### Abstract

Original language | English |
---|---|

Pages (from-to) | 1527–1553 |

Number of pages | 27 |

Journal | Fractional Calculus and Applied Analysis |

Volume | 19 |

Issue number | 6 |

DOIs | |

Publication status | E-pub ahead of print - 16 Dec 2016 |

### Fingerprint

### Keywords

- non-linear noise excitamath-phtion
- space-time fractional stochastic equations
- fractional Duhamel’s principle
- noise excitability
- Caputo derivatives

### Cite this

*Fractional Calculus and Applied Analysis*,

*19*(6), 1527–1553. https://doi.org/10.1515/fca-2016-0079

}

*Fractional Calculus and Applied Analysis*, vol. 19, no. 6, pp. 1527–1553. https://doi.org/10.1515/fca-2016-0079

**Non-linear noise excitation for some space-time fractional stochastic equations in bounded domains.** / Foondun, Mohammud; Mijena, Jebessa B.; Nane, Erkan.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Non-linear noise excitation for some space-time fractional stochastic equations in bounded domains

AU - Foondun, Mohammud

AU - Mijena, Jebessa B.

AU - Nane, Erkan

N1 - 23 pages. arXiv admin note: substantial text overlap with arXiv:1505.04615

PY - 2016/12/16

Y1 - 2016/12/16

N2 - In this paper we study non-linear noise excitation for the following class of space-time fractional stochastic equations in bounded domains: ∂βtut(x)=−ν(−Δ)α/2ut(x)+I1−βt[λσ(u)F⋅(t,x)] in (d+1) dimensions, where ν > 0, β ∈ (0, 1), α ∈ (0, 2]. The operator ∂βt is the Caputo fractional derivative, −(−Δ)α/2 is the generator of an isotropic stable process and I1−βt is the fractional integral operator. The forcing noise denoted by F⋅(t,x) is a Gaussian noise. The multiplicative non-linearity σ : ℝ → ℝ is assumed to be globally Lipschitz continuous. These equations were recently introduced by Mijena and Nane [32]. We first study the existence and uniqueness of the solution of these equations and under suitable conditions on the initial function, we also study the asymptotic behavior of the solution with respect to the parameter λ. In particular, our results are significant extensions of those in [14], [16], [32], and [33].

AB - In this paper we study non-linear noise excitation for the following class of space-time fractional stochastic equations in bounded domains: ∂βtut(x)=−ν(−Δ)α/2ut(x)+I1−βt[λσ(u)F⋅(t,x)] in (d+1) dimensions, where ν > 0, β ∈ (0, 1), α ∈ (0, 2]. The operator ∂βt is the Caputo fractional derivative, −(−Δ)α/2 is the generator of an isotropic stable process and I1−βt is the fractional integral operator. The forcing noise denoted by F⋅(t,x) is a Gaussian noise. The multiplicative non-linearity σ : ℝ → ℝ is assumed to be globally Lipschitz continuous. These equations were recently introduced by Mijena and Nane [32]. We first study the existence and uniqueness of the solution of these equations and under suitable conditions on the initial function, we also study the asymptotic behavior of the solution with respect to the parameter λ. In particular, our results are significant extensions of those in [14], [16], [32], and [33].

KW - non-linear noise excitamath-phtion

KW - space-time fractional stochastic equations

KW - fractional Duhamel’s principle

KW - noise excitability

KW - Caputo derivatives

UR - https://www.degruyter.com/view/j/fca.2016.19.issue-6/issue-files/fca.2016.19.issue-6.xml

U2 - 10.1515/fca-2016-0079

DO - 10.1515/fca-2016-0079

M3 - Article

VL - 19

SP - 1527

EP - 1553

JO - Fractional Calculus and Applied Analysis

JF - Fractional Calculus and Applied Analysis

SN - 1311-0454

IS - 6

ER -