On Space-Time Fractional Heat Type Non-Homogeneous Time-Fractional Poisson Equation
Ejighikeme McSylvester Omaba *
Department of Mathematics, College of Science, University of Hafr Al Batin, P.O.Box 1803 Hafr Al Batin 31991, KSA.
*Author to whom correspondence should be addressed.
Abstract
Consider the following space-time fractional heat equation with Riemann-Liouville derivative of
non-homogeneous time-fractional Poisson process
where 
The operator 
with 
(t) the Riemann-Liouville non-homogeneous fractional integral process, 
is the Caputo fractional derivative, 
is the generator of an isotropic stable process, 
is the fractional integral operator, and σ : R → R is Lipschitz continuous. The above time fractional stochastic heat type equations may be used to model sequence of catastrophic events 
for some specific
rate functions were computed. Consequently, the growth moment bounds for the class of heat equation perturbed with the non-homogeneous fractional time Poisson process were given and we show that the solution grows exponentially for some small time interval 
and t0> 1; that is, the result establishes that the energy of the solution grows atleast as c4(t + t0) 
exp(c5t) and at most as c1t 
exp(c3t) for different conditions on the initialdata, where c1, c3, c4 and c5 are some positive constants depending on T. Existence and uniqueness result for the mild solution to the equation was given under linear growth condition on σ.
Keywords: Caputo derivative, energy moment bounds, fractional heat kernel, fractional Duhamel’s principle, riemann-Liouville derivative, riemann-Liouville integral process