First passage time distribution of a modified fractional diffusion equation in the semi-infinite interval

Guo, Gang; Chen, Bin; Zhao, Xinquan; Zhao, Fang; Wang, Quanmin

doi:10.1016/j.physa.2015.04.005

Cited by 11 publications

(20 citation statements)

References 42 publications

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“…3) and > * . This result can not hold for the corresponding first-order model of (16). Thus, the fractional order can expand the region of the stability.…”

Section: The Characteristic Equation Ofmentioning

confidence: 87%

“…In this region, the characteristic roots of (18) have positive real parts. If the system (16) is first-order one, then E 2 is unstable for arbitrary and r. However, by virtue of Lemma 2.6, we can find a parameter region such that E 2 is also stable for the system (16). To this end, let the characteristic roots of (18) = P + Qi, P, Q ∈ R. Hence,…”

Section: The Characteristic Equation Ofmentioning

confidence: 99%

“…From the above discussion, we can conclude the following result. (16) and independent of the parameters and r. For this case, the result of the stability is the same as the corresponding first-order system of (16). When ∈ (0, 1∕ √ 3), E 2 is unstable for the corresponding first-order system of (16) and arbitrary parameters and r. However, when the fractional order ∈ (0, 1), E 2 for (16) is stable in the case of ∈ (0, 1∕ √…”

Section: The Characteristic Equation Ofmentioning

confidence: 99%

“…Fractional‐order derivatives allow us to deal comfortably with memory effects in dynamical system, and thus, it can be successfully applied in some field, such as physics, control engineering, biochemical reaction, and signal processing. At present, a large numbers of monographs and papers are devoted to fractional dynamical systems. Based on this consideration, introducing time fractional‐order derivatives into yields

({, \begin{matrix} right \frac{\partial^{α} u}{\partial t^{α}} & left = d_{1} Δ u + r u (1 - u) - \sqrt{u} v, & right \\ right \frac{\partial^{α} v}{\partial t^{α}} & left = d_{2} Δ v + β v (\sqrt{u} - δ), \end{matrix})

where

\frac{\partial^{α}}{\partial t^{α}}

is a standard Caputo's derivative, α ∈(0,1).…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Hopf bifurcation of a diffusive Gause‐type predator‐prey model induced by time fractional‐order derivatives

Yin

Wen

2018

Math Methods in App Sciences

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Since population behaviors possess the characteristic of history memory, we, in this paper, introduce time fractional‐order derivatives into a diffusive Gause‐type predator‐prey model, which is time fractional‐order reaction‐diffusion equations and a generalized form of its corresponding first‐derivative model. For this kind of model, we prove the existence and uniqueness of a global positive solution by using the theory of evolution equations and the comparison principle of time fractional‐order partial differential equations. Besides, we obtain the stability and Hopf bifurcation of the Gause‐type predator‐prey model in the forms of the time fractional‐order ordinary equations and of the time fractional‐order reaction‐diffusion equations, respectively. Our results show that the stable region of the parameters in these 2 models can be enlarged by the time fractional‐order derivatives. Some numerical simulations are made to verify our results.

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“…3) and > * . This result can not hold for the corresponding first-order model of (16). Thus, the fractional order can expand the region of the stability.…”

Section: The Characteristic Equation Ofmentioning

confidence: 87%

Section: The Characteristic Equation Ofmentioning

confidence: 99%

Section: The Characteristic Equation Ofmentioning

confidence: 99%

({, \begin{matrix} right \frac{\partial^{α} u}{\partial t^{α}} & left = d_{1} Δ u + r u (1 - u) - \sqrt{u} v, & right \\ right \frac{\partial^{α} v}{\partial t^{α}} & left = d_{2} Δ v + β v (\sqrt{u} - δ), \end{matrix})

where

\frac{\partial^{α}}{\partial t^{α}}

is a standard Caputo's derivative, α ∈(0,1).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Hopf bifurcation of a diffusive Gause‐type predator‐prey model induced by time fractional‐order derivatives

Yin

Wen

2018

Math Methods in App Sciences

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“…Employing a Hankel transform and the properties of the Fox-H functions [15,21,22], Equation (36) can be written as…”

Section: Analytical Solution With a Frictional Memory Kernel Of Powermentioning

confidence: 99%

The Analytical Solution of Parabolic Volterra Integro-Differential Equations in the Infinite Domain

Zhao

2016

Entropy

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This article focuses on obtaining analytical solutions for d-dimensional, parabolic Volterra integro-differential equations with different types of frictional memory kernel. Based on Laplace transform and Fourier transform theories, the properties of the Fox-H function and convolution theorem, analytical solutions for the equations in the infinite domain are derived under three frictional memory kernel functions. The analytical solutions are expressed by infinite series, the generalized multi-parameter Mittag-Leffler function, the Fox-H function and the convolution form of the Fourier transform. In addition, graphical representations of the analytical solution under different parameters are given for one-dimensional parabolic Volterra integro-differential equations with a power-law memory kernel. It can be seen that the solution curves are subject to Gaussian decay at any given moment.

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Extinction Time of Non-Markovian Self-Similar Processes, Persistence, Annihilation of Jumps and the Fréchet Distribution

2019

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We start by providing an explicit characterization and analytical properties, including the persistence phenomena, of the distribution of the extinction time T of a class of non-Markovian self-similar stochastic processes with two-sided jumps that we introduce as a stochastic time-change of Markovian self-similar processes. For a suitably chosen timechanged, we observe, for classes with two-sided jumps, the following surprising facts. On the one hand, all the T's within a class have the same law which we identify in a simple form for all classes and reduces, in the spectrally positive case, to the Fréchet distribution. On the other hand, each of its distribution corresponds to the law of an extinction time of a single Markov process without positive jumps, leaving the interpretation that the time-changed has annihilated the effect of positive jumps. The example of the non-Markovian processes associated to Lévy stable processes is detailed.

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First passage time distribution of a modified fractional diffusion equation in the semi-infinite interval

Cited by 11 publications

References 42 publications

Hopf bifurcation of a diffusive Gause‐type predator‐prey model induced by time fractional‐order derivatives

Hopf bifurcation of a diffusive Gause‐type predator‐prey model induced by time fractional‐order derivatives

The Analytical Solution of Parabolic Volterra Integro-Differential Equations in the Infinite Domain

Extinction Time of Non-Markovian Self-Similar Processes, Persistence, Annihilation of Jumps and the Fréchet Distribution

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