Continuity Result on the Order of a Nonlinear Fractional Pseudo-Parabolic Equation with Caputo Derivative

In this paper, we consider a problem of continuity fractional-order for pseudo-parabolic equations with the fractional derivative of Caputo. Here, we investigate the stability of the problem with respect to derivative parameters and initial data. We also show that uω′ → uω in an appropriate sense as ω′ → ω, where ω is the fractional order. Moreover, to test the continuity fractional-order, we present several numerical examples to illustrate this property.

In this work, we focus on the time-fractional pseudo-parabolic equation as follows ∂ ω t u(x, t) + κRu(x, t) + Ru(x, t) = G((x, t, u(x, t)), (x, t) ∈ Ω × (0, T], u(x, t) = 0, (x, t) ∈ ∂Ω × (0, T], u(x, 0) = g 0 (x), x ∈ Ω, u t (x, 0) = 0, where Ω is a bounded domain in R n with sufficiently smooth boundary ∂Ω, 1 < ω < 2 and the Caputo fractional derivative operator of order ω is given by the notation as ∂ ω t . Assume that the function u is definitely continuous in time, then the definition in [19] make less to the traditional form where ∂ 2 u ds is the second order integer derivative of function u(s) with respect to the independent variable s and Γ is the Gamma function. The operator R is defined in Section 2.2. Some functions G, g 0 (x) are defined later. The Equation (1) defines the mechanism of mass transfer in fractal arrangement structures.
In case ω = 1 with the derivative of integer order, the problem (1) becomes the well known pseudo-parabolic as follows u t + κRu t + Ru = G(u).
The Equation (3) is called the pseudo-parabolic, which has many real-world applications, a case in this point is that the seepage of homogeneous fluids from a broken rock, unidirectional distribution of long waves of nonlinear dispersion [20,21] and Population aggregation [22] (where u is the population density). Moreover, there are a lot of works on well-posedness of the pseudo-parabolic equation with classical derivative, for instance, we can see in [23][24][25][26][27][28][29][30][31][32][33][34][35] and the references therein. In particular, in fractional calculus, investigating the existence, uniqueness, stability of fractional differential equations, has been the important goal in the scientific community. To the best of our knowledge, there are a few papers which consider the fractional-order of the pseudo-parabolic partial differential equation. Recently, the authors in [36] generalized Ulam-Hyers-Rassias's stability results for FPPDE solution. M. Beshtokov [37][38][39] considered a boundary value problem for FPPDE. However, the regularity of mild solutions for FPPDE has not been investigated.
It follows that continuity of the solutions with respect to these parameters is important for modeling purposes. This paper comes from the motivation of [40] for considering the continuity of the solutions on fractional order. In practical problem, the parameter is defined or computed by experiments. Therefore, we only know its value incorrectly. Even if the parameters are known exactly but are irrational, then we also get only its approximate value. Assume that ω → ω, a natural question is as follows Does u ω → u ω in an appropriate sense as ω → ω ?
The main purpose of our paper is answer above question. The difficulty in the problem occurs when we have to evaluate the upper and lower quantities by the terms independent of fractional order α. The question (4) for linear fractional wave equation has been recent studied in [41]. Due to the nonlocal and nonlinearity of our problem, we have to choose an effective method to give suitable estimation. This paper is organized as follows. Section 2 shows the premilinaries of the Mittag-Leffler function and mild solution. The key findings are discussed in Section 3 which demonstrates the continuous dependency of the problem solution (1) on input data and the fractional parameter. In the last section, we show some numerical examples to illustrate the property which is called that the continuity fractional-order.

The Mittag-Leffler Function
The Mittag-Leffler function is denoted and defined as following for ω > 0 and φ ∈ R. We recall the following lemmas (we can see in [7,42,43]), that would be helpful for the primary analysis of Sections 3 and 4.
with for all ξ > 0 and C is a positive number which only depends on ω, such that For λ > 0 the following identities holds true where ξ is a positive real number.
Proof. We can use Lemma 2.2 in [41] to prove the above Lemma.
Lemma 3. Assume 1 < ω < 2 and if T is a number that large enough then we get For all j is a natural number, there are always two constants: C 1 ω and C 2 ω such that Proof. This Lemma can be found in [44].
From Lemma 2.3 in [40], we have the following Lemmas.
We have the following lemmas by applying Lemmas 3.3 and 3.4 from Section 3 [41].
Proof. Using Lemma 3.3 from Section 3 in [41] to prove above Lemma.
which has the norm, we can find it in [40], For ζ > 0 and L ∞ ζ (0, T, H s (Ω)) is a Banach space with norm Now, suppose that the (1) problem has a unique solution, so we find the shape of it. Let u(x, t) = ∑ ∞ n=1 u(·, t), ψ n (·) ψ n (x) be the Fourier series in L 2 (Ω). From (1), we can deduce that Using (12), we have Therefore, we obtain By using the theory of fractional ordinary differential equations (see e.g., [43,47]), we have the following unique function u n u n (t) = E ω,1 −λ n t ω 1 + κλ n g 0,n where we denote g 0,n = g 0 , ψ n and u n (t) = (·, t, u(·, t)), ψ n . Deduce, the solution (1) can be shown as by Fourier series u(x, t) = ∑ ∞ n=1 u(·, t), ψ n (·) ψ n (x) and then given by We will rely on the results of Section 2.1 and the calculation of some variations in the Mittag-Leffler function to investigate the stability of the solution to Problem (1.1) with respect to the parameter. From (21) we get the solution of Problem (1) is given by where Therefore, we also have The following inequalities hold: Proof. Using Lemma 4, with assumption 0 < θ < 1, we get Similarly, using Lemma 4, we have the following estimate Therefore, we deduce where Thus, we complete the proof of Lemma 8. Lemma 9. Let 1 < ω < 2, α > 0 with 0 < ρ < 1 and Θ ∈ H α (Ω). The following inequalities hold: where D 1 , D 2 are independent of ω and also defined in the proof.
Proof. Parseval's equality implies that the following equality Using the estimate of Lemma 6, we can deduce which allows us to get Here, we note 0 < ρ < 1 and . By a similarly argument as above, applying Lemma 7, we also get We have all the Lemma 9 estimate and this completes the proof.
Next, we state the main results in the following section.

Stability of a Nonlinear Fractional Pseudo-Parabolic Equation Regarding
Fractional-Order of the Time In this section, we propose the continuous dependency of the problem solution (1) on input data (the fractional-order ω and the initial state g 0 ). The forcing terms are assumed to satisfy the following assumptions: In the following theorem, we present results about the unique solution of Problem (1) and the stability of solution regarding fractional order. Theorem 1. Let 0 < θ < 1, θ < α < θ + 1. Assume that 1 < σ < ω < ω < ν < 2 and let g 0 ∈ H α (Ω), and G(0) = 0. Then the Problem (1) has unique solution u(x, t) ∈ L ∞ ζ (0, T, H α (Ω)). Moreover, let u ω and u ω be the solutions to Problem (1) for fractional-orders ω and ω respectively. If existing two positive numbers ρ, satisfy 0 < ρ < 1, and 0 < < and where C 2 (σ, ν, κ, θ), R, W 1 , W 2 are positive constants which are independent of ω and ω .

Numerical Experiments
In this section, we employ our proposed numerical scheme for the continuity fractional order for the pseudo-parabolic equation with the fractional Caputo derivative. For more detail, we present some examples to illustrate the property which are shown in Theorem 1. It means that the solution u ω (the solution u with order ω ) converge on u ω (the solution u with order ω) when ω tends to ω, where ω is the fractional order. To do this, firstly, we use finite difference to discrete the spatial and time variable on the domain (x, t) ∈ [0, 1] × [0, 1] as follows where N x and N t are given positive constants. Secondly, in calculations with Python software, we use some numerical approximation methods as follows.
We use the approximation of the Mittag-Leffler function E ω,φ as follows and we evaluate the function Γ(·) by gamma(·) function which defined in the Matplotlib library. The integral approximation method by sum Rieamann. This approach can be used to find a numerical approximation for a definite integral where M is large enough positive number. Next, by taking the operator R = −∆. Then we have an orthonormal eigenbasis in L 2 (0, 1) is and the respective eigenvalues as follows where λ n = n 2 π 2 , for n ∈ Z + .
On the domain [0, 1] × [0, 10], we have a performance below where [A] T is the transposition matrix of the matrix [A] and u 1 (t q ) = E ω,1 −n 2 π 2 t α p 1 + n 2 π 2 1 0 g 0 (τ)ψ n (τ)dτ Then the solution (49) can be written in the matrix form u pq , where the representative element u pq of this matrix is u(x p , t q ).
Finally, by fixing t, we have the relative error estimation (RE E ω i ω ) and percent error estimation (P E E ω i ω ) between the solutions u ω and u ω as follows Let get started, we consider (x, t) on the domain [0, 1] × [0, 1] with the functions are given by In the following subsection, by choosing N x = N t = 50 and N = 10, we present the result of the numerical implementations for the problem (45) with the conditions (53) and (54) in some cases as follows. The numerical result is shown that the performance of the proposed method is acceptable. Indeed, the convergent estimate between the solutions for the fractional orders ω and ω i at t ∈ {0.1, 0.5, 0.9} is shown in Tables 1-3. Table 1. The relative error and percent error estimations for ω = 1.1.  Table 2. The relative error and percent error estimations for ω = 1.5.  Table 3. The relative error and percent error estimations for ω = 1.9.  1]. For detail, the first case ω = 1.1 is shown in Figures 1-4. The second case ω = 1.5 is shown in Figures 5-8. And the final case, ω = 1.9 is shown in Figures 9-12.

Fractional Order
From the observations on these tables and figures, it concludes that the smaller the error output between the solutions for the fractional orders ω and ω i when the smaller in the fractional orders.      In this subsection, we consider the asymptotic behaviour of the solutions for some values of ω = 1.5 and ω i ∈ {1.55, 1.53, 1.51}.       In final case, we consider the asymptotic behaviour of the solutions for some values of ω = 1.9 and ω i ∈ {1.95, 1.93, 1.91}.