Reconstructing the right-hand side of the Rayleigh–Stokes problem with nonlocal in time condition

In this paper, the problem of finding the source function for the Rayleigh–Stokes equation is considered. According to Hadamard’s definition, the sought solution of this problem is both unstable and independent of continuous data. By using the fractional Tikhonov method, we give the regularized solutions and then deal with a priori error estimate between the exact solution and its regularized solutions. Finally, the proposed regularized methods have been verified by simple numerical experiments to check error estimate between the sought solution and the regularized solution.


Introduction
Equation (1.1) below arises in Newtonian fluids and magnetohydrodynamic flows in porous media [1], and initial value problems for fractional Rayleigh-Stokes were studied, for example, in [2][3][4][5]. In this study, we are interested in dealing with the Rayleigh-Stokes problem associated with fractional derivative as follows: where ⊂ R d (d = 1, 2, 3) is a boundary domain with the boundary ∂ smooth enough, and T > 0. τ > 0 is a constant, u 0 in L 2 ( ), the notations ∂ t = ∂/∂t, and ∂ α t is the Riemann-Liouville fractional derivative of order β ∈ (0, 1) defined by [6,7]  The Rayleigh-Stokes introduced as above has much practical importance, see in [8,9], and in describing the behavior of some non-Newtonian fluids [10]. The numerical solutions of the Rayleigh-Stokes problem with fractional derivatives have been considered and developed by Dehghan or Zaky, see [3-5, 11, 12]. According to our understanding, in recent times, the study of this problem begins to receive the attention of mathematicians, such as M. Kirane [13] and S. Tatar [14]. In [15], authors studied a Rayleigh-Stokes equation in the simple bounded domain by using the fractional Landweber method. Besides that, the study of problem (1.1) with random noise data also began to receive the attention of mathematicians. In [16], using the truncation method and some new techniques, the authors showed the regularized solution, and convergence rates were established. In [17], Triet et al. investigated an inverse source problem (1.1) by a general filter method for random noise, the results for the study of problem (1.1) were rare. However, articles about the survey of source functions for problem (1.1) were rarer than the results. See in [18], the authors investigated problem (1.1) by the Tikhonov regularization method, attached was a simple numerical calculation example to simulate research results in theoretical way. Besides, we also find relevant applications in a broad sense with problem (1.1), please see [19][20][21][22][23][24][25]. In most of these studies, mathematicians are interested in the final condition as follows: u(x, T) = (x).
Recently, a few papers mentioned the nonlocal condition T 0 u(x, t) dt = (x), for example, two papers [26,27]. We repeat that if the source function F(x, t) = ϕ(t)f (x) is given, then problem (1.1) is called the forward problem. The problem of determining the source function is understood as defining a function f when we know that T 0 u(x, t) dt = (x) and the function ϕ. It is worth pointing out that our article is one of the first results to study this problem with nonlocal in time condition. This work can be considered a development step of the results in the article [18]. In this paper, the couple functions ( , ϕ) are approximated by ( , ϕ ) such that This paper is organized as follows. In Sect. 2, we introduce some preliminaries. The main results are given in Sect. 3 which presents the non-well-posedness of our problem (1.1). Next, in Sect. 4, we propose the fractional Tikhonov regularization method to find the regularized solution and the convergent rate. In Sect. 5, we present a simple numerical example to verify the results proved in our theory section. The conclusion is presented in Sect. 6.

Preliminaries
Definition 2.1 ([28]) Let {λ j , e j } be the eigenvalues and corresponding eigenvectors of the Laplacian operator -in . The family of eigenvalues {λ j } ∞ j=1 satisfy 0 < λ 1 ≤ λ 2 ≤ · · · ≤ λ j ≤ · · · , where λ j → ∞ as j → ∞: Based on [2], we can know that the solution of the Rayleigh-Stokes problem is as follows: Here, C j (β, t) satisfies the following equation: From the condition T 0 u(x, t) dt = (x) and u 0 = 0, we can check that where we note that F j (z) = ϕ(z)f j . A simple calculation gives From the result of [2], we obtain

Lemma 2.4
Let us assume that β ∈ ( 1 2 , 1). For all t ∈ [0, T], we have Besides, there exists M such that Proof We can see that in [29].

Lemma 2.6 Assume that there exist positive constants
Proof From now on, for short, we denote B 1 0 = A 1 + 4 -1 A 0 . For the proof of this lemma, readers can see document [30].

The non-well-posedness of problem (1.1)
Proof Let P : L 2 ( ) → L 2 ( ) be a linear operator as follows: , we know that P is a self-adjoint operator. Define the finite rank operators P N by From (3.1) and (3.2), we have

From (3.3), we can know that
Hence, we can deduce that So, we get immediately that P is a compact operator. From (3.1), the inverse source problem can be formulated as an operator equation Pf (x) = (x), and by Kirsch [32], it is unstable. Next, we propose an example, with input final data If we choose = 0, then f = 0, an error in L 2 -norm between k and is And an error in L 2 norm between f k and f is From (3.7) and combining with Lemma 2.4, one has By choosing β > 1 2 , we get Combining (3.6) and (3.10), this implies that problem (1.1) is non-well-posed.
Next, we give the following theorem which shows the conditional stability of the function f .
Proof From (2.5) and using the Hölder inequality, one has , and this implies that Combining (3.12) and (3.13), we have (3.14)

The fractional Tikhonov regularization method
In this section, we solve problem (1.1) by using the fractional Tikhonov method. The ideas of this method are based on the work of Hochstenbach in [33] or Yang in [31]. We use two kinds of fractional Tikhonov regularization methods to solve (1.1) as follows: With the Tikhonov minimization problem (4.1) with Y defined by (4.2) given by the solution of (4.3) is uniquely determined for any γ > 0 and a > 0. It is obvious to see that the formula of f [γ ( )] 1 is as follows: , e j e j (x). (4.4) We have the fractional Tikhonov regularized solution , e j e j (x). (4.5) Refer to [34], another type of fractional Tikhonov regularized solution is given by the following formula: where [γ ( )] 2 is the regularized parameter, with 1 2 ≤ a < 1. For the noisy data, we get , e j e j (x). (4.7) Putting and Next, we continue to investigate the convergence rates in two various cases.

Simulation
In this section, we consider the problem as follows: The couple of ( , ϕ ) plays as observed data as follows: (·) = (·) + 2 rand(·) -1 , In (5.1) with u(x, t) = t 2 sin(x), we get (x) = T 3 3 sin(x) and ϕ(t) = 2t -1 -2t t 2-β (3-β) . Next, we can write the term B j (β, tz) as follows, see Lemma 2.3: , e j e j (x) Similarly, in the formula finding the methodological seriousization for fractional Tikhonov method type one and the regularized solution for fractional Tikhonov method type two, we just need replace with and ϕ with ϕ .
Step 1: As the discretization level, a uniform grid of mesh-point (x i ) is used to discrete the space interval In this example, with N = 121, we take the following calculation steps.
Step 2: Set f [γ ( )] (x j ) = f γ ,j and f (x j ) = f j , construct two vectors containing all discrete values of f γ ,j and f denoted by γ ,j and j , respectively.
Step 3: Error estimate From the results of the above calculations, Table 1 points out the relative error estimates for a regularized solution using the fractional Tikhonov method, see formula (4.8), and the fractional Tikhonov solution type two, see formula (4.9), respectively. In this table, the values are as follows: In the case of the regularization solution fractional Tikhonov type one, since a ∈ (0, 1), then we choose a 1 = 0.65; in case of the regularization solution fractional Tikhonov type two, because of a 2 ∈ ( 1 2 , 1), we choose a 2 = 0.75 and values T = 1, β = 0.5, τ = 1.2, δ = 1. Table 1 shows the relative error estimates between the exact solution and its regularized solution for both FT 1 and FT 2 with = 5 * 10 -k , k = 1, 2, 3, 4, respectively. Table 2 shows the error estimate with values β in the first column. Similarity, with different τ , this error can be found in Table 3. In general, it shows that with both fractional Tikhonov methods, the convergence rate is of almost the same level. From the  results obtained in the number table, we conclude that when tends to 0, the tensile test will converge the accuracy, although this convergence is relatively slow.

Conclusion
In this article, we consider problem (1.1) for the Rayleigh-Stokes problem. In this article, by using the fractional Tikhonov method, we establish an approximate solution. Then, we show the rate of convergence between the sought solution and the regularized one and provide a simple numerical experiment. In the future work, we may use the condition θ u(x, T) + θ 2 T 0 u(x, t) dt = (x) to study problem (1.1).