Abstract
The main goal of the paper is to approximate two types of inverse problems for conformable heat equation (or called parabolic equation with conformable operator); as follows, we considered two cases: the right hand side of equation such that and . Up to now, there are very few surveys working on the results of regularization in spaces. Our paper is the first work to investigate the inverse problem for conformable parabolic equations in such spaces. For the inverse source problem and the backward problem, use the Fourier truncation method to approximate the problem. The error between the regularized solution and the exact solution is obtained in under some suitable assumptions on the Cauchy data.
1. Introduction
Partial differential equations (PDEs) have applications in many branches of science and engineering; see for example [1–8]. In this paper, for , we consider the initial value problem for the conformable heat equation (or called parabolic equation with conformable operator)
Here, () is a bounded domain with the smooth boundary , and is a given positive number. Here, is called the conformable time derivative with order (Khalil et al. [9]) for a given function ; the of order is defined by for all . For some , and the exist, then . Some properties of can be found in more detail in [10, 11]. is a natural extension of usual derivative, it preserves basic properties of the classical derivative [10, 11], and it is a local and limit-based operator. In [10, 12, 13], we saw some applications. For the convenience of the reader, we will consider two models related to Problem (1) that most mathematicians often study. (i)The first part of the paper deals with the final value problem for Problem (1) with a linear source function. The new feature of this part is the appearance of observed data, namely, . This result is well described in Theorem 3. We investigated the problem of restoring the temperature function , in the fact that the couple are noised by the measurement data such that:(ii)The second part of the paper deals with the final value problem for Problem (1) with is a linear source function as follows: , where both functions are perturbed by in , respectively
The main contributions and novelties of this paper are stated as follows. As we know, two inverse problems are ill-posed in the sense of Hadamard. The well-posed problem satisfies three conditions above: the solution is existence, the solution is uniqueness, and the solution continues on data The problem that violates one of the above three conditions is an ill-posed problem. We need to regularize this problem, to give a good approximation. The number of works on the regularized problem with input data in is quite abundant. The results of this study can be found in the following documents, attached to the regularization methods: the Tikhonov method, see [14, 15], the Fractional Tikhonov method, see [16], the fractional Landweber method, see [17, 18], the Quasi Boundary method, see [19], the truncation method, see [20], and their references.
However, for , results for regularized problem in are quite rare. We confirm that our paper is the first result for the inverse problem for the conformable parabolic equation when the observed data is in the space with . If the data is not in , the use of Parseval equality is not feasible. In this case, we used the embedding between and Hilbert scales spaces . These results are well described in Theorem 3 and Theorem 5. The main analytical technique in our paper is to use some embeddings and some analysis estimators related to Hölder inequality. To do this, we learn many interesting techniques from N.H. Tuan [21].
This paper is organized as follows. In Section 2, we state some function spaces and embeddings. In Section 3, we deal with the regularized solution for the inverse source problem for (1). Section 4 gives the mild solution of backward problem in case . After that, we solve two problems in the case of observed data in space.
2. Preliminary Results
Let us recall that the spectral problem admits the eigenvalues with as . The corresponding eigenfunctions are .
Definition 1 (Hilbert scale space). We recall the Hilbert scale space, which is given as follows: for any . It is well-known that is a Hilbert space corresponding to the norm
Lemma 2 (See [22]). The following statement is true:
3. Regularization of Backward Problem
In order to find a precise formulation for solutions, we consider the mild solution in Fourier series , with . Taking the inner product of the equations of Problem (1) with gives
The first equation of (9) is a differential equation with a conformable derivative as follows:
Because of the result in [23], the solution of Problem (1) is
Letting , we follow from (11) that
From (12), we have
Substituting (13) into (12), we obtain
This leads to
4. The Mild Solution of Backward Problem in Case
In this section, we investigate the existence and regularity of mild solutions of Problem (1). Firstly, we consider the following initial value problem
According to (15), in this case, we have
4.1. The Ill-Posedness of Problem (1)
In order to prove that the solution to the backward problem is unstable at, let us take the perturbed final data, by choosing. For , let us choose input final data ; we know that an error in norm between two input final data as follows:
Therefore, we obtain
First of all, we have , and ; this implies that
Next, using the inequality , for , this leads to:
For , and from (21), we get
Thus, Problem (1), in general, ill-posed in the Hadamard sense in -norm.
4.2. Regularization of inverse Problem (1) in space
From (15), we know that the explicit formula of the mild solution
By applying the Fourier truncation method, we have its approximation
Here, is parameter regularization which is defined later.
Theorem 3. For , taking for any for any , assume that is observed by the couple such that Let us assume that for and . With such that Then, the error estimate
Remark 4. One choice for such that then .
Proof. Let
It is clear that
We continue to consider the two components of the right hand side.
Step 1:
For , it is easy to see that . The first term on is bounded by
Since the Sobolev space embedding , we have
This follows from (32) that
The second term is estimated as follows:
By the same arguments as above, we find that
We can see that and . From (36), using Holder’s inequality, we get
This latter inequality together with Sobolev embedding gives us
Combining (36) and (38), we get
Taking the square root on the both sides, we have
From (34) and (40), we deduce that
Step 2: Estimate of .
From the definition (23) and (29), we have
Therefore, we get
Combining two steps and noting that , we deduce that
The proof of Theorem 3 is completed. In the following theorem, we give a regularization result in the case that has a split form .
Theorem 5. For , let us assume that the input data such that Assume that for any , then we construct a regularized solution defined by Then, the error is of order
Remark 6. , then the error
Proof. Since , we know that
The triangle inequality allows us to obtain that
Next, we will evaluate the right side of (49), by the same way as demonstrated in (42),
It is easy to see that
whereby
We will divide this review into several steps as follows:
Step 1: Estimate of , we obtain that
Step 2: Due to Parseval’s equality, the term can be bounded as follows:
Thank to Holder’s inequality, we derive that for and , one has
The latter inequality leads to
where . In view of Sobolev embedding , we derive that the following estimate
where
Step 3: Let us now to consider the term . By applying Hölder’s inequality, we get that for and This inequality together with Parseval’s equality allows us to derive that
By the fact that , we deduce that
Combining Step 1 to Step 3, we get
Finally, combining the reviews from above, we conclude that
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no competing interests. The corresponding author is a full-time member of the School of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran
Authors’ Contributions
All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.
Acknowledgments
The author Le Dinh Long is supported by Van Lang University.