Two-Parameter RegularizationMethod for aNonlinear Backward Heat Problem with a Conformable Derivative

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.


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 timeinverse problem from the final data, the main goal is to reconstruct the whole structure in previous times. ese 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. e BHP has been considered in the literature using different methods (see [3][4][5][6][7][8][9][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]. e 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 Ω � [0, a], T is a positive number, the functions f(x, t) and g(x) are given, and D α t is the conformable derivative of order α with respect to t defined by From the information given at final time t � T, the goal of the problem is to recover the information u(x, t) for 0 ≤ t < T. 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. ere exists a solution of the problem. (2) Uniqueness. e solution must be unique.
(3) Stability. e 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 eorem 1 in Section 2, we see that the solution of problems (1)-(3) is given by where the terms in the above equation are given in eorem 1. We observe that exp(k n (b α − t α /α)) ⟶ n ⟶ ∞ ∞, so this yields an instability of the solution of problems (1)-(3). is 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 is fixed, and c 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, α ∈ (1/2, 1) and α ∈ (0, 1). 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 that where D is specified below, u(x, t) is a solution of problems (1)-(3), and u ε,c (x, t) is a solution of the regularized problem (6). In Section 4, we provide numerical tests to illustrate the theoretical results in the paper.
2 Complexity . en, the solution of the original problems (1)-(3) has the following form: where k n � (nπ/a) 2 and Proof. By choosing the orthogonal basis φ n (x) ≔ sin(nπx/a), n � 1, 2, . . ., in the Hilbert space L 2 (Ω) and by taking the inner product in L 2 (Ω) on the two sides of (1), we obtain On the other hand, by using boundary conditions (2), we also get en, it follows from (11)-(13) that Solving problem (14), we get Basing on (3), we have that where g n � u n (b). en, (15) yields that erefore, 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 n tends to infinity, the term exp(k n (b α − t α )/α) is increasing rather quickly. Hence, the exact solution given in (18) of problems (1)-(3) is unstable. us, 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 exp(k n (b α − t α )/α) is replaced by a stable term which depends on two regularization parameters defined by (εk n + e − k n ( where the first one (ε) captures the measuring error and the second one (c) captures the regularity of the solution. erefore, in this paper, we shall use integral equation (6) to approximate or to regularize problems (1)-(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.
, and z > 0. en, the following inequalities hold: Complexity and exp Proof. Let b > 0, ε ∈ D, c ≥ 0, and z > 0, and let α ∈ (0, 1) be fixed, then we observe that the function en, we obtain the following inequality: Furthermore, we have From (23), we obtain is yields estimate (20).  (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.
Proof. To prove the theorem, we shall divide the proof into two steps. e first step shows that regularized problem (6) has a unique solution u ε,c ∈ C([0, b]; L 2 (Ω)) provided that α ∈ ((1/2), 1). In the second step, the continuous dependence of the solution on the data g will be verified.

Error Estimates.
e following theorem presents the error estimate in the case of exact data between the solution u 8 Complexity of (1)-(3) with the conditional stable and the regularized solution u ε,c of (6) without the conditional stable.
Proof. Assume that u is a unique solution of (1)-(3), and then based on eorem 1, one observes that u is represented by where Multiplying both sides of (58) with where Λ 1 and Λ 2 are denoted as in the proof of eorem 2. Moreover, from (6), we have Using the inequality (z 1 + z 2 ) 2 ≤ 2z 2 1 + 2z 2 2 and the in- Let C n (c, t, b) ≔ k 2 n exp(2k n ((b α + c)/α))|u n (t)| 2 . By employing Hölder's inequality and then from Lemma 1 and (61), one obtains Let where α ∈ (0, 1) is fixed. It is clear that for all z ≥ εα. erefore, for ε ∈ (0, (b α + c)/α), one gets Hence, It follows from (62) that Remark 3. According to Figures 6-8, the regularized solution will be closer to the exact one with a higher value of c. It is very useful if we want to obtain a better approximation while the measurement process cannot be improved. The exact solution and the regularized solution Exact sol Reg. sol. ε = 10 -1 Reg. sol. ε = 10 -3 Reg. sol. ε = 10 -5 Figure 5: e graphs of u(t) and u ε,c (t).

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. ese estimates are supported by several numerical examples.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.   14 Complexity