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

Mohammud Foondun, Jebessa B. Mijena, Erkan Nane

Research output: Contribution to journalArticle

8 Citations (Scopus)

Abstract

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].
Original languageEnglish
Pages (from-to)1527–1553
Number of pages27
JournalFractional Calculus and Applied Analysis
Volume19
Issue number6
DOIs
Publication statusE-pub ahead of print - 16 Dec 2016

Fingerprint

Stochastic Equations
Bounded Domain
Fractional
Excitation
Space-time
Derivatives
Fractional Integral Operator
Caputo Fractional Derivative
Stable Process
Gaussian Noise
Forcing
Lipschitz
Multiplicative
Existence and Uniqueness
Asymptotic Behavior
Nonlinearity
Generator
Operator
Class

Keywords

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

Cite this

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

In: Fractional Calculus and Applied Analysis, Vol. 19, No. 6, 16.12.2016, p. 1527–1553.

Research output: Contribution to journalArticle

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