Adomian Decomposition Method and Fractional Poisson Processes: A Survey

  • K. K. Kataria
  • P. VellaisamyEmail author
Conference paper
Part of the Trends in Mathematics book series (TM)


This paper gives a survey of recent results related to the applications of the Adomian decomposition method (ADM) to certain fractional generalizations of the homogeneous Poisson process. First, we briefly discuss the ADM and its advantages over existing methods. As applications, this method is employed to obtain the state probabilities of the time fractional Poisson process (TFPP), space fractional Poisson process (SFPP) and Saigo space–time fractional Poisson process (SSTFPP). Usually, the Laplace transform technique is used to obtain the state probabilities of fractional processes. However, for certain state-dependent fractional Poisson processes, the Laplace transform method is difficult to apply, but the ADM method could be effectively used to obtain the state probabilities of such processes.


Adomian decomposition method Fractional derivatives Fractional point processes 


Primary: 60G22 Secondary: 60G55 


  1. 1.
    Adomian, G.: Nonlinear Stochastic Operator Equations. Academic Press, Orlando (1986)zbMATHGoogle Scholar
  2. 2.
    Adomian, G.: Solving Frontier Problems of Physics: The Decomposition Method. Kluwer Academic, Dordrecht (1994)CrossRefGoogle Scholar
  3. 3.
    Beghin, L., Orsingher, E.: Fractional Poisson processes and related planar random motions. Electron. J. Probab. 14, 1790–1826 (2009)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Biard, R., Saussereau, B.: Fractional Poisson process: long-range dependence and applications in ruin theory. J. Appl. Probab. 51, 727–740 (2014)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cahoy, D.O., Uchaikin, V.V., Woyczynski, W.A.: Parameter estimation for fractional Poisson processes. J. Stat. Plann. Inference 140(11), 3106–3120 (2010)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Garra, R., Orsingher, E., Polito, F.: State-dependent fractional point processes. J. Appl. Probab. 52, 18–36 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Kataria, K.K., Vellaisamy, P.: Simple parametrization methods for generating Adomian polynomials. Appl. Anal. Discret. Math. 10, 168–185 (2016)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Kataria, K.K., Vellaisamy, P.: Saigo space-time fractional Poisson process via Adomian decomposition method. Stat. Probab. Lett. 129, 69–80 (2017)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Kataria, K.K., Vellaisamy, P.: Correlation between Adomian and partial exponential Bell polynomials. C. R. Math. Acad. Sci. Paris 355(9), 929–936 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kataria, K.K., Vellaisamy, P.: On distributions of certain state dependent fractional point processes. J. Theor. Probab. (2019). Scholar
  11. 11.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theroy and Applications fo Fractional Differential Equations. Elsevier, Amsterdam (2006)Google Scholar
  12. 12.
    Laskin, N.: Fractional Poisson Process. Commun. Nonlinear Sci. Numer. Simul. 8, 201–213 (2003)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Laskin, N.: Some applications of the fractional Poisson probability distribution. J. Math. Phys. 50(11), 113513 (2009)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Maheshwari, A., Vellaisamy, P.: On the long-range dependence of fractional Poisson and negative binomial processes. J. Appl. Probab. 53(4), 989–1000 (2016)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Meerschaert, M.M., Nane, E., Vellaisamy, P.: The fractional Poisson process and the inverse stable subordinator. Electron. J. Probab. 16, 1600–1620 (2011)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Meerschaert, M.M., Scheffler, H.-P.: Limit theorems for continuous-time random walks with infinite mean waiting times. J. Appl. Probab. 41, 623–638 (2004)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Meerschaert, M.M., Straka, P.: Inverse stable subordinators. Math. Model. Nat. Phenom. 8, 1–16 (2013)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Orsingher, E., Polito, F.: Fractional pure birth processes. Bernoulli 16, 858–881 (2010)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Orsingher, E., Polito, F.: The space-fractional Poisson process. Stat. Probab. Lett. 82, 852–858 (2012)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Orsingher, E., Ricciuti, C., Toaldo, B.: On semi-Markov processes and their Kolmogorov’s integro-differential equations. J. Funct. Anal. 275(4), 830–868 (2018)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Pillai, R.N.: On Mittag-Leffler functions and related distributions. Ann. Inst. Stat. Math. 42(1), 157–161 (1990)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Polito, F., Scalas, E.: A generalization of the space-fractional Poisson process and its connection to some Lévy processes. Electron. Commun. Probab. 21, 1–14 (2016)CrossRefGoogle Scholar
  23. 23.
    Rach, R.: A convenient computational form for the Adomian polynomials. J. Math. Anal. Appl. 102, 415–419 (1984)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Rao, A., Garg, M., Kalla, S.L.: Caputo-type fractional derivative of a hypergeometric integral operator. Kuwait J. Sci. Eng. 37, 15–29 (2010)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Ricciuti, C., Toaldo, B.: Semi-Markov models and motion in heterogeneous media. J. Stat. Phys. 169, 340–361 (2017)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Saigo, M.: A remark on integral operators involving the Gauss hypergeometric functions. Math. Rep. Kyushu Univ. 11, 135–143 (1978)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Watanabe, S.: On discontinuous additive functionals and Lévy measures of a Markov process. Japan. J. Math. 34, 53–70 (1964)CrossRefGoogle Scholar
  28. 28.
    Zhu, Y., Chang, Q., Wu, S.: A new algorithm for calculating Adomian polynomials. Appl. Math. Comput. 169, 402–416 (2005)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology BhilaiRaipurIndia
  2. 2.Department of MathematicsIndian Institute of Technology BombayMumbaiIndia

Personalised recommendations