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Volumn , Issue , 2008, Pages 93-127

Continuous Time Random Walk, Mittag-Leffler Waiting Time and Fractional Diffusion: Mathematical Aspects

Author keywords

Continuous time random walk; Diffusion limit in space; Fractional diffusion; Gnedenko Kovalenko theory of thinning; Manipulations; Mathematical aspects; Mittag Leffler waiting time; Mittag Leffler Waiting Time Density asymptotic universality; Mittag Leffler Waiting Time Density power laws; Rescaling; Respeeding; Time fractional drift process

Indexed keywords


EID: 65549151413     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1002/9783527622979.ch4     Document Type: Chapter
Times cited : (71)

References (68)
  • 1
    • 0000993220 scopus 로고
    • Anomalous diffusion in one dimension
    • V. Balakrishnan, Anomalous diffusion in one dimension, Physica A 132 (1985), 569-580.
    • (1985) Physica A , vol.132 , pp. 569-580
    • Balakrishnan, V.1
  • 2
    • 0033742723 scopus 로고    scopus 로고
    • Fractional Kramers equation
    • E. Barkai and R.J. Silbey, Fractional Kramers equation, J. Phys. Chem. B 104 (2000), 3866-3874.
    • (2000) J. Phys. Chem. B , vol.104 , pp. 3866-3874
    • Barkai, E.1    Silbey, R.J.2
  • 3
    • 33750488103 scopus 로고    scopus 로고
    • On Hilfer's objection to the fractional time diffusion equation
    • E. Barkai and I.M. Sokolov, On Hilfer's objection to the fractional time diffusion equation, Physica A 373 (2007), 231-236.
    • (2007) Physica A , vol.373 , pp. 231-236
    • Barkai, E.1    Sokolov, I.M.2
  • 4
    • 84977255207 scopus 로고
    • Linear models of dissipation whose Q is almost frequency independent
    • M. Caputo, Linear models of dissipation whose Q is almost frequency independent, Part II. Geophys. J. R. Astr. Soc. 13 (1967), 529-539.
    • (1967) Part II. Geophys. J. R. Astr. Soc. , vol.13 , pp. 529-539
    • Caputo, M.1
  • 5
    • 0003954624 scopus 로고
    • Elasticità e Dissipazione
    • Bologna, Zanichelli
    • M. Caputo, Elasticità e Dissipazione. Bologna, Zanichelli (1969).
    • (1969)
    • Caputo, M.1
  • 7
    • 0004291893 scopus 로고
    • Renewal Theory
    • 2nd edn., Methuen, London
    • D.R. Cox, Renewal Theory, 2nd edn., Methuen, London (1967).
    • (1967)
    • Cox, D.R.1
  • 8
    • 0003767059 scopus 로고
    • Integral Transforms and Representations of Functions in the Complex Plane
    • Moscow, Nauka, In Russian. [Note that there is also the transliteration as Djrbashyan]
    • M.M. Dzherbashyan, Integral Transforms and Representations of Functions in the Complex Plane, Moscow, Nauka (1966). In Russian. [Note that there is also the transliteration as Djrbashyan]
    • (1966)
    • Dzherbashyan, M.M.1
  • 9
    • 0003031851 scopus 로고
    • Fractional derivatives and the Cauchy problem for differential equations of fractional order
    • In Russian
    • M.M. Dzherbashyan and A.D. Nersesian, Fractional derivatives and the Cauchy problem for differential equations of fractional order. Izv. Acad. Nauk Armjanskvy SSR, Matematika 3 (1968), 3-29. In Russian.
    • (1968) Izv. Acad. Nauk Armjanskvy SSR, Matematika , vol.3 , pp. 3-29
    • Dzherbashyan, M.M.1    Nersesian, A.D.2
  • 10
    • 0003864328 scopus 로고
    • Higher Transcendental Functions
    • McGraw-Hill, New York
    • A. Erdélyi, W. Magnus, F. Oberhettinger, and F.G. Tricomi, Higher Transcendental Functions, Vol. 3, McGraw-Hill, New York (1953-1954).
    • (1953) , vol.3
    • Erdélyi, A.1    Magnus, W.2    Oberhettinger, F.3    Tricomi, F.G.4
  • 11
    • 0002087137 scopus 로고
    • On a generalization of Marcel Riesz' potentials and the semi-groups generated by them, Meddelanden Lunds Universitets Matematiska Seminarium
    • Comm. Sém. Mathém. Université de Lund
    • W. Feller, On a generalization of Marcel Riesz' potentials and the semi-groups generated by them, Meddelanden Lunds Universitets Matematiska Seminarium (Comm. Sém. Mathém. Université de Lund). Tome suppl. dédié a M. Riesz, Lund, 1952, pp. 73-81.
    • (1952) Tome suppl. dédié a M. Riesz, Lund , pp. 73-81
    • Feller, W.1
  • 12
    • 0003421261 scopus 로고
    • An Introduction to Probability Theory and its Applications
    • 2nd edn., Wiley, New York, 1st edn. 1966
    • W. Feller, An Introduction to Probability Theory and its Applications, Vol. 2, 2nd edn., Wiley, New York (1971). [1st edn. 1966]
    • (1971) , vol.2
    • Feller, W.1
  • 13
    • 0003625475 scopus 로고
    • Generalized Functions
    • Academic Press,New York and London
    • I.M. Gelfand and G.E. Shilov, Generalized Functions, Vol. I. Academic Press,New York and London (1964).
    • (1964) , vol.I
    • Gelfand, I.M.1    Shilov, G.E.2
  • 14
    • 0003513648 scopus 로고
    • Limit Distributions for Sums of Independent Random Variables
    • Addison-Wesley, Cambridge, MA, Translated from the Russian edition, Moscow 1949, with notes by K.L. Chung, revised 1968
    • B.V. Gnedenko and A.N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables, Addison-Wesley, Cambridge, MA (1954). [Translated from the Russian edition, Moscow 1949, with notes by K.L. Chung, revised 1968]
    • (1954)
    • Gnedenko, B.V.1    Kolmogorov, A.N.2
  • 15
    • 0010219143 scopus 로고
    • Introduction to Queueing Theory
    • Israel Program for Scientific Translations, Jerusalem
    • B.V. Gnedenko and I.N. Kovalenko, Introduction to Queueing Theory, Israel Program for Scientific Translations, Jerusalem (1968).
    • (1968)
    • Gnedenko, B.V.1    Kovalenko, I.N.2
  • 16
    • 32544454465 scopus 로고    scopus 로고
    • From power laws to fractional diffusion: the direct way
    • R. Gorenflo and E. Abdel-Rehim, From power laws to fractional diffusion: the direct way, Vietnam J. Math. 32 SI (2004), 65-75.
    • (2004) Vietnam J. Math. , vol.32 , Issue.SI , pp. 65-75
    • Gorenflo, R.1    Abdel-Rehim, E.2
  • 17
    • 0000009397 scopus 로고    scopus 로고
    • Analytical properties and applications of the Wright function
    • E-print arXiv:math-ph/0701069
    • R. Gorenflo, Yu. Luchko, and F. Mainardi, Analytical properties and applications of the Wright function, Fractional Calculus Appl. Anal. 2 (1999), 383-414. [E-print arXiv:math-ph/0701069]
    • (1999) Fractional Calculus Appl. Anal. , vol.2 , pp. 383-414
    • Gorenflo, R.1    Luchko, Y.2    Mainardi, F.3
  • 18
    • 0000103589 scopus 로고    scopus 로고
    • Wright functions as scale-invariant solutions of the diffusion-wave equation
    • R. Gorenflo, Yu. Luchko, and F. Mainardi, Wright functions as scale-invariant solutions of the diffusion-wave equation, J. Comput. Appl. Math. 118 (2000), 175-191.
    • (2000) J. Comput. Appl. Math. , vol.118 , pp. 175-191
    • Gorenflo, R.1    Luchko, Y.2    Mainardi, F.3
  • 19
    • 0002847893 scopus 로고    scopus 로고
    • Fractional calculus: integral and differential equations of fractional order
    • A. Carpinteri and F. Mainardi (Editors), Springer, Wien and New York
    • R. Gorenflo and F. Mainardi, Fractional calculus: integral and differential equations of fractional order. In A. Carpinteri and F. Mainardi (Editors), Fractals and Fractional Calculus in Continuum Mechanics, Springer, Wien and New York (1997), pp. 223-276. [http://www.fracalmo.org]
    • (1997) Fractals and Fractional Calculus in Continuum Mechanics , pp. 223-276
    • Gorenflo, R.1    Mainardi, F.2
  • 20
    • 0000361678 scopus 로고    scopus 로고
    • Random walk models for space fractional diffusion processes
    • R. Gorenflo and F. Mainardi, Random walk models for space fractional diffusion processes, Fractional Calculus Appl. Anal. 1 (1998), 167-191.
    • (1998) Fractional Calculus Appl. Anal. , vol.1 , pp. 167-191
    • Gorenflo, R.1    Mainardi, F.2
  • 21
    • 0141974618 scopus 로고    scopus 로고
    • Fractional diffusion processes: probability distributions and continuous time random walk
    • G. Rangarajan and M. Ding (Editors), Springer, Berlin, Lecture Notes in Physics, No. 621
    • R. Gorenflo and F. Mainardi, Fractional diffusion processes: probability distributions and continuous time random walk, in: G. Rangarajan and M. Ding (Editors), Processes with Long Range Correlations, Springer, Berlin (2003), pp. 148-166. [Lecture Notes in Physics, No. 621]
    • (2003) Processes with Long Range Correlations , pp. 148-166
    • Gorenflo, R.1    Mainardi, F.2
  • 22
    • 33646113530 scopus 로고    scopus 로고
    • Simply and multiply scaled diffusion limits for continuous time random wal
    • S. Benkadda, X. Leoncini, and G. Zaslavsky (Editors) (France) 20-25 June 2004, IOP (Institute of Physics) Journal of Physics: Conference Series
    • R. Gorenflo and F. Mainardi, Simply and multiply scaled diffusion limits for continuous time random wal, in: S. Benkadda, X. Leoncini, and G. Zaslavsky (Editors), Proceedings of the International Workshop on Chaotic Transport and Complexity in Fluids and Plasmas Carry Le Rouet (France) 20-25 June 2004, IOP (Institute of Physics) Journal of Physics: Conference Series 7 (2005), 1-16.
    • (2005) Proceedings of the International Workshop on Chaotic Transport and Complexity in Fluids and Plasmas Carry Le Rouet , vol.7 , pp. 1-16
    • Gorenflo, R.1    Mainardi, F.2
  • 23
    • 0002898176 scopus 로고    scopus 로고
    • Fractional calculus and continuous-time finance III: the diffusion limit
    • M. Kohlmann and S. Tang (Editors), Birkhäuser, Basel
    • R. Gorenflo, F. Mainardi, E. Scalas, and M. Raberto, Fractional calculus and continuous-time finance III: the diffusion limit, in: M. Kohlmann and S. Tang (Editors), Mathematical Finance, Birkhäuser, Basel (2001), pp. 171-180.
    • (2001) Mathematical Finance , pp. 171-180
    • Gorenflo, R.1    Mainardi, F.2    Scalas, E.3    Raberto, M.4
  • 24
    • 0012793682 scopus 로고    scopus 로고
    • Special functions in fractional relaxation-oscillation and fractional diffusion-wave phenomena
    • D. Bainov (Editor), Plovdiv 1997 VSP, Utrecht
    • R. Gorenflo, F. Mainardi, and H.M. Srivastava, Special functions in fractional relaxation-oscillation and fractional diffusion-wave phenomena, in: D. Bainov (Editor), Proceedings VIII International Colloquium on Differential Equations, Plovdiv 1997 VSP, Utrecht, 1998, pp. 195-202.
    • (1998) Proceedings VIII International Colloquium on Differential Equations , pp. 195-202
    • Gorenflo, R.1    Mainardi, F.2    Srivastava, H.M.3
  • 25
    • 34047155650 scopus 로고    scopus 로고
    • Continuous time random walk and parametric subordination in fractional diffusion
    • E-print arXiv:cond-mat/0701126
    • R. Gorenflo, F. Mainardi, and A. Vivoli, Continuous time random walk and parametric subordination in fractional diffusion, Chaos, Solitons and Fractals 34 (2007), 87-103. [E-print arXiv:cond-mat/0701126]
    • (2007) Chaos, Solitons and Fractals , vol.34 , pp. 87-103
    • Gorenflo, R.1    Mainardi, F.2    Vivoli, A.3
  • 26
    • 15544368331 scopus 로고    scopus 로고
    • Discrete and continuous random walk models for space-time fractional diffusion
    • R. Gorenflo, A. Vivoli, and F. Mainardi, Discrete and continuous random walk models for space-time fractional diffusion, Nonlinear Dynamics 38 (2004), 101-116.
    • (2004) Nonlinear Dynamics , vol.38 , pp. 101-116
    • Gorenflo, R.1    Vivoli, A.2    Mainardi, F.3
  • 27
    • 0000254982 scopus 로고
    • Exact solutions for a class of fractal time random walks
    • R. Hilfer, Exact solutions for a class of fractal time random walks, Fractals 3 (1995), 211-216.
    • (1995) Fractals , vol.3 , pp. 211-216
    • Hilfer, R.1
  • 28
    • 0141955949 scopus 로고    scopus 로고
    • On fractional diffusion and continuous time random walks
    • R. Hilfer, On fractional diffusion and continuous time random walks, Physica A 329 (2003), 35-39.
    • (2003) Physica A , vol.329 , pp. 35-39
    • Hilfer, R.1
  • 29
    • 0000437484 scopus 로고
    • Fractional master equations and fractal time random walks
    • R. Hilfer and L. Anton, Fractional master equations and fractal time random walks, Phys. Rev. E 51 (1995), R848-R851.
    • (1995) Phys. Rev. E , vol.51
    • Hilfer, R.1    Anton, L.2
  • 30
    • 0037166166 scopus 로고    scopus 로고
    • Renewal processes and the Hurst effect
    • T. Huillet, Renewal processes and the Hurst effect, J. Phys. A 35 (2002), 4395-4413.
    • (2002) J. Phys. A , vol.35 , pp. 4395-4413
    • Huillet, T.1
  • 31
    • 0036344199 scopus 로고    scopus 로고
    • On the waiting time paradox and related topics
    • T. Huillet, On the waiting time paradox and related topics, Fractals 10 (2002), 1-20.
    • (2002) Fractals , vol.10 , pp. 1-20
    • Huillet, T.1
  • 32
    • 0003654359 scopus 로고
    • Generalized Fractional Calculus and Applications
    • Harlow, Longman, Pitman Research Notes in Mathematics
    • V. Kiryakova, Generalized Fractional Calculus and Applications. Harlow, Longman (1994). [Pitman Research Notes in Mathematics, Vol. 301]
    • (1994) , vol.301
    • Kiryakova, V.1
  • 33
    • 0007045475 scopus 로고
    • A Cauchy problem for evolution equations of fractional order
    • English translation from the Russian Journal Differentsial'nye Uravneniya
    • A.N. Kochubei, A Cauchy problem for evolution equations of fractional order, Diff. Equ. 25 (1989), 967-974. [English translation from the Russian Journal Differentsial'nye Uravneniya]
    • (1989) Diff. Equ. , vol.25 , pp. 967-974
    • Kochubei, A.N.1
  • 34
    • 0000332009 scopus 로고
    • Fractional order diffusion
    • English translation from the Russian Journal Differentsial'nye Uravneniya
    • A.N. Kochubei, Fractional order diffusion, Diff. Equ. 26 (1990), 485-492. [English translation from the Russian Journal Differentsial'nye Uravneniya]
    • (1990) Diff. Equ. , vol.26 , pp. 485-492
    • Kochubei, A.N.1
  • 35
    • 21844503941 scopus 로고
    • Asymptotic distributions of continuous-time random walks: a probabilistic approach
    • M. Kotulski, Asymptotic distributions of continuous-time random walks: a probabilistic approach, J. Stat. Phys. 81 (1995), 777-792.
    • (1995) J. Stat. Phys. , vol.81 , pp. 777-792
    • Kotulski, M.1
  • 37
    • 0030464353 scopus 로고    scopus 로고
    • Fractional relaxationoscillation and fractional diffusion-wave phenomena. Chaos
    • F. Mainardi, Fractional relaxationoscillation and fractional diffusion-wave phenomena. Chaos, Solitons and Fractals 7 (1996), 1461-1477.
    • (1996) Solitons and Fractals , vol.7 , pp. 1461-1477
    • Mainardi, F.1
  • 38
    • 0001983732 scopus 로고    scopus 로고
    • Fractional calculus: some basic problems in continuum and statistical mechanics
    • A. Carpinteri and F. Mainardi (Editors), Springer, Wien and New York
    • F. Mainardi, Fractional calculus: some basic problems in continuum and statistical mechanics, in A. Carpinteri and F. Mainardi (Editors), Fractals and Fractional Calculus in Continuum Mechanics, Springer, Wien and New York, 1997, pp. 291-348.
    • (1997) Fractals and Fractional Calculus in Continuum Mechanics , pp. 291-348
    • Mainardi, F.1
  • 39
    • 0000145521 scopus 로고    scopus 로고
    • On Mittag-Leffler-type functions in fractional evolution processes
    • F. Mainardi and R. Gorenflo, On Mittag-Leffler-type functions in fractional evolution processes, J. Comput. Appl. Math. 118 (2000), 283-299.
    • (2000) J. Comput. Appl. Math. , vol.118 , pp. 283-299
    • Mainardi, F.1    Gorenflo, R.2
  • 40
    • 32544461435 scopus 로고    scopus 로고
    • A fractional generalization of the Poisson processes
    • E-print arXiv:math-PR/0701454
    • F. Mainardi, R. Gorenflo, and E. Scalas, A fractional generalization of the Poisson processes. Vietnam J. Mathematics 32 SI (2004), 53-64. [E-print arXiv:math-PR/0701454]
    • (2004) Vietnam J. Mathematics , vol.32 , Issue.SI , pp. 53-64
    • Mainardi, F.1    Gorenflo, R.2    Scalas, E.3
  • 41
    • 33645175085 scopus 로고    scopus 로고
    • Renewal processes of Mittag-Leffler and Wright type
    • E-print arXiv:math-PR/0701455
    • F. Mainardi, R. Gorenflo, and A. Vivoli, Renewal processes of Mittag-Leffler and Wright type, Fractional Calculus Appl. Anal., 8 (2005), 7-38. [E-print arXiv:math-PR/0701455]
    • (2005) Fractional Calculus Appl. Anal. , vol.8 , pp. 7-38
    • Mainardi, F.1    Gorenflo, R.2    Vivoli, A.3
  • 42
    • 0001407424 scopus 로고    scopus 로고
    • The fundamental solution of the space-time fractional diffusion equation
    • E-print arXiv:cond-mat/0702419
    • F. Mainardi, Yu. Luchko, and G. Pagnini, The fundamental solution of the space-time fractional diffusion equation, Fractional Calculus Appl. Anal. 4 (2001), 153-192. [E-print arXiv:cond-mat/0702419]
    • (2001) Fractional Calculus Appl. Anal. , vol.4 , pp. 153-192
    • Mainardi, F.1    Luchko, Y.2    Pagnini, G.3
  • 43
    • 0037457788 scopus 로고    scopus 로고
    • The Wright functions as solutions of the time fractional diffusion equations
    • F. Mainardi and G. Pagnini, The Wright functions as solutions of the time fractional diffusion equations, Appl. Math. Comput. 141 (2003), 51-62.
    • (2003) Appl. Math. Comput. , vol.141 , pp. 51-62
    • Mainardi, F.1    Pagnini, G.2
  • 44
    • 0034502929 scopus 로고    scopus 로고
    • Fractional calculus and continuous-time finance II: the waiting time distribution
    • F. Mainardi, M. Raberto, R. Gorenflo, and E. Scalas, Fractional calculus and continuous-time finance II: the waiting time distribution, Physica A 287 (2000), 468-481.
    • (2000) Physica A , vol.287 , pp. 468-481
    • Mainardi, F.1    Raberto, M.2    Gorenflo, R.3    Scalas, E.4
  • 45
    • 23744510704 scopus 로고    scopus 로고
    • Continuous time random walk and time fractional diffusion: a numerical comparison between the fundamental solutions
    • F. Mainardi, A. Vivoli, and R. Gorenflo, Continuous time random walk and time fractional diffusion: a numerical comparison between the fundamental solutions, Fluct. Noise Lett. 5 (2005), L291-L297.
    • (2005) Fluct. Noise Lett. , vol.5
    • Mainardi, F.1    Vivoli, A.2    Gorenflo, R.3
  • 47
    • 18144408075 scopus 로고    scopus 로고
    • Anomalous diffusion and relaxation close to thermal equilibrium: a fractional Fokker-Planck equation approach
    • R. Metzler, E. Barkai, and J. Klafter, Anomalous diffusion and relaxation close to thermal equilibrium: a fractional Fokker-Planck equation approach, Phys. Rev. Lett 82 (1999), 3563-3567.
    • (1999) Phys. Rev. Lett , vol.82 , pp. 3563-3567
    • Metzler, R.1    Barkai, E.2    Klafter, J.3
  • 48
    • 0000338623 scopus 로고
    • Random walks on lattices. IV: Continuous-time walks and influence of absorbing boundaries
    • E.W. Montroll and H. Scher, Random walks on lattices, IV: Continuous-time walks and influence of absorbing boundaries, J. Stat. Phys. 9 (1973), 101-135.
    • (1973) J. Stat. Phys. , vol.9 , pp. 101-135
    • Montroll, E.W.1    Scher, H.2
  • 49
    • 0002567920 scopus 로고
    • Random walks on lattices II
    • E.W. Montroll and G.H. Weiss, Random walks on lattices, II, J. Math. Phys. 6 (1965), 167-181.
    • (1965) J. Math. Phys. , vol.6 , pp. 167-181
    • Montroll, E.W.1    Weiss, G.H.2
  • 50
    • 0002609677 scopus 로고
    • On an enriched collection of stochastic processes
    • E.W. Montroll and J. Leibowitz (Editors), North-Holland, Amsterdam, Studies in Statistical Mechanics
    • E.W. Montroll and D.J. West, On an enriched collection of stochastic processes, in E.W. Montroll and J. Leibowitz (Editors), Fluctuation Phenomena, North-Holland, Amsterdam, 1979, pp. 61-175. [Studies in Statistical Mechanics, Vol. VII]
    • (1979) Fluctuation Phenomena , vol.VII , pp. 61-175
    • Montroll, E.W.1    West, D.J.2
  • 51
    • 0000495328 scopus 로고
    • On Mittag-Leffler functions and related distributions
    • R.N. Pillai, On Mittag-Leffler functions and related distributions, Ann. Inst. Statist. Math. 42 (1990), 157-161.
    • (1990) Ann. Inst. Statist. Math. , vol.42 , pp. 157-161
    • Pillai, R.N.1
  • 52
    • 13444309297 scopus 로고    scopus 로고
    • Models of anomalous diffusion: the subdiffusive case
    • A. Piryatinska, A.I. Saichev, and W.A. Woyczynski, Models of anomalous diffusion: the subdiffusive case, Physica A 349 (2005), 375-420.
    • (2005) Physica A , vol.349 , pp. 375-420
    • Piryatinska, A.1    Saichev, A.I.2    Woyczynski, W.A.3
  • 53
    • 0003797958 scopus 로고    scopus 로고
    • Fractional Differential Equations
    • Academic Press, San Diego
    • I. Podlubny, Fractional Differential Equations. Academic Press, San Diego (1999).
    • (1999)
    • Podlubny, I.1
  • 54
    • 84932210307 scopus 로고
    • A characteristic of the Poisson stream
    • In Hungarian
    • A. Renyi, A characteristic of the Poisson stream, Proc. Math. Inst. Hungarica Acad. Sci. 1 (4) (1956), 563-570. [In Hungarian]
    • (1956) Proc. Math. Inst. Hungarica Acad. Sci. , vol.1 , Issue.4 , pp. 563-570
    • Renyi, A.1
  • 55
    • 0003897447 scopus 로고    scopus 로고
    • Introduction to Probability Models
    • 6th edn., Academic Press, New York
    • S.M. Ross, Introduction to Probability Models, 6th edn., Academic Press, New York, (1997).
    • (1997)
    • Ross, S.M.1
  • 56
    • 0000599032 scopus 로고    scopus 로고
    • Fractional kinetic equations: solutions and applications
    • A.I. Saichev and G.M. Zaslavsky, Fractional kinetic equations: solutions and applications, Chaos 7 (1997), 753-764.
    • (1997) Chaos , vol.7 , pp. 753-764
    • Saichev, A.I.1    Zaslavsky, G.M.2
  • 57
    • 0003598080 scopus 로고
    • Fractional Integrals and Derivatives: Theory and Applications
    • Gordon and Breach, New York, Translation from the Russian edition, Nauka i Tekhnika, Minsk (1987)
    • S.G. Samko, A.A. Kilbas, and O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, New York (1993) [Translation from the Russian edition, Nauka i Tekhnika, Minsk (1987)].
    • (1993)
    • Samko, S.G.1    Kilbas, A.A.2    Marichev, O.I.3
  • 58
    • 32544456755 scopus 로고    scopus 로고
    • The application of continuoustime random walks in finance and economics
    • E. Scalas, The application of continuoustime random walks in finance and economics, Physica A 362 (2006), 225-239.
    • (2006) Physica A , vol.362 , pp. 225-239
    • Scalas, E.1
  • 59
    • 0034275979 scopus 로고    scopus 로고
    • Fractional calculus and continuous-time finance
    • E. Scalas, R. Gorenflo, and F. Mainardi, Fractional calculus and continuous-time finance, Physica A 284 (2000), 376-384.
    • (2000) Physica A , vol.284 , pp. 376-384
    • Scalas, E.1    Gorenflo, R.2    Mainardi, F.3
  • 60
    • 1842531858 scopus 로고    scopus 로고
    • Uncoupled continuous-time random walks: Solution and limiting behavior of the master equation
    • E. Scalas, R. Gorenflo, and F. Mainardi, Uncoupled continuous-time random walks: Solution and limiting behavior of the master equation, Phys. Rev. E 69 (2004), 011107-1/8.
    • (2004) Phys. Rev. E , vol.69
    • Scalas, E.1    Gorenflo, R.2    Mainardi, F.3
  • 61
    • 0001553919 scopus 로고
    • Fractional diffusion and wave equations
    • W.R. Schneider and W. Wyss, Fractional diffusion and wave equations, J. Math. Phys. 30 (1989) 134-144.
    • (1989) J. Math. Phys. , vol.30 , pp. 134-144
    • Schneider, W.R.1    Wyss, W.2
  • 62
    • 0242556344 scopus 로고    scopus 로고
    • Do strange kinetics imply unusual thermodynamics?
    • I.M. Sokolov, J. Klafter, and A. Blumen, Do strange kinetics imply unusual thermodynamics? Phys. Rev. E. 64 (2001), 021107/1-4.
    • (2001) Phys. Rev. E. , vol.64
    • Sokolov, I.M.1    Klafter, J.2    Blumen, A.3
  • 63
    • 0042518233 scopus 로고
    • Limiting distribution for the sums of random number of random variables concerning the rarefaction of recurrent events
    • T. Szàntai, Limiting distribution for the sums of random number of random variables concerning the rarefaction of recurrent events, Studia Scientiarum Mathematicarum Hungarica 6 (1971), 443-452.
    • (1971) Studia Scientiarum Mathematicarum Hungarica , vol.6 , pp. 443-452
    • Szàntai, T.1
  • 64
    • 69949164932 scopus 로고
    • On an invariance problem related to different rarefactions of recurrent events
    • T. Szàntai, On an invariance problem related to different rarefactions of recurrent events. Studia Scientiarum Mathematicarum Hungarica 6 (1971), 453-456.
    • (1971) Studia Scientiarum Mathematicarum Hungarica , vol.6 , pp. 453-456
    • Szàntai, T.1
  • 65
    • 0142009320 scopus 로고    scopus 로고
    • Stochastic solution of partial differential equations of fractional orders
    • V.V. Uchaikin and V.V. Saenko, Stochastic solution of partial differential equations of fractional orders, Siberian J. Num. Math. 6 (2003), 197-203.
    • (2003) Siberian J. Num. Math. , vol.6 , pp. 197-203
    • Uchaikin, V.V.1    Saenko, V.V.2
  • 66
    • 0003572301 scopus 로고
    • Aspects and Applications of Random Walks
    • North-Holland, Amsterdam
    • G.H. Weiss, Aspects and Applications of Random Walks, North-Holland, Amsterdam (1994).
    • (1994)
    • Weiss, G.H.1
  • 67
    • 0030264775 scopus 로고
    • On the Cole-Cole relaxation function and related Mittag-Leffler distribution
    • K. Weron and M. Kotulski, On the Cole-Cole relaxation function and related Mittag-Leffler distribution, Physica A 232 (1995), 180-188.
    • (1995) Physica A , vol.232 , pp. 180-188
    • Weron, K.1    Kotulski, M.2
  • 68
    • 0004201980 scopus 로고
    • The Laplace Transform
    • Princeton University Press, Princeton
    • D.V. Widder, The Laplace Transform, Princeton University Press, Princeton (1946).
    • (1946)
    • Widder, D.V.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.