Abstract

In this paper, we study a diffusion equation of the Kirchhoff type with a conformable fractional derivative. The global existence and uniqueness of mild solutions are established. Some regularity results for the mild solution are also derived. The main tools for analysis in this paper are the Banach fixed point theory and Sobolev embeddings. In addition, to investigate the regularity, we also further study the nonwell-posed and give the regularized methods to get the correct approximate solution. With reasonable and appropriate input conditions, we can prove that the error between the regularized solution and the search solution is towards zero when tends to zero.

1. Introduction

The aim of this study is to investigate the final value for the space fractional diffusion equationwhere the symbol is called the conformable derivative which is defined clearly in Section 2. Here, is a bounded domain with the smooth boundary , and is a given positive number. The function represents the external forces or the advection term of a diffusion phenomenon, etc., and the function is the final datum which will be specified later.

The applications of the conformable derivative are interested in various models such as the harmonic oscillator, the damped oscillator, and the forced oscillator (see, e.g., [1]), electrical circuits (see, e.g., [2]), chaotic systems in dynamics (see, e.g., [3]), and quantum mechanics (see, e.g., [4]). From the paper, see, e.g., [5], we must confirm that the study of the ODE problem with the conformable derivative is very different from the study of the PDE problem with a conformable derivative. Results and research methods of the well-posedness for the ODE and PDE model are not the same and are completely different. The following two remarks confirm what we have just pointed out.

Remark 1. Let us first discuss conformable ODEs. Let be the functions whose domain of its value is . If , becomes the classical derivative. If , by the paper of [6], we know that the relation between the conformable derivative and the classical derivative by the following lemma.

Lemma 2. If , then a conformable derivative of order at of exists if and only if it is differentiable at , and the following equality is true:

Remark 3. In the following, we mention the PDEs with conformable derivative where is a Sobolev space, such as , , and . When we study the PDE model, we often do with a multivariable function , where is a Sobolev space. This means that, for each , can take on values in many classes of spaces with . Some illustrated examples given in [5] say that (2) may be not true on Sobolev spaces.

Let us mention some recent works on diffusion equations with a conformable derivative, for example, [2, 5, 716]. Some interesting papers on fractional diffusion equations can be found in [1724] and the references therein.

When , the main equation of Problem (1) appears in many population dynamics. By the work of Chipot and Lovat [25], we know that the diffusion coefficient is dependent on the entire population in the domain instead of local density; that is, the moves are guided by considering the global state of the vehicle. The function is a descriptive population density (e.g., bacteria) spread. According to article [26], we find that model (1) is a type of Kirchhoff equation, arising in vibration theory; see, for example, [27].(i)This paper is the first study on the final value problem for a diffusion equation with a Kirchhoff-type equation and conformable derivative. Since our models are nonlinear, in order to establish the existence and uniqueness of solutions, we have to use the Banach contracting mapping theorem combined with some techniques to evaluate inequality and some Sobolev embeddings. One of the most difficult points is finding the appropriate functional spaces for the solution(ii)The second result is to investigating the regularized solution for our problem. We show the ill-posedness of the problem and give Fourier regularization. The most difficult thing that we have to overcome is finding the appropriate space, to prove that the regularized solution converges with the exact solution

It can be said that our article is one of the first results, giving a general and comprehensive picture, considering both the frequency and the inaccuracy of Kirchhoff’s diffusion equation with fractional time and space derivative. Using complex and interoperable assessment techniques, we find the right keys and tools to achieve both of our goals.

This paper is organized as follows. In Section 3, we present the existence of the backward Problem (1) with the simple case . In the appropriate terms of the terminal data , we show that the mild solution of (1) in the case converges with the mild solution of the same problem in the case when . Finally, in Section 4, we consider a backward problem with an inhomogeneous source term. The first part of this section discusses the existence of a mild solution under the appropriate conditions of the source function . Furthermore, we also give an example, which shows that the problem is not stable, and then look for the approximate solution. Using the Fourier truncation method, we involve the regularized solution. Convergence error between the regularized solution and the correct solution has also been established, with some suitable conditions of input value data.

2. Preliminaries

2.1. Conformable Derivative Model

Let the function , where is a Banach space.

If for each , the limitationfinitely exists, then it is called the conformable derivative of order of . We can refer the reader to [6, 8, 14, 28, 29].

We introduce fractional powers of as follows:

The space is a Banach space in the following with the corresponding norm:

The information for negative fractional power can be provided by [30]. For any , we introduce the following Hölder continuous space of exponent corresponding to the following norm:

For any , let us introduce the following space:corresponding to the norm .

Let us define the space as follows:

3. Backward Problem for Homogeneous Case

In this section, we consider the final value problem for the homogeneous equation with a space fractional derivative as follows:where and . The following theorem states the existence and uniqueness of the solution of Problem (10).

Theorem 4. Let . Then, Problem (10) has a unique mild solution which satisfies that

Furthermore, this solution is not stable in the norm.

Proof. We express a mild solution of (10) by Fourier series as follows:It follows from Problem (10) and the equality thatNote that this formulais correct; we get thatMultiply both sides of equation (15) by the quantity , we reach the following assertion:where we have used the fact thatIntegrating the two sides of the latter equation 0 to , we obtain the following confirmation:It yields thatTherefore, we find thatFor , we consider the following function:We shall prove by induction if , thenFor , using the inequality for any , we haveAssume that (22) holds for . We show that (22) holds for . Indeed, we haveBy the theory of the induction principle, (22) holds for all . Since the fact thatthere exists a positive integer number such that is a contraction. It follows that the equation has a unique solution . It is easy to see that is also a fixed point of .

Theorem 5. Assume that for any . Let us choose such thatLet any . Then, there exists such thatwhere

Proof. Let be the solution of Problem (11). Let be the solution to Problem (11) with . Then, we getWe havewherebyThe term is bounded byThe term is estimated as follows:Consider the following subset:If , then using the inequality , we getwhich allows us to obtainIf , then using the inequality , we findHence, we obtainCombining (36) and (38), we find thatLet us choose . Then, we follow from (33) and the latter equality thatThis above inequality together with (32) and (3) yields thatIt is easy to get thatIt follows from the inequalitywe getThe inequality leads towhich allows us to get immediately thatIt follows from (41) and (42) that for any Since the right-hand side of (47) is independent of , we deduce thatThen, we find that

4. Backward Problem for Inhomogeneous Case

In this section, we consider the final value problem for homogeneous equation as follows:where is defined later.

4.1. Existence and Uniqueness of the Mild Solution

In this subsection, we state the existence and uniqueness of the mild solution. In order to give the main results, we require the condition which belongs to the space .

Theorem 6. Let and be the source function that belongs to for any . Let be the functions which satisfy , andThen, Problem (50) has a unique mild solution , where is small enough. The function satisfies thatFurthermore, this solution is not stable in the norm.

Proof. By a simple calculation, we get the following equality:By letting and noting that , we find thatTherefore, we obtainCombining (53) and (55), we deduce thatLet us denote by the functional subspace of corresponding to the normwhereSet the following function:and we letSo, using Parseval’s equality, we get thatUsing the inequality , we continue to treat the term as follows:Therefore, applying the Hölder inequality, we getInserting (61) and (63) yields the following inequality:Take any . By a similar explanation as (46), we find thatBy applying the Hölder inequality, we also obtain thatFrom some observations as above, we deduce thatSince the right-hand side of the latter estimate is independent of , we find thatLet us choose such thatThen, we can conclude that is a contraction mapping in the space . Next, we continue to show that if , then . If , thenHence, from Parseval’s equality, we find thatThis says that belongs to the space . Using (68), we arrive at the confirmation that belongs to if . For any , let be the function that satisfies the following integral equation:Let us assume thatIt is not difficult to verify that , so we get that .
Using Theorem 6, we conclude that equation (72) has a unique solution . By the fact that , we obtain the following estimate:The estimate is true for all , so it is easy to see thatWhen tends to +∞, we can check that go to zero when andThis shows that Problem (50) is ill-posed in the sense of Hadamard in the L2-norm.☐

4.2. Fourier Truncation Method

In this section, we will provide a regularized solution and solve the problem by the Fourier truncation method as follows:

Here, goes to infinity as tends to zero which is called a parameter regularization. The function is disturbed by the observed data provided by

The main results of this subsection are given by the theorem below.

Theorem 7. Let such that belongs to the space . Let be as above. Let us assume that Problem (50) has a unique mild solution for . Let us choose such thatHere . Then, there exists a positive large enough such that Problem has a unique solution . Moreover, we have the following estimate:

Remark 8. Since , we can choose a natural number such that

Proof. Part 1: prove that the nonlinear integral equation (77) has a unique mild solution.
Let any , we denote by the following functionBy applying Parseval’s equality, we follow from (82) thatIf , then we have in view of (63) thatThe above two observations (65) lead toBecause the right-hand side of the latter estimate is independent of and noting the Sobolev embedding , we arrive atLet us choose such thatIt is easy to see that is a contracting mapping on the space . Therefore, we can conclude that there exists a uniqueness solution for Problem (77).
Next, we continue to give the upper bound of the term . First, we haveThe above equality and Parseval’s equality allow us to get thatSince the condition and applying Hölder inequality, the quantity is bounded bywhere we have used the fact that .
The quantity is estimated as follows:We have in view of (63) thatThis leads to the following estimate:The term is estimated as follows:Combining (89), (90), (93), and (94), we find thatWe choose such that both the following inequalities are satisfied:Some observations above give us the following confirmation:Since the fact thatWe easily obtain the desired result (80).☐

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

The authors contributed equally to the work. The four authors read and approved the final manuscript.