Moment Bound of Solution to a Class of Conformable Time-Fractional Stochastic Equation

We study a class of conformable time-fractional stochastic equation Ta α,tu(x, t) = σ(u(x, t))Ẇt, x ∈ R, t ∈ [a, T], T < ∞, 0 < α < 1. The initial condition u(x, 0) = u0(x), x ∈ R is a non-random function assumed to be non-negative and bounded, Ta α,t is a conformable time-fractional derivative, σ : R → R is Lipschitz continuous and Ẇt a generalized derivative of Wiener process. Some precise condition for the existence and uniqueness of a solution of the class of equation is given and we also give an upper bound estimate on the growth moment of the solution. Unlike the growth moment of stochastic fractional heat equation with Riemann–Liouville or Caputo–Dzhrbashyan fractional derivative which grows in time like tc1 exp(c2t), c1, c2 > 0; our result also shows that the energy of the solution (the second moment) grows exponentially in time for t ∈ [a, T], T < ∞ but with at most c1 exp(c2(t− a)2α−1) for some constants c1, and c2.


Introduction
The use of fractional derivative, which is a generalization of derivative to any arbitrary order, has received a tremendous attention due to its physical and modelling applications in Science, Engineering and Mathematics, see [1] and the references.There are various definitions and generalizations of the fractional derivative, not limited to the Riemann-Liouville fractional derivative and Caputo-Dzhrbashyan fractional derivative, with their respective limitations, see [2].One of the limitations of the above two fractional derivatives, is that they do not satisfy the classical chain rule; hence, the need for a better definition of fractional derivative.In [3], Khalil and his co-authors, introduced a better and a new well-behaved definition of a fractional derivative known as the conformable fractional derivative, satisfying the usual chain rule, the Rolle's and the mean value theorem, conformable integration by parts, fractional power series expansion and the conformal fractional derivative of the real function is zero, etc.; which has given a new research direction.Thus conformable fractional derivative is a natural extension of the classical derivative (since it can be expressed as a first derivative multiplied by a fractional factor or power) and has many advantages over other fractional derivatives as enumerated above.
Conformable fractional derivatives are applied in certain classes of conformable differentiable linear systems subject to impulsive effects and establish quantitative behaviour of the nontrivial solutions (stability, disconjugacy, etc.), see [4,5]; and used to develop the Swartzendruber model for description of non-Darcian flow in porous media [5][6][7].Conformable derivatives are also used to solve approximate long water wave equation with conformable fractional derivative and conformable fractional differential equations via radial basis functions collocation method [8,9].Though there has been a significant contribution in the study of class of stochastic heat equations with Riemann-Liouville (R-L) and Caputo-Dzhrbashyan (C-D) fractional derivatives [10][11][12][13][14][15][16][17], but not much has been done in the study of stochastic Cauchy equation with conformable fractional derivative, see [2], see also [18] for a recent study on conformable fractional differential equations.The challenge in studying stochastic Cauchy equation with conformable fractional derivative is that there is no singular kernel of the form (t − s) −α generated for 0 < α < 1 which reflect the nonlocality and the memory in the fractional operator as in the case of R-L and C-D fractional derivatives.Thus, despite the fact that conformable fractional derivative combines the best characteristics of known fractional derivatives, seems more appropriate to describe the behaviour of classical viscoelastic models under interval uncertainty, see [19] and gives models that agree and are consistent with experimental data, see [20], it does not possess or satisfy a semigroup property unlike the R-L and C-D fractional operators that have well-behaved semigroup properties.
We are motivated by the fact that the conformable fractional derivative can be used to solve fractional differential equation more easily, see [18,21], and therefore consider the following class of conformable fractional stochastic equation with an initial condition u(x, 0) = u 0 (x); where T a α,t is a conformable fractional derivative, σ : R → R is a Lipschitz continuous function and Ẇt is a generalized derivative of Wiener process (Gaussian white noise).The existence and uniqueness result is given and we also give the moment growth bound estimate on the solution of the above equation.Similar models have been considered for Caputo derivatives [22][23][24] where existence and uniqueness results were studied.To the best of our knowledge, we are the first to consider this model for the conformable fractional derivative.We prove the existence and uniqueness result and also give the moment growth bound estimate on the solution of the above equation.
The paper is outlined as follows.In Section 2, we give a brief overview of basic concepts used in this paper.The problem formulation, the main results, their proofs are given in Section 3, and Section 4 contains some theoretical examples to illustrate our result.We end with a short conclusion in Section 5.
and if u t (x, t) exists then T a α,t u(x, t) = (t − a) 1−α u t (x, t).
Definition 2. The fractional integral starting from a of a function u We first give a generalized derivative for a deterministic function w(t).
Definition 3. Given that g(t) is any smooth and compactly supported function, then we define the generalized derivative ẇ(t) of w(t) (not necessarily differentiable) as Similarly, the generalized derivative Ẇt of Wiener process with a smooth function g(t) as follows: We give the gamma function as follows where γ(z, x) is an incomplete gamma function given by γ(z, x) = x 0 e −t t z−1 dt, x > 0 and Γ(z, x) is the complement of the incomplete gamma function given by Next, we give some estimates (bounds) on the incomplete gamma function.

Main Results
Assume the following condition on σ; that is, σ is globally Lipschitz: Following similar idea in [18], we give the following results: Lemma 1.Given that Condition 2 holds, then a function u in L 2 (P) is a solution of Equation (1) if and only if it is a solution of the integral equation Thus, the solution to Equation ( 1) is given as follows We start by defining the operator and the fixed point of the operator gives the solution of Equation ( 1).
The proof of the theorem is based on the following lemmas: Lemma 2. Suppose u is a predictable random solution such that u 2,α,β < ∞ and Conditions 1 and 2 hold.Then there exists a positive constant C α,β,T such that Proof.By the assumption that u 0 is bounded, we obtain Multiply through by e − β α (t−a) α to obtain Thus taking sup over t ∈ [a, T] and x ∈ R and evaluating the integral we have By the estimate on the incomplete gamma function in Theorem 1, we obtain and therefore, since 0 < t − a < T − a, we have Lemma 3. Suppose u and v are predictable random solutions such that u 2,α,β + v 2,α,β < ∞ and Conditions 1 and 2 hold.Then there exists a positive constant C α,β,T such that Remark 1.By Fixed point theorem we have u(x, t) = Au(x, t) and The existence and uniqueness result follows by Banach's contraction principle.
Next, we give the growth moment bound (upper bound estimate) on the solution: Theorem 3. Given that Conditions 1 and 2 hold, then for t ∈ [a, T], 0 < T < ∞ we have for some positive constant c 1 and c 2 = Proof.Assume that the initial condition u 0 (x) is bounded, then by Itó isometry, we have and the result follows.

Some Examples
Here, we give some theoretical examples.
2. Consider also the conformable fractional equation

Condition 1 .Condition 2 . 2
There exist a finite positive constant, Lip σ such that for all x, y ∈ R, we have|σ(x) − σ(y)| ≤ Lip σ |x − y|, with σ(0) = 0 for convenience.Also, the assumption on u: The random solution u : D → R is L 2 -continuous (or continuous in probability).Define the following L 2 (P) norm u