Abstract
In this paper, we consider the nonlinear inverse-time heat problem with a conformable derivative concerning the time variable. This problem is severely ill posed. A new method on the modified integral equation based on two regularization parameters is proposed to regularize this problem. Numerical results are presented to illustrate the efficiency of the proposed method.
1. Introduction
Partial differential equations (PDEs) with different types of boundary conditions play an essential tool in modelling natural phenomena. For time-dependent phenomena, one usually adds an initial time condition or a final time condition, which can be considered as the data. For the time-inverse problem from the final data, the main goal is to reconstruct the whole structure in previous times. These problems were widely studied in the papers by Tikhonov and Arsenin [1], Glasko [2], and the references cited therein. An example is the backward heat problem (BHP) where the goal is to recover the previous status of a physical field from the present information. It is well known that the BHP is a classical ill-posed problem, and it is quite difficult to consider since the solution does not always exist. Furthermore, even if the solution does exist, the continuous dependence of the solution on the data is not guaranteed. The BHP has been considered in the literature using different methods (see [3–10] and the references cited therein). Fu et al. [3] applied a wavelet dual least square method to investigate a BHP with constant coefficients, in [4], Hao et al. gave an approximation for this problem using a nonlocal boundary value problem method, Hao and Duc [5] used the Tikhonov regularization method to give an approximation for this problem in a Banach space, and Tautenhahn in [6] established an optimal error estimate for a backward heat equation with constant coefficients. Using the stabilized quasireversibility method, the final value problem for a class of nonlinear parabolic equations is investigated by Trong and Tuan [7], and in [8], the authors used the integral equation method to regularize the backward heat conduction problem and they obtained some error estimates. Tuan and Ngo [10] introduced the truncation method for solving the BHP and presented new error estimates for investigating the stability of the given problem. Also, the modified integral equation method and the modified quasiboundary value are extended to investigate inverse-time problems for axisymmetric backward heat equations in [11, 12] and the nonlinear spherically symmetric backward heat equation in [13].
The concept of the so-called conformable derivative was proposed by Khalil et al. [14] and discussed by Atangana et al. [15] and Abdeljawad [16]. Anderson and Ulness in [17] provided a potential application of the conformable derivative in quantum mechanics, Hammad and Khalil [18] used a conformable fourier series to interpret the solution of the conformable heat equation, and Chung [19] employed the conformable derivative concept to investigate the problem of Newtonian mechanics, and the Euler–Lagrange equation was also constructed. Eslami [20] employed the Kudryashov method to obtain the traveling wave solutions to the coupled nonlinear Schrodinger equation with a conformable derivative, Çenesiz et al. [21, 22] studied Burgers’ equation, the modified Burgers’s equation, and the Burgers–Korteweg–de Vries equation with a conformable derivative version, Çenesiz et al. [23] investigated the stochastic solution of conformable Cauchy problems where the space operators may correspond to Brownian motion or a Levy process, and Vu et al. [24] employed the quasiboundary value method to regularize the inverse-time problem for the nonhomogeneous heat equation with a conformable derivative, and a Hölder-type estimation error for the whole time interval was obtained. In this paper, we consider the following backward heat equations:where , is a positive number, the functions and are given, and is the conformable derivative of order with respect to defined by
From the information given at final time , the goal of the problem is to recover the information for . Unfortunately, BHP (1)–(3) is ill posed in the sense of Hadamard, i.e., it violates at least one of the following conditions:(1)Existence. There exists a solution of the problem.(2)Uniqueness. The solution must be unique.(3)Stability. The solution must depend continuously on the data, i.e., any small error in given data must lead to a corresponding small error in the solution.
Using Theorem 1 in Section 2, we see that the solution of problems (1)–(3) is given bywhere the terms in the above equation are given in Theorem 1. We observe that , so this yields an instability of the solution of problems (1)–(3). This violates condition (3), so problems (1)–(3) are ill posed. In this paper, to stabilize problems (1)–(3), we shall apply the modified integral equation method via a two-parameter regularization to regularize problems (1)–(3). To do this, we shall replace the above instability term by the term , where , , is fixed, and is a positive constant. From the proposed term, we use the following modified integral equation to approximate or to regularize the solution of problems (1)–(3):
In Section 2, we show that problems (1)–(3) can be transformed into an integral equation (5). In Section 3.1, we prove that the regularized problem (6) is well posed in the sense of Hadamard in two cases, namely, and . In Section 3.2, the error estimates between the regularized solution of problem (6) and the solution of problems (1)–(3) with the prior condition on the solution in two cases of exact data and nonexact data are presented. In particular, we show thatwhere is specified below, is a solution of problems (1)–(3), and is a solution of the regularized problem (6). In Section 4, we provide numerical tests to illustrate the theoretical results in the paper.
2. Statement of the Problem
Throughout this paper, we denote by , where , the Hilbert space of Lebesgue measurable functions on . and represent the inner product and norm on , respectively. Specifically, the norm and inner product in are defined as follows:where . Denoting by the space of all continuous functions and denoting by the sup norm in defined as
The following theorem establishes the formula of the solution to problems (1)–(3).
Theorem 1. Let and let . Then, the solution of the original problems (1)–(3) has the following form:where and
Proof. By choosing the orthogonal basis , in the Hilbert space and by taking the inner product in on the two sides of (1), we obtainOn the other hand, by using boundary conditions (2), we also getThen, it follows from (11)–(13) thatSolving problem (14), we getBasing on (3), we have thatwhere . Then, (15) yields thatTherefore, the representation of solution of problems (1)–(3) can be written as the infinite series:
Remark 1. As stated in Section 1, we observe from (18) that when tends to infinity, the term is increasing rather quickly. Hence, the exact solution given in (18) of problems (1)–(3) is unstable. Thus, problems (1)–(3) are ill posed, and the above term is the unstable factor. So, to regularize the problem or to obtain a stable approximation for problems (1)–(3), we shall replace this unstable factor by a stable one. In this paper, the term is replaced by a stable term which depends on two regularization parameters defined by , where the first one captures the measuring error and the second one () captures the regularity of the solution. Therefore, in this paper, we shall use integral equation (6) to approximate or to regularize problems (1)–(3).
3. Regularization and Error Estimates
Before investigating the uniqueness and stability of the solution of problem (6), we present the following two inequalities, which will be useful in the proof of the next theorems.
Lemma 1. Let be fixed, and , , , and . Then, the following inequalities hold:and
Proof. Let , , , and let be fixed, then we observe that the functionhas the maximum value at . This yields thatThen, we obtain the following inequality:Furthermore, we haveFrom (23), we obtainThis yields estimate (20).
3.1. The Well Posedness of Regularized Problem (6)
In the following theorem, we show that regularized problem (6) is well posed in the sense of Hadamard, i.e., problem (6) has a unique solution, and this solution continuously depends on the given data.
Theorem 2. Let , and let satisfy the globally Lipschitz property with respect to the third variable, i.e., there exists a constant independent of such thatwhere is the norm in . Then, regularized problem (6) is well posed in the sense of Hadamard provided that .
Proof. To prove the theorem, we shall divide the proof into two steps. The first step shows that regularized problem (6) has a unique solution provided that . In the second step, the continuous dependence of the solution on the data will be verified. Step 1. Consider the operator given by where We set It follows from Lemma 1 that Therefore, by Hölder’s inequality, one obtains, for and , where is fixed, , and . So, from Lipschitz condition (26), one has that We observe that the function is decreasing on . Hence, for all . With the same calculation, one also gets where for , Using the mathematical induction method, for , one obtains Now, if we assume that (38) holds for , then by Hölder’s inequality and from Lemma 1, for , we have that Then, by (38) and Lipschitz condition (26), we have that which is inequality (38) for . Thus, (38) is satisfied for all . In addition, inequality (38) is similar to the estimate as follows: where is given by (31). Moreover, we have where is given and is fixed. Therefore, there exists such that is a contraction. Thus, problem (6) has a unique solution. Step 2. Let be a solution of (6) with the data and be a solution of (6) with the data . From problem (6), we setwhere . In view of the inequality , the estimates (32) and (33), and then by Hölder’s inequality, we obtainUsing Lipschitz condition (26), we have thatThen, by setting and by using Gronwall’s inequality, one gets the following estimate:This verifies that the solution of (6) depends continuously on the given data. Therefore, the proof is completed.
As stated in Theorem 2, problem (6) is well posed with the condition . In the following theorem, we shall extend the restriction of from the subinterval to the interval . To achieve our aim, we replace the global Lipschitz condition in Theorem 2 with a new Lipschitz condition in the next theorem.
Theorem 3. Let , and let satisfy the following Lipschitz condition:where is fixed, , is a constant, and . Then, regularized problem (27) is well posed in the sense of Hadamard.
Proof. We follow the ideas in the proof of Theorem 2. We reconsider the operator given by (27). By Hölder’s inequality, Lipschitz condition (47), and from (28), we have that, for and ,where is fixed, , and . With the same calculation and using the mathematical induction method, one also getsIndeed, if we assume that (49) holds for , then by using Lemma 1, the Hölder’s inequality, and Lipschitz condition (47), for , we have thatwhich is inequality (49) for . Thus, inequality (49) is satisfied for all . Furthermore, inequality (49) yields thatHence, where is given and is fixed. So, there exists such that is a contraction. Thus, problem (6) has a unique solution in the case of . Finally, in order to show that the solution of problem (6) depends continuously on , we assume that and are two solutions of problem (6) corresponding to the final values and , respectively. Similar to Step 2 in the proof of Theorem 2, by using Lipschitz condition (5), we also obtainBy putting and and by Gronwall’s inequality, we getThis yieldsTherefore, we can conclude that the solution of (6) depends continuously on the data . The proof is completed.
3.2. Error Estimates
The following theorem presents the error estimate in the case of exact data between the solution of (1)–(3) with the conditional stable and the regularized solution of (6) without the conditional stable.
Theorem 4. Let be given, and satisfy the conditions of Theorem 3. Suppose that the unique solution of (1)–(3) satisfies the conditional stable , then we obtain the following estimate:where , is fixed, , , , and is a unique solution of regularized problem (6).
Proof. Assume that is a unique solution of (1)–(3), and then based on Theorem 1, one observes that is represented bywhereMultiplying both sides of (58) with , one getswhere and are denoted as in the proof of Theorem 2. Moreover, from (6), we haveUsing the inequality and the inequality , where , one has thatLet . By employing Hölder’s inequality and then from Lemma 1 and (61), one obtainsLetwhere is fixed. It is clear thatfor all . Therefore, for , one getsHence,It follows from (62) thatThen, by using Lipschitz condition (47), we get the following estimate:By putting and , we get the inequality as follows:Using Gronwall’s inequality, we haveThis infers thatBased on the results of Theorems 3 and 4, the following theorem presents the error estimate in the case of nonexact data between the solution of (1)–(3) with the conditional stable and regularized solution (6) without the conditional stable.
Theorem 5. Let , where is a positive parameter, and let satisfy condition (47). Assume that is a unique solution of (1)–(3) with the data which satisfies the conditional stable as in Theorem 4. In addition, let be the measured data which satisfies
Then, there is a function corresponding to the measured data such thatwhere and is given as in Theorem 4.
Proof. Denote by and the solutions of (6) with the final values and , respectively. We observe thatBased on estimates (55) and (56) of Theorems 3 and 4, we have thatIt follows from (72) that, for every ,where . The proof is complete.
4. Numerical Illustration
Consider the following linear backward heat problem involving the conformable derivative:
It is not difficult to check that is an exact solution of (77)–(79). It is well known that errors always occur during measurement, and so in this section, we assume the measured data are perturbed by a “noise” with the level as follows:where and is a sequence of random numbers with mean 0. Basing on regularized problem (6), we obtain the regularized solution corresponding to the data :
Using the concept of the relative error given bywe shall present the difference between the exact solution and the approximate solution for each . Now, the following situations are considered: Situation 1. We focus on the regularization parameter . In our numerical example, we take , and let , , and . The graphs of the solution and the regularized solution are shown in Figures 1–4 with various values of . In addition, we also obtain the error of the proposed method in this paper with random measured data (80) given in Table 1 at . Furthermore, let , then the graphs of the regularized solution and the exact solution are shown in Figure 5 with the various values of .
Remark 2. From Figures 1–4 and Table 1, we observe that the smaller the measurement error is, the closer the regularized solution becomes to the exact solution. Situation 2. We focus on the parameter . In our numerical example, we take fixed and let . We consider the values of : . The graphs of the regularized solution in this case are shown in Figures 6 and 7. Furthermore, let , then the graphs of the regularized solution and the exact solution are shown in Figure 8 with the various values of .
Remark 3. According to Figures 6–8, the regularized solution will be closer to the exact one with a higher value of . It is very useful if we want to obtain a better approximation while the measurement process cannot be improved.
5. Conclusion
In this paper, we have discussed the modified integral equation method involving two regularization parameters for the backward heat problem with a conformable derivative. We have also established error estimates between exact and regularized solutions in the cases of exact data and inexact data. These estimates are supported by several numerical examples.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors would like to thank Dr. Tra Quoc Khanh for his help in doing numerical examples.