Continuity with respect to fractional order of the time fractional diffusion-wave equation

This paper studies a time-fractional diffusion-wave equation with a linear source function. First, some stability results on parameters of the Mittag-Leffler functions are established. Then, we focus on studying the continuity of the solution of both the initial problem and the inverse initial value problems corresponding to the fractional-order in our main results. One of the difficulties encounteblack comes from estimating all constants independently of the fractional orders. Finally, we present some numerical results to confirm the effectiveness of our methods.


1.
Introduction. Fractional calculus has many applications in mechanic, physics and engineering science. For example, a fractional diffusion equation is a generalization of a classical diffusion equation which models anomalous diffusive phenomena. Many ideas and methods have been developed to deal with fractional partial differential equations. We refer the reader to [9,1,8,36,3,4,37,38,12,26,13,6,14,15,16,29] and the references therein.
Let Ω be an open and bounded domain in R N with the boundary ∂Ω. Given a function G, we seek u such that where T is a positive number. Here, the operator A is a symmetric and uniformly elliptic operator on Ω defined by where a kl ∈ C 1 Ω , a kl = a lk , 1 ≤ k, l ≤ N , and a 0 ∈ C Ω; [0, +∞) . Suppose that there exists a > 0 such that, for x ∈ Ω, z = (z 1 , z 2 , ..., z N ) ∈ R N , 1≤k,l≤N z k a kl (x)z l ≥ a|z| 2 , see, e.g., [30]. The fractional derivative in time ∂ α t for 1 < α < 2 is understood as the left-sided Caputo fractional derivative of order α with respect to t and defined by s v(s)(t − s) 1−α ds, where Γ is the Gamma function. For α = 2, we consider the usual time derivative ∂ 2 t . The first equation of (1) is called a fractional wave equation and can be used to describe evolution processes between diffusion and wave propagation [23,24,25]. In [23] the author shows that the fractional wave equation governs the propagation of mechanical diffusion-waves in viscoelastic media. The physical background for a time-space fractional diffusion-wave equation can be found in [5].
We are interested in the two following problems for time fractional wave equations (1). The first one is to identify u satisfying equations (1) and the initial condition u(x, 0) = h(x). ( It is called an initial value problem (IVP). The second one is to determine u satisfying equations (1) and the final condition which is called an inverse initial value problem (inverse IVP) or a final value problem (FVP). The initial value problem (1)- (2) has been extensively consideblack in the literature see, for example, [11,17,22,27,30,21,2,10]. The inverse initial value problem (1),(3) and its applications have been studied in [32]. To the best of our knowledge, there are only a few papers on the inverse initial value problem for the time fractional diffusion-wave equation; see [34,35,32,39,18,33]. In practice, many problems on time-space fractional equations depend on fractional parameters, i.e., fractional orders. However, these fractional parameters are not known a priori in the modeling process. Hence the continuity of solutions on these parameters is very important for modeling purposes. Furthermore, numerical computations are not allowed if this continuity does not hold. Motivated from [7,31], this paper studies the continuity of the solution of both the initial problem (1)-(2) and the inverse initial value problems (1),(3) with respect to the fractional order α. Namely, this work focuses on the question Does u αn → u α in an appropriate sense as n → ∞?
To answer this question, one of the difficulties encounteblack is estimating all constants independently on the fractional orders.
This paper is organized as follows. Section 2 provides some basic definitions and preliminaries. In Section 3, we present some stability results on parameters of the Mittag-Leffler functions which help to establish our main results in the next sections. The continuity of the solution of the initial problem (1)-(2) and the inverse initial value problems (1),(3) with respect to the fractional-order α will be shown in Sections 4, 5 respectively. Finally, we also present some numerical results to confirm the effectiveness of our method. Lemma 3.3. Let 1 < ν 0 < α < α < η 0 < 2 and > 0. Then there exists a positive constant D 2 (ν 0 , η 0 , , β, T ) such that for any 0 ≤ β ≤ 1 and 0 < t ≤ T .
Proof. First, it is easy to see that Estimating I 1 : From Part a of Lemma 2.3 in [7], there exists a positive constant C(ν 0 , η 0 ) depending only on ν 0 , η 0 such that for all 1 < ζ < 2 and t > 0. The latter estimates together with the Fundamental Theorem of Calculus gives Estimating I 2 : Lemma 2.2 implies that ∂ t E α,1 (−λ j t α ) = −λ j t α−1 E α,α (−λ j t α ), which combined with the Fundamental Theorem of Calculus yields Using Lemma 3.1, we obtain where we have used that (λj r α ) 2−β 1+(λj r α ) 2 ≤ 1. Hence, we deduce that Next, we will find a bound for J 1 . If t ≥ 1, then applying Lemma 3.2 yields If 0 < t ≤ 1, then by the same techniques as in Lemma 3.2 one has The above observations yield where From the above steps, we deduce that This completes the proof.
Lemma 3.4. Let α, α be as defined in Lemma 3.3. For any 0 ≤ β ≤ 1 and > 0, there exists a positive constant D 6 (ν 0 , η 0 , , β, T ) such that Moreover, the following inequality holds: By taking (12) and (30) together, we obtain where Estimating J 3 : Using the same argument as in (5) we deduce that where Estimating J 4 : Using Lemma 2.2 and applying the product rule of differentiation we see that Therefore, by Part (a) of Lemma 2.3 in [7], we can find a constant C(ν 0 , η 0 ) > 0 such that .

Hence, the Fundamental Theorem of Calculus gives
where we have used the estimates (7), (8) for J 1 . Here the latter constant is given by By combining (11), (14), (15), and (16), we deduce that where This completes the proof.

4.
Continuity with respect to fractional order of the IVP. In this section, we present the continuous dependence of the solution of Problem (1)-(2) on the input data (the fractional order α, and the initial condition h).
Let u α and u α be the solutions of Problem (1)-(2) with respect to the fractional orders α and α . If the numbers β, satisfy where Proof. The proof will be based on the results given in Section 3 which helps to estimate some differences of Mittag-Leffler functions. From [30], the solution of Problem (1)-(2) is given by Therefore, we have where By applying Lemma 3.3, the term F (1) α,α can be bounded as follows: In order to bound the term F α,α (., t) H ν (Ω) , we will use Lemma 3.4. Indeed, by applying this lemma and using some direct computations, one can see that Assumption 0 < < ν 0 β yields that the number ν 0 β − 1 − is strictly greater than −1. Hence, we now observe from assumption G ∈ L ∞ (0, T ; H ν (Ω)) that This estimate implies that where we denote by Combining the equation (19) and the estimates (21) and (22), there exists a positive constant C 1 (ν 0 , η 0 , , β, T ) such that Therefore, we can find a positive constant C 1 (ν 0 , η 0 , , β, T ) satisfying which directly implies inequality (18).
Remark 2. We now present an example to simulate the theory. Let us consider the IVP of finding u = u(x, t), (x, t) ∈ (0, π) × (0, 1), such that We consider the negative Laplace operator −∆ associated with the Dirichlet boundary condition on H 1 0 (0, π)∩H 2 (0, π). Then, it has the eigenvalues b j = j 2 , j ≥ 1, and corresponding eigenfunctions ϕ j (x) = 2/π sin(jx), j ≥ 1. The analytic solution of this problem is given by Let us recall the composite Simpson's rule. Suppose that the interval [a, b] is split up into n sub-intervals, with n being an even number. Then, the composite Simpson's rule is given by where z j = a + jh for j = 0, 1, ..., n with h = b − a n , and in particular, z 0 = a and z n = b. In the following simulation results, we will use the finite difference method to discretize the time and spatial variables as follows where N x ≥ 1, N t ≥ 1 are two given integer numbers.
Proof. In order to prove the desiblack result, we will apply the results given in Section 3, and then we make some suitable choices of solution spaces. We first refer the reader to the formula (20), page 5, in [32] to see the precise formulations of u α and u α which gives where we let Here N x = N t = 40. and Note that notation H 1 was given by (20).