Reconstructing the Unknown Source Function of a Fractional Parabolic Equation from the Final Data with the Conformable Derivative

this work. Abstract: The paper’s main purpose is to ﬁnd the unknown source function for the conformable heat equation. In the case of ( Φ , g ) ∈ L 2 ( 0, T ) × L 2 ( Ω ) , we give a modiﬁed Fractional Landweber solution and explore the error between the approximate solution and the desired solution under a priori and a posteriori parameter choice rules. The error between the regularized and exact solution is then calculated in L q ( D ) , with q (cid:54) = 2 under some reasonable Cauchy data assumptions.

Conformable derivative model: Let us take B as a Banach space, and f as a B-valued function on [0, ∞). Let C ∂ β ∂t β be the conformable derivative of order 0 < β ≤ 1 locally defined by: C ∂ β f (t) for each t > 0. For more knowledge about the above definition, we refer the reader to [22][23][24]. There are two interesting points regarding the relationship between conformable and classical derivatives: • Let us assume that B ≡ R, if f is a real function and s > 0, then f has a conformable fractional derivative of order β, and If B is not R, for example B are Sobolev spaces. There are not many conformable related results in Banach spaces, see [25].
The inverse source problem for (1) is described as follows. The final time condition u(x, T) = g(x), together with the additional condition u(x, 0) = 0, x ∈ D. The inverse source problem for (1) is understood as finding the function f when the input data g, Φ is given. As we know, the Problem (1) is ill-posed, and according to our experience and understanding, the frequent infringement is the continuity of the solution according to data. Therefore, to provide a good approximation, we need to regularize these problems. Before going into adjustment, we would like to review a bit of the history of the Problem (1).

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In case β = 1, the above equation becomes the classical pseudo-parabolic equation; this type of equation has received much attention from mathematicians, see [26][27][28]; • In case β = 1, with Caputo derivative model, we find the following documents, see [4]. Luc and co-authors studied the existence and uniqueness of a class of mild solutions of these equations. In [29], the authors considered the non-local Problem (1) for a pseudo-parabolic equation with fractional time and space. In [30], Tuan and his group considered a class of pseudoparabolic equations with the nonlocal condition in two cases: the nonlinear source function ad linear source function. For the first case, by using the Sobolev embeddings, they established the existence, the uniqueness, and some regularity results for the mild solution of Problem (1). For the second case, using the Banach fixed-point theorem, they proved the existence and the uniqueness of the mild solution for (1). In [31], the authors considered two problems. For the first problem with the source term satisfying the globally Lipschitz condition, we establish the local well-posedness theory, and the further local existence theory related to the finite time blow-up are also obtained for the problem with logarithmic nonlinearity. For the second problem, they proved the global existence theorem. • We have not seen any findings for this kind in case β = 1 with the conformable derivative model, and the source function survey problem is much more sparse, thus our study concentrates on this topic.
The regularization problem is a very interesting problem, with common regularization methods such as the Tikhonov method [32], Quasi Boundary Value method [33], Fractional Tikhonov method [34] and mollification method [35]. In [36], T. Wei with the Tikhonov regularization method, in [37], Ting Wei and her group considered a time-dependent source term by using a boundary element method combined with a generalized Tikhonov regularization. However, in this article, we use a modified fractional Landweber method to solve the unknown source Problem (1). Besides, there is a new point in this paper; we evaluate the error of exact and normalized solutions in L q (D) space, with q = 2. In this case, the equality Parseval was not used. One way to overcome this weakness is to use the embedding between L q (D) and Hilbert scales spaces X r (D). The main analytical technique in our paper is to use some embeddings and some analysis estimators related to Hölder inequality. To complete our proofs, we learn many interesting techniques from N.H. Tuan [38]. For the reader's convenience, we would like to outline the main results of the paper.

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We give the ill-posedness of Problem (1); • Showing the regularization of Problem (1), with the two subsections; -Uncertainty of Problem (1) of determining the source function; - The conditional stability of Problem (1); • Using the modified fractional Landweber method to solve the Problem (1). We obtain the convergence rate as follows: -In Section 4.1, under a priori parameter choice rule; -In Section 4.2, under a posteriori parameter choice rule.
• It gives the error estimate in L q space, with q = 2. This paper is organized as follows. Section 2 introduces some function spaces and embeddings. In Section 4, we deal with the regularized solution for the inverse source problem for (1) by the Modified Fractional Landweber method under a priori and a posteriori parameter choice rules. In Section 5, we solve the Problem (1) in the case of observed data in L q space. Finally, we present a numerical experiment.

Definition 1.
(Hilbert scale space). We recall the Hilbert scale space, which is given as follows: for any r ≥ 0. It is well-known that X r (D) is a Hilbert space corresponding to the norm, Then the following estimates are true: and Proof. First of all, we have the estimate Since Φ(t) ≥ Φ 0 > 0, we have: Let us consider the following function: which is equal to: G (x) = 1 (1 + kx) 2 > 0. This implies that G is an increasing function on (0, +∞). Therefore, we get: It follows from (6) that: The Lemma is proven.

Regularization of Inverse Source Problem
We consider the mild solution in Fourier series, u( Taking the inner product of the equations of Problem (1) with e n gives: The first equation of (10) is a differential equation with a conformable derivative as follows: In view of the result in (Theorem 5, " [41], p. 318") and (Theorem 3.3, " [42], p. 318"), the solution of Problem (1) is: To find the formula of the mild solution to Problem (1), with u n (0) = 0 and F n (s) = Φ(s) f (x). Letting t = T, we know that: After a simple transformation, we get: This leads to: 3.1. Uncertainty of Source Problem Proof of Theorem 1. We defined a linear operator S : L 2 (D) → L 2 (D) as follows. where Due to k(x, ω) = k(ω, x), S is a self-adjoint operator. We defined the finite rank operators S N and considered its compactness : From (16), through some basic calculations and using the Lemma 1, we have: From (17), we have: Additionally, S is a compact operator. The SVDs for the linear self-adjoint compact operator S are: and corresponding eigenvectors are e n is an orthonormal basis in L 2 (D). Therefore, the inverse source Problem (1) can be formulated as an operator equation S f (x) = g(x) where, by g(x), and by Kirsch, it is ill-posed. Next, we show an example, with final time data i e i and u 0 = 0. By (14), the source term corresponding to g i is: The input final data g = 0 then the source term corresponding to g is f = 0. We have error in L 2 norm between g i and g, Then the error in L 2 norm between f i and f From (21) and (22), we deduce that the solution to Problem (1) is unstable in L 2 (D).

The Conditional Stability
Theorem 2. Assume that f ∈ X r (D), g ∈ L 2 (D), and f X r (D) ≤ B then we have: Proof of Theorem 2. The proof of this theorem can be conducted similar to the articles [39]. We omit this here.

A Modified Fractional Landweber Method and Convergent Rate
Based on a modified Fractional Landweber method, we show the error estimate under the a priori regularization parameter choice rule and the a-posteriori regularization parameter choice rule, respectively. Now, we use the modified Fractional Landweber iterative method to obtain the regularization solution for Problem (1). We give the following iterative form: where τ is the iterative step number and the regularization parameter is τ −1 . The coefficient ξ is called the relaxation factor and satisfies 0 < ξ < S −(α+1) , by denoting an operator with (19), it gives : With how to choose ξ in above, by denoting z = ξ Qλ 1 α+1 , we have Proof of Lemma 4. A complete demonstration of this inequality can be found in [9].

A Priori Parameter Choice Rule
Before we go into proving the main theorem, we need the following lemmas: Lemma 5. For 0 < α ≤ 1, then we get: Proof of Lemma 5. In here, we denote λ n = z, this implies that: Taking the derivative of the variable z and solving the G 1 (z 0 ) = 0, we can find that: maximuizes the function G(z). Hence, we get: Finally, one has: Theorem 3. Let f ∈ X r (D), for any g, g δ ∈ L 2 (D) such that g δ − g L 2 (D) ≤ δ. The regularization parameter m is chosen then we get: whereby m denotes the largest integer less than or equal to m.

Proof of Theorem 3.
We have: Applying the inequality (a + b) 2 ≤ 2(a 2 + b 2 ), we estimate at f m,δ − f m L 2 (D) as follows: We will divide the evaluation (36) into two steps as follows: Step 1: By means of the Lemma 4, we have estimate of H 1 as follows: Step 2: Next, H 2 can be bounded as follows: In here, we have: Next, we give the estimate of the approximation error f m − f L 2 (D) as follows: From (41), using the Lemma 5, we can know that: Combining (36) to (42), we get: Let by choose m as follows: This leads to

A Posteriori Parameter Choice Rule
In this section, we look at the following regulatory parameter choices in Morozov's difference principle. We construct the regular solution sequence g δ , f m,δ equals the Landweber iteration method. Stop algorithm at the first occurrence of m = m(δ).
. Then we declare that: d. Y (m) is a strictly increasing function, for any m ∈ (0, +∞).
The proof of the above lemma is simple and completely similar to that in [39].
Proof of Lemma 6. From (15) and (26), our proof starts with the observation that: Lemma 7. Assume that (47) holds, the regularization parameter m satisfies Proof of Lemma 7. From (26), we have: and It is easy to see that 1 − ξ|Qλ −1 n | α+1 ≤ 1, by using the (46), we get : On the other hand, from the reviews above, add the Lemma 6, we receive: From (53), we conclude that: Theorem 4. Let the condition g δ − g L 2 (D) ≤ δ and f ∈ X r (D) hold, and the parameter regularization m is found in the Formula (46), then it gives: Proof of Theorem 4. Using the triangle inequality, we have: The proof in the Formula (39) has given us: Applying the Lemma 7, we get: Now, using the Holder's inequality and the Formula (46), and results obtained from the Lemma 2, we get the estimate of error f m − f L 2 (Ω) as follows: Next, H 4 can be bounded.
Combining (59) to (61), it gives : From (58) and (62), we conclude that: whereby In the next section, we provide the error estimation between the exact solution and the regularized solution by the Fourier truncation method.

Regularization of Inverse Source in
Let us assume that Φ δ ∈ X r+γ (D) for r > 0 and 0 < γ < N 4 . Constructing a regularized solution as follows: Then the error estimate is bounded by: here N δ satisfies that: Proof of Theorem 5. It is clear that: where we denote some following functions: and Now, we need to establish the upper bound of the expressions on the right of (13). For convenience, we consider the following step.
Step 1: Estimate of f 2,δ − f X γ (D) , let us recall the function f as (14). This expression together with the Formula (71) gives us the claim of the following difference: Using the Parseval equality, the left hand side of (72) is calculated as follows: It is obvious to see that λ −2r n ≤ |N δ | −2r if λ n > N δ and m > 0. Therefore, we have: which leads to Step 2: Estimate of f 2,δ − f 1,δ X γ (D) , we get From (76), we know that: By applying the Holder inequality, we receive: Next, we have the estimate where q * = 1 + 1 s−1 , this implies that: and through some basic calculations, we obtain: whereby we note that q > 1 β and we also have used the fact that exp q * λ n 1 + kλ n Combining three evaluations (79), (80) and (81), we derive that the following estimate: Next, applying the Lemma 1 and the Lemma 2, we have estimated From the two observation above, we assert that : whereby P 1 depends on Φ 0 , λ 1 , T, β, q and Combining (78) to (84), it gives: We assume that the finite sum ∑ Therefore, we conclude that: where P 1 is defined in (85).
Step 3: Estimate of f 1,δ − f X γ (D) , due to Formulas (14) and (70), so Using the Parseval's equality, and taking the norm of both sides, we obtain that: Because of the inequality (83), one has : Continuing to deal with the finite series on the right above, we have: (D). Therefore, we get that: Combining (91) to (93), one has: where Step 1 to Step3, we can conclude that: By using the Lemma 3, since 0 < γ < N 4 , with Sobolev embedding X γ (D) → L 2N N−4γ (D), we have the results as in (67).

Simulation
In this section, we present one numerical example. By choosing Ω = (0, π), T = 1, β = 0.5, k = 0.8, and α = 0.5, and r = 1 are shown in this section, respectively. In this section, we consider the problem as follows: where C ∂ β ∂t β is the conformable derivative is given by [23]. In this calculation, we chose the operator ∆u = ∂ 2 ∂x 2 u, we have chosen λ n = n 2 , n = 1, 2, . . . and e n (x) = 2 π sin(nx), respectively. We have the function, In general, the numerical procedure is summarized in the following steps: Step 1: Finite difference to discretize the time and spatial variable for x ∈ (0, π) as follows: Step 2: The input data g is noised by observation data g δ such that: From (14) From (26) and (27), by choosing the regularization parameter as in the number Formula (33), in the case of a priori parameter choice rule, and Formula (48), in the case of a posteriori parameter choice rule, where N is a large enough truncation number, we have the regularized solution with Modified Fractional Landweber as follows: We choose N = 50 and δ = 0.5, δ = 0.25 and δ = 0.125. Figure 1a shows the 2D graphs of the source function with the exact solution and its approximation for the case for the a priori parameter choice rule. Figure 1b shows the error estimate between the exact solution and regularized solution for the a posteriori parameter choice rule. Figure 2a-c shows the 2D graphs comparing the convergent rate between the exact solution and its approximation under a priori and a posteriori parameter choice rules with noise levels δ = 0.5, δ = 0.25, and δ = 0.125. From the observations above, the comparison with the results developed in theory (see evaluation (34) and (55)) shows that the convergence in these two cases is almost equivalent, illustrating that the proposed method is effective.