Volume 2022, Issue 1 8008838

Research Article

Open Access

Regularization of Inverse Initial Problem for Conformable Pseudo-Parabolic Equation with Inhomogeneous Term

L. D. Long

orcid.org/0000-0001-8805-4588

Division of Applied Mathematics, Science and Technology Advanced Institute, Van Lang University, Ho Chi Minh City, Vietnam vanlanguni.edu.vn

Faculty of Applied Technology, School of Engineering and Technology, Van Lang University, Ho Chi Minh City, Vietnam vanlanguni.edu.vn

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Reza Saadati,

Corresponding Author

Reza Saadati

[email protected]

orcid.org/0000-0002-6770-6951

School of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran iust.ac.ir

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L. D. Long,

L. D. Long

orcid.org/0000-0001-8805-4588

Division of Applied Mathematics, Science and Technology Advanced Institute, Van Lang University, Ho Chi Minh City, Vietnam vanlanguni.edu.vn

Faculty of Applied Technology, School of Engineering and Technology, Van Lang University, Ho Chi Minh City, Vietnam vanlanguni.edu.vn

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Reza Saadati,

Corresponding Author

Reza Saadati

[email protected]

orcid.org/0000-0002-6770-6951

School of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran iust.ac.ir

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First published: 08 August 2022

https://doi.org/10.1155/2022/8008838

Citations: 1

Academic Editor: Yusuf Gurefe

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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 F(x, t) and F(x, t) = φ(t)f(x). 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 s > 1, we consider the initial value problem for the conformable heat equation (or called parabolic equation with conformable operator)

(1)

Here,

(N ≥ 1) is a bounded domain with the smooth boundary

, and T > 0 is a given positive number. Here,

is called the conformable time derivative with order β ∈ (0, 1) (Khalil et al. [9]) for a given function f : [0, ∞)⟶ℝ; the

of order β ∈ (0, 1] is defined by

(2)

for all t > 0. For some (0, t₀), t₀ > 0 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 y(x, t), in the fact that the couple (y_T, F) are noised by the measurement data (y_T,δ, F_δ) such that:

(3)

(ii)
The second part of the paper deals with the final value problem for Problem (1) with F is a linear source function as follows: F(x, t) = Φ(t)f(x), where both functions (Φ, f, g) are perturbed by (Φ_δ, f_δ, g_δ) in , respectively

(4)

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 p ≠ 2, 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 p ≠ 2. 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 F = 0. After that, we solve two problems in the case of observed data in space.

2. Preliminary Results

Let us recall that the spectral problem

(5)

admits the eigenvalues 0 < λ₁ ≤ λ₂ ≤ ⋯≤λ_m ≤ ⋯ with λ_m⟶∞ as m⟶∞. The corresponding eigenfunctions are

Definition 1 (Hilbert scale space). We recall the Hilbert scale space, which is given as follows:

(6)

for any n ≥ 0. It is well-known that is a Hilbert space corresponding to the norm

(7)

Lemma 2 (See [22]). The following statement is true:

(8)

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 e_m gives

(9)

The first equation of (9) is a differential equation with a conformable derivative as follows:

(10)

Because of the result in [23], the solution of Problem (1) is

(11)

Letting t = T, we follow from (11) that

(12)

From (12), we have

(13)

Substituting (13) into (12), we obtain

(14)

This leads to

(15)

4. The Mild Solution of Backward Problem in Case F = 0

In this section, we investigate the existence and regularity of mild solutions of Problem (1). Firstly, we consider the following initial value problem

(16)

According to (15), in this case, we have

(17)

4.1. The Ill-Posedness of Problem (1)

In order to prove that the solution to the backward problem is unstable atF(x, t) = 0, let us take the perturbed final data

, by choosing

. For s > 3/2, let us choose input final data y_T(x) = 0; we know that an error in

norm between two input final data as follows:

(18)

Therefore, we obtain

(19)

First of all, we have

, and ((T^β − t^β)/β) ≥ 0; this implies that

(20)

Next, using the inequality exp(x) ≥ x, for x > 0, this leads to:

(21)

For s > 3/2, and from (21), we get

(22)

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

(23)

By applying the Fourier truncation method, we have its approximation

(24)

Here, is parameter regularization which is defined later.

Theorem 3. For s > 1, taking for any 0 ≤ t ≤ T for any 1/β < p < 2, assume that (y_T, F) is observed by the couple (y_T,δ, F_δ) such that

(25)

Let us assume that for σ > 0 and 0 < n < N/4. With such that

(26)

Then, the error estimate

(27)

Remark 4. One choice for such that

(28)

then

Proof. Let

(29)

It is clear that

(30)

We continue to consider the two components of the right hand side.

Step 1:

(31)

For s > 1, it is easy to see that . The first term on is bounded by

(32)

Since the Sobolev space embedding , we have

(33)

This follows from (32) that

(34)

The second term is estimated as follows:

(35)

By the same arguments as above, we find that

(36)

We can see that and . From (36), using Holder’s inequality, we get

(37)

This latter inequality together with Sobolev embedding gives us

(38)

Combining (36) and (38), we get

(39)

Taking the square root on the both sides, we have

(40)

From (34) and (40), we deduce that

(41)

Step 2: Estimate of .

From the definition (23) and (29), we have

(42)

Therefore, we get

(43)

Combining two steps and noting that , we deduce that

(44)

The proof of Theorem 3 is completed. In the following theorem, we give a regularization result in the case that F has a split form F(x, t) = Φ(t)f(x).

Theorem 5. For s > 1, let us assume that the input data Φ_δ, g_δ, f_δ such that

(45)

Assume that for any σ > 0, then we construct a regularized solution defined by

(46)

Then, the error is of order

Remark 6. , then the error

(47)

Proof. Since F(x, t) = Φ(t)f(x), we know that

(48)

The triangle inequality allows us to obtain that

(49)

Next, we will evaluate the right side of (49), by the same way as demonstrated in (42),

(50)

It is easy to see that

(51)

whereby

(52)

We will divide this review into several steps as follows:

Step 1: Estimate of , we obtain that

(53)

Step 2: Due to Parseval’s equality, the term can be bounded as follows:

(54)

Thank to Holder’s inequality, we derive that for p > 1 and p^∗ = 1 + (1/(p − 1)), one has

(55)

The latter inequality leads to

(56)

where s > 1/β. In view of Sobolev embedding

, we derive that the following estimate

(57)

where

(58)

Step 3: Let us now to consider the term . By applying Hölder’s inequality, we get that for p > 1 and p^∗ = 1/(p − 1)·

(59)

This inequality together with Parseval’s equality allows us to derive that

(60)

By the fact that , we deduce that

(61)

Combining Step 1 to Step 3, we get

(62)

Finally, combining the reviews from above, we conclude that

(63)

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.

Open Research

Data Availability

No data were used to support this study.

References

1 Alharbia F. M., Baleanu D., and Ebaid A., Physical properties of the projectile motion using the conformable derivative, Chinese Journal of Physics. (2019) 58, 18–28, https://doi.org/10.1016/j.cjph.2018.12.010, 2-s2.0-85061979994.
10.1016/j.cjph.2018.12.010
Web of Science® Google Scholar
2 Kilbas A. A., Marichev O. I., and Samko S. G., Fractional Integrals and Derivatives (Theory and Applications), 1993.
Google Scholar
3 Tuan N. H., Aghdam Y. E., Jafari H., and Mesgarani H., A novel numerical manner for two-dimensional space fractional diffusion equation arising in transport phenomena, Numerical Methods for Partial Differential Equations. (2021) 37, no. 2, 1397–1406, https://doi.org/10.1002/num.22586.
10.1002/num.22586
Web of Science® Google Scholar
4 Karapinar E., Binh H. D., Luc N. H., and Can N. H., On continuity of the fractional derivative of the time-fractional semilinear pseudo-parabolic systems, Adv. Difference Equ.(2021) 2021, no. 1, article 70, https://doi.org/10.1186/s13662-021-03232-z.
10.1186/s13662-021-03232-z
Web of Science® Google Scholar
5 Adigüzel R. S., Aksoy Ü., Karapinar E., and Erhan İ. M., On the solution of a boundary value problem associated with a fractional differential equation, Mathematical Methods in the Applied Sciences. (2020) https://doi.org/10.1002/mma.6652.
10.1002/mma.6652
Google Scholar
6 Sevinik-Adıgüzel R., Aksoy Ü., Karapınar E., and Erhan İ. M., Uniqueness of solution for higher-order nonlinear fractional differential equations with multi-point and integral boundary conditions, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas. (2021) 115, no. 3, article 155, https://doi.org/10.1007/s13398-021-01095-3.
10.1007/s13398-021-01095-3
Web of Science® Google Scholar
7 Adiguzel R. S., Aksoy U., Karapinar E., and Erhan I. M., On the solutions of fractional differential equations via Geraghty type hybrid contractions, Applied and Computational Mathematics. (2021) 20, no. 2, 313–333.
Web of Science® Google Scholar
8 Phuong N. D., Note on a Allen-Cahn equation with Caputo-Fabrizio derivative, Results in Nonlinear Analysis. (2021) 4, no. 3, 179–185, https://doi.org/10.53006/rna.962068.
10.53006/rna.962068
Google Scholar
9 Khalil R., Al Horani M., Yousef A., and Sababheh M., A new definition of fractional derivative, Journal of Computational and Applied Mathematics. (2014) 264, 65–70, https://doi.org/10.1016/j.cam.2014.01.002, 2-s2.0-84893186929.
10.1016/j.cam.2014.01.002
Web of Science® Google Scholar
10 Abdeljawad T., On conformable fractional calculus, Journal of Computational and Applied Mathematics. (2015) 279, 57–66, https://doi.org/10.1016/j.cam.2014.10.016, 2-s2.0-84911406110.
10.1016/j.cam.2014.10.016
Web of Science® Google Scholar
11 Jaiswal A. and Bahuguna D., Semilinear conformable fractional differential equations in Banach spaces, Differential Equations and Dynamical Systems. (2019) 27, no. 1-3, 313–325, https://doi.org/10.1007/s12591-018-0426-6, 2-s2.0-85061970248.
10.1007/s12591-018-0426-6
Web of Science® Google Scholar
12 Acan O., Al Qurashi M. M., and Baleanu D., New exact solution of generalized biological population model, The Journal of Nonlinear Sciences and Applications. (2017) 10, no. 7, 3916–3929, https://doi.org/10.22436/jnsa.010.07.44.
10.22436/jnsa.010.07.44
Google Scholar
13 Bouaouid M., Hilal K., and Melliani S., Nonlocal telegraph equation in frame of the conformable time-fractional derivative, Advances in Mathematical Physics. (2019) 2019, 7, 7528937, https://doi.org/10.1155/2019/7528937, 2-s2.0-85063143597.
10.1155/2019/7528937
Web of Science® Google Scholar
14 Nguyen H. T., Le D. L., and Nguyen V. T., Regularized solution of an inverse source problem for a time fractional diffusion equation, Applied Mathematical Modelling. (2016) 40, no. 19-20, 8244–8264, https://doi.org/10.1016/j.apm.2016.04.009, 2-s2.0-84975123166.
10.1016/j.apm.2016.04.009
Web of Science® Google Scholar
15 Qian A. and Li Y., Optimal error bound and generalized Tikhonov regularization for identifying an unknown source in the heat equation, Journal of Mathematical Chemistry. (2011) 49, no. 3, 765–775, https://doi.org/10.1007/s10910-010-9774-3, 2-s2.0-79651471606.
10.1007/s10910-010-9774-3
CAS Web of Science® Google Scholar
16 Yang S., Xiong X., and Nie Y., Iterated fractional Tikhonov regularization method for solving the spherically symmetric backward time-fractional diffusion equation, Applied Numerical Mathematics. (2021) 160, 217–241, https://doi.org/10.1016/j.apnum.2020.10.008.
10.1016/j.apnum.2020.10.008
Web of Science® Google Scholar
17 Yang F., Fu J.-L., Fan P., and Li X.-X., Fractional Landweber iterative regularization method for identifying the unknown source of the time-fractional diffusion problem, Acta Applicandae Mathematicae. (2021) 175, no. 1, https://doi.org/10.1007/s10440-021-00442-1.
10.1007/s10440-021-00442-1
PubMed Web of Science® Google Scholar
18 Phuong N. D., Luc N. H., and Long L. D., Modified quasi boundary value method for inverse source problem of the bi-parabolic equation, Advances in Theory of Nonlinear Analysis and its Applications. (2020) 4, no. 3, 132–142.
Google Scholar
19 Wei T. and Wang J., A modified quasi-boundary value method for an inverse source problem of the time-fractional diffusion equation, Applied Numerical Mathematics. (2014) 78, 95–111, https://doi.org/10.1016/j.apnum.2013.12.002, 2-s2.0-84892589642.
10.1016/j.apnum.2013.12.002
Web of Science® Google Scholar
20 Yang F., Zhang P., and Li X.-X., The truncation method for the Cauchy problem of the inhomogeneous Helmholtz equation, Applicable Analysis. (2019) 98, no. 5, 991–1004, https://doi.org/10.1080/00036811.2017.1408080, 2-s2.0-85035761108.
10.1080/00036811.2017.1408080
Web of Science® Google Scholar
21 Tuan N. H., On some inverse problem for bi-parabolic equation with observed data in L^p spaces, Opuscula Mathematica. (2022) 42, no. 2, 305–335, https://doi.org/10.7494/OpMath.2022.42.2.305.
10.7494/OpMath.2022.42.2.305
Web of Science® Google Scholar
22 Tuan N. H. and Caraballo T., On initial and terminal value problems for fractional nonclassical diffusion equations, Proceedings of the American Mathematical Society. (2021) 149, no. 1, 143–161, https://doi.org/10.1090/proc/15131.
10.1090/proc/15131
Web of Science® Google Scholar
23 Tuan N. H., Ngoc T. B., Baleanu D., and O’Regan D., On well-posedness of the sub-diffusion equation with conformable derivative model, Communications in Nonlinear Science and Numerical Simulation. (2020) 89, article 105332, https://doi.org/10.1016/j.cnsns.2020.105332.
10.1016/j.cnsns.2020.105332
Web of Science® Google Scholar

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Regularization of Inverse Initial Problem for Conformable Pseudo-Parabolic Equation with Inhomogeneous Term

Abstract

1. Introduction

2. Preliminary Results

3. Regularization of Backward Problem

4. The Mild Solution of Backward Problem in Case F = 0

4.1. The Ill-Posedness of Problem (1)

4.2. Regularization of inverse Problem (1) in space

Conflicts of Interest

Authors’ Contributions

Acknowledgments

Open Research

Data Availability

References

Citing Literature

References

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Regularization of Inverse Initial Problem for Conformable Pseudo-Parabolic Equation with Inhomogeneous Term

Abstract

1. Introduction

2. Preliminary Results

3. Regularization of Backward Problem

4. The Mild Solution of Backward Problem in Case F = 0

4.1. The Ill-Posedness of Problem (1)

4.2. Regularization of inverse Problem (1) in space

Conflicts of Interest

Authors’ Contributions

Acknowledgments

Open Research

Data Availability

References

Citing Literature

References

Related

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