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Progress in Fractional Differentiation & Applications

Author Country (or Countries)

Saudi Arabia

Abstract

The present paper addresses the following stochastic heat fractional integral equation (SHFIE): ∂ u(x,t)=−(−∆)α/2u(x,t)+σ(u(x,t))N β,ν(t), x∈Rd, t ≥0, ∂t λ with β > 0, ν ∈ (0,1], α ∈ (0,2]. The operator −(−∆)α/2 is the generator of an isotropic stable process and N β,ν(t) is the λ Riemann–Liouville non-homogeneous fractional integral process. The mean and variance for the process N β,ν(t) for some specific λ rate functions were computed. Also, the growth moment bounds for the class of heat equation perturbed with the non-homogeneous fractional time Poisson process were given. In addition, the paper shows that the solution grows exponentially for some small time interval t ∈ [t0,T], T < ∞ and t0 > 1. To explain, the result establishes that the energy of the solution grows at least as c3(t +t0)−(β+aν) exp(c4t) and at most as c1t−(β+aν) exp(c2t) for different conditions on the initial data, where c1, c2, c3 and c4 are some positive constants depending on T .

Digital Object Identifier (DOI)

http://dx.doi.org/10.18576/pfda/060107

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