S. MacNamara, {\it Functions of difference matrices are Toeplitz plus Hankel}, SIAM Rev., 56 (2014), pp. 525-546, . · Zbl 1304.65195
[57] L. N. Trefethen and M. Embree, {\it Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators}, Princeton University Press, Princeton, NJ, 2005. · Zbl 1085.15009
[58] G. Wanner and E. Hairer, {\it Analysis by Its History}, Springer, Berlin, 1995. · Zbl 0842.26002
[59] Q. Yang, T. Moroney, K. Burrage, I. Turner, and F. Liu, {\it Novel numerical methods for time-space fractional reaction diffusion equations in two dimensions}, ANZIAM J., 52 (2011), pp. 395-409. · Zbl 1390.65093
[60] K. Yosida, {\it Functional Analysis}, Springer, Berlin, 1995. · Zbl 0842.92020
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.