Saigo space–time fractional Poisson process via Adomian decomposition method

Kataria, K. K.; Vellaisamy, P.

doi:10.1016/j.spl.2017.05.007

Cited by 18 publications

(9 citation statements)

References 9 publications

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“…In the absence of the nonlinear term H(u), the ADM can be used effectively as the recursive relationship (2.4) then simply reduces to u n = L(u n−1 ) with u 0 = f . Kataria and Vellaisamy (2017) obtained the distribution of Saigo space time fractional Poisson process (SSTFPP) via ADM, which was otherwise difficult to obtain using the prevalent method of inverting Laplace transform. As special case, the state probabilities of TFPP, SFPP and STFPP follow easily.…”

Section: Adomian Decomposition Methodsmentioning

confidence: 99%

On Distributions of Certain State-Dependent Fractional Point Processes

Kataria

Vellaisamy

2018

J Theor Probab

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We obtain the explicit expressions for the state probabilities of various state dependent fractional point processes recently introduced and studied by Garra et al. (2015). The inversion of the Laplace transforms of the state probabilities of such processes is rather cumbersome and involved. We employ the Adomian decomposition method to solve the difference differential equations governing the state probabilities of these state dependent processes. The distributions of some convolutions of the Mittag-Leffler random variables are derived as special cases of the obtained results.The first version of the state dependent time fractional Poisson process (SDTFPP-I) {N 1 (t, λ)} t≥0 , λ > 0, is defined as the stochastic process whose probability mass function (pmf) p αn (n, t) = Pr{N 1 (t, λ) = n}, satisfies (see Eq. (1.1), Garra et al. (2015)) ∂ αn t p αn (n, t) = −λ(p αn (n, t) − p α n−1 (n − 1, t)), 0 < α n ≤ 1, n ≥ 0, (1.2) with p α −1 (−1, t) = 0, t ≥ 0, and the initial conditions p αn (0, 0) = 1 and p αn (n, 0) = 0, n ≥ 1. For each n ≥ 0, ∂ αn t denotes the fractional derivative in Caputo sense which is defined as ∂ αn t f (t) :=    1 Γ(1−αn) t 0 (t − s) −αn f ′ (s) ds, 0 < α n < 1,

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Section: Adomian Decomposition Methodsmentioning

confidence: 99%

On Distributions of Certain State-Dependent Fractional Point Processes

Kataria

Vellaisamy

2018

J Theor Probab

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“…There are many effective methods to solve this problem, like Adomian decomposition method [8][9][10], variation iteration method [11], differential transform method [12], residual power series method [13,14], iteration method [15], homotopy perturbation method [16], homotopy analysis method [17], and so on. Furthermore, for the nonlinear problem, the multiple exp-function method [18,19], the transformed rational function method [20][21][22], and invariant subspace method [23,24] are three systematical approaches to handle the nonlinear terms.…”

Section: Discrete Dynamics In Nature and Societymentioning

confidence: 99%

Homotopy Series Solutions to Time-Space Fractional Coupled Systems

Zhang

Cai

Chen

et al. 2017

Discrete Dynamics in Nature and Society

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We apply the homotopy perturbation Sumudu transform method (HPSTM) to the time-space fractional coupled systems in the sense of Riemann-Liouville fractional integral and Caputo derivative. The HPSTM is a combination of Sumudu transform and homotopy perturbation method, which can be easily handled with nonlinear coupled system. We apply the method to the coupled Burgers system, the coupled KdV system, the generalized Hirota-Satsuma coupled KdV system, the coupled WBK system, and the coupled shallow water system. The simplicity and validity of the method can be shown by the applications and the numerical results.

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“…The time fractional versions of the Poisson process are obtained by replacing the time derivative in the governing difference-differential equations of the state probabilities of Poisson process by certain fractional derivatives. These include Riemann-Liouville fractional derivative (see Laskin, 2003), Caputo fractional derivative (see Beghin and Orsingher, 2009), Prabhakar derivative (see Polito and Scalas, 2016), Saigo fractional derivative (see Kataria and Vellaisamy, 2017b) etc. These time fractional models is further generalized to state-dependent fractional Poisson processes (see Garra et al, 2015) and the mixed fractional Poisson process (see Beghin, 2012 andAletti et al, 2018).…”

Section: Introductionmentioning

confidence: 99%

Convoluted Fractional Poisson Process

Kataria¹,

Khandakar²

2021

ALEA

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In this paper, we introduce and study a convoluted version of the time fractional Poisson process by taking the discrete convolution with respect to space variable in the system of fractional differential equations that governs its state probabilities. We call the introduced process as the convoluted fractional Poisson process (CFPP). It has integer valued jumps of size j ≥ 1. The explicit expression for the Laplace transform of its state probabilities are obtained whose inversion yields its one-dimensional distribution. A special case of the CFPP, namely, the convoluted Poisson process (CPP) is studied and its time-changed subordination relationships with CFPP are discussed. It is shown that the CPP is a Lévy process using which the long-range dependence property of CFPP is established. Moreover, we show that the increments of CFPP exhibits short-range dependence property.

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Saigo space–time fractional Poisson process via Adomian decomposition method

Cited by 18 publications

References 9 publications

On Distributions of Certain State-Dependent Fractional Point Processes

On Distributions of Certain State-Dependent Fractional Point Processes

Homotopy Series Solutions to Time-Space Fractional Coupled Systems

Convoluted Fractional Poisson Process

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