Adomian Decomposition Method and Fractional Poisson Processes: A Survey

Abstract

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.