Progress in Fractional Differentiation & Applications

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Saudi Arabia


Consider a Ψ –Caputo fractional stochastic differential equation of order 0 < ν < 1 given by CDν,Ψ φ(x,t) = γ 􏰀 θ(φ(y,t))w ̇(y,t)dy, t > 0. 0 B(0,tν) Assume a non-negative and bounded function φ (x, 0) = φ0 (x), x ∈ B(0, t ν ) ⊂ R2 , C D ν ,Ψ is a generalized Ψ –Caputo fractional ν0 derivative operator, θ : B(0,t ) → R is Lipschitz continuous, w ̇(y,t) a space-time white noise and γ > 0 the noise level. Under some http://dx.doi.org/10.18576/pfda/080304 precise conditions, we present the existence and uniqueness of solution to the class of equation and give upper moment growth bound and the long-term behaviour of the mild solution for the parameter ν such that 1 < ν < 1. The result shows that the second moment of 2 the solution to the Ψ–Caputo-type fractional stochastic differential equation exhibits an exponential growth in time at most c exp􏰃c γ 2 Ψ(t)􏰄, ∀t > 0; and at a rate of 2 as the noise level grows large. 5 6 2ν−1 2ν−1

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