Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

IMPACT FACTOR 2018: 0.726
5-year IMPACT FACTOR: 0.869

CiteScore 2018: 0.90

SCImago Journal Rank (SJR) 2018: 0.323
Source Normalized Impact per Paper (SNIP) 2018: 0.821

Mathematical Citation Quotient (MCQ) 2018: 0.34

ICV 2017: 161.82

Open Access
See all formats and pricing
More options …
Volume 16, Issue 1


Volume 13 (2015)

Regularization and error estimates for an inverse heat problem under the conformable derivative

Ho Vu
  • Faculty of Mathematical Economics, Banking University of Ho Chi Minh City, Ho Chi Minh City, Vietnam
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Donal O’Regan
  • School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Ngo Van Hoa
  • Corresponding author
  • Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam
  • Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2018-08-24 | DOI: https://doi.org/10.1515/math-2018-0084


In this paper we study an inverse time problem for the nonhomogeneous heat equation under the conformable derivative which is a severely ill-posed problem. Using the quasi-boundary value method with two regularization parameters (one related to the error in a measurement process and the other is related to the regularity of the solution) we regularize this problem and obtain a Hölder-type estimation error for the whole time interval. Numerical results are presented to illustrate the accuracy and efficiency of the method.

Keywords: Conformable derivative; Inverse heat problem; Ill-posed problem; Nonhomogeneous heat; Two regularization parameters

MSC 2010: 35R30; 35R35; 58J35

1 Introduction

Partial differential equations (PDEs) arise in the natural sciences, and various boundary value problems for these were widely studied including inverse and ill-posed problems (see, e.g., Tikhonov and Arsenin [1] and Glasko [2]). An example is the backward heat conduction problem (BHCP) and the aim is to detect the previous status of a physical area from present information. The BHCP is a classical ill-posed problem that is difficult to solve since, in general, 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 and numerical calculations are difficult. The BHCP has been considered by many authors using different methods [3], [4], [5], [6], [7], [8], [9], [10]. In [5], Hao, Duc and Lesnic gave an approximation for this problem using a non-local boundary value problem method, Hao and Duc in [6] used the Tikhonov regularization method to give an approximation for this problem in a Banach space, and Trong and Tuan in [11] used the method of integral equations to regularize the BHCP with a nonlinear right hand side.

Fractional calculus arises in many areas in science and engineering such as aerodynamics and control systems, signal processing, bioengineering and biomedical, viscoelasticity, finance and plasma physics, etc. (see [12], [13], [14]). For basic information and results we refer the reader to the monographs of Samko et al. [15], Podlubny [16] and Kilbas et al. [17]. Mathematical modeling of many real world phenomena based on definitions of fractional order integrals and derivatives is regarded as more appropriate than ones depending on integer order operators, so as a result fractional differential equations and fractional partial differential equations are important fields of research [18], [19], [20], [21]. In the above works the definition of the fractional used is either the Riemann-Liouville or the Caputo fractional derivative and most works use an integral form for the fractional derivative. Many researchers are interested in the time-inverse problem for the heat equation where the time-derivative is in the Caputo fractional sense. In particular, they consider the problem


where α ∈ (0, 1) is the fractional order of derivative and


By a time-inverse problem, we mean that, given information at a specific point of time, say t = T, the goal is to recover the corresponding structure at an earlier time t < T. When α = 1, the problem (1) turns back to the classical ill-posed problem for the well-known heat equation (BCHP). Many researchers have applied different methods to regularize this problem. For example, in [9], [10] the authors successfully applied various methods to stabilize BCHP and obtained many results on the convergent of the regularized solution to the exact one. In [7], [8], the authors consider BCHP where the frequency domain is ℝ. Problem (1) with 0 < α < 1 was studied in [22], [23], [24] where fundamental contributions were made for problem (1) on existence and uniqueness of solution for this problem. In [25], the authors simplified the Tikhonov regularization method to stabilize problem (1). In [26] the authors consider problem (1) where the data is discrete.

However, there are some setbacks in the approaches of the Riemann-Liouville fractional and the Caputo fractional derivative when modeling real world phenomena (see [27] for a discussion). In [27] the authors gave a new well-behaved simple fractional derivative called “the conformable derivative” depending just on the basic limit definition of the derivative and this concept seems to satisfy all the requirements of the standard derivative. For a function u: (0, ∞) → ℝ the conformable derivative of order α ∈ (0, 1) of u at t > 0 is defined by


Note that if u is differentiable, then Dtα u (t) = t1–α u′(t), where u′(t) = limh → 0 [u(t + h) – u(t)]/ h. This concept overcomes the setbacks of the previous concept and this new theory is discussed by Atangana [28] and Abdeljawad [29]. In addition, Anderson and Ulness in [30] provide a potential application of the conformable derivative in quantum mechanics.

For PDEs concerning the conformable derivative there are several studies. In [31], Hammad and Khalil used conformable fourier series to interpret the solution for the conformable heat equation, which is a fundamental equation in mathematical physics. In [32], Chung used the conformable fractional derivative and integral to study fractional Newtonian mechanics, and in addition, the fractional Eule-Lagrange equation was constructed. In [33], Eslami applied the Kudryashov method to obtain the traveling wave solutions to the conformable fractional coupled nonlinear Schrodinger equation. In [34, 35], Çenesiz et al. studied the conformable version of the time-fractional Burgers’ equation, the modified Burgers’s equation, the Burgers-Korteweg-de Vries equation and the solutions of conformable derivative heat equation. Çenesiz, Kurt and Nane in [36] studied stochastic solutions of conformable fractional Cauchy problems where the space operators may correspond to fractional Brownian motion, or a Levy process. Motivated by the above studies, it is natural to consider the time-inverse problem for the heat equation under the conformable derivative. Throughout this paper, we let Ω = [0, a], T is a positive number and Dtα is the conformable derivative of order α with respect to t. We begin with the inverse problem in the conformable heat equation.

1.1 The direct problem

Consider the following conformable heat equation




Solving this equation with the given information f(x, t) and u0(x) is called the direct problem.

1.2 The inverse problem

Consider the following conformable heat equation




where f(x, t) ∈ C(0, T;L2(Ω)) and g(x) ∈ L2(Ω). From the information given at final time t = T, the goal of the inverse problem is to recover the information u(x, t) for 0 ≤ t < T. Unfortunately, the inverse problem is usually an ill-posed problem in the sense of Hadamard. An ill-posed problem in the sense of Hadamard is the one which violates at least one of the following conditions:

  • Existence: There exists a solution of the problem.

  • Uniqueness: The solution must be unique.

  • 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.

Problems which satisfy these conditions are called well-posed problems. We will show that the conformable backward heat problem is an ill-posed problem.

First, let us make clear what a solution of the Problem (6) - (8) is. We call a function uC2, 1((0, a) × (0, T);L2 (Ω)) a solution for Problem (6) - (8) if


for all functions wL2(Ω). In fact, it is enough to choose w in the orthogonal basis sinnπaxn=1 and then (9) reduces to


and as a result, the solution of (6) - (8) can be represented by


where kn = (n π/a)2 and


It is noted that the term eknTαtαα tends to infinity as n tends to infinity. Hence, it causes instability in the solution.

In this paper, we will apply the quasi-boundary value method with a small modification to regularize (6) - (8). In fact, rather than using the original information, we will consider problem (6) - (8) with adjusted information so that the adjusted problem is well-posed and approximates the original one. Consider the following problem






Lemma 1.1

Let 0 ≤ tT, τ > 0, ε ∈ 𝔻:= (0,(Tα+τ)α) and x > 0. Then, for α ∈ (0, 1) the following inequality holds



For any ε ∈ 𝔻, x > 0, α ∈ (0, 1] and T > 0, the function


maximizes at x = ln(Tα+ταε)/(Tα+τα). Therefore,


Then, we obtain the following estimation


The proof is complete. □

The rest of the paper is organized as follows. In Section 2, we study the well-posedness of problem (12) - (14) and provide an error estimation between solutions of these two problems. Section 3 provides a numerical example to illustrate the efficiency of our method.

2 Well-posedness of the regularized problem (12)-(14)

Theorem 2.1

Let f(x, t) ∈ C(0, T;L2(Ω)) and g(x)∈ L2(Ω). Let τ > 0 and ε ∈ 𝔻 be given. Then, (12)(14) has a unique solution uε, τ satisfying


The solution depends continuously on g in L2(Ω).


First we prove the existence and uniqueness of a solution of the regularized problem (12) - (14).

Existence of solution

For all 0 ≤ tT, we have




It follows that


On the other hand, we have


Hence, u ε, τ is the solution of the regularized problem (12)(14), so the existence of a solution of the regularized problem (12)(14) is proved.

Uniqueness of solution

Let uε, τ(x, t) and vε, τ(x, t) be two solutions of (12) - (14). We denote w(x, t) = uε, τ(x, t) - vε, τ(x, t). It is clear that w(x, T) = 0. We expand w n(x, t) = n=1wn(t)sinnπax with the coefficient


Multiply both sides of equation (12) by sin nπax and integrate by parts with respect to x, and use the boundary condition to obtain


where kn = (n π /a)2 and


The condition w(x, T) = 0 yields wn(T) = 0. Then, the well-posedness for the fractional differential equation (21) with the boundary condition w n(T) = 0 yields wn(t) ≡ 0 in t ∈ [0, T]. This infers that w(x, t) ≡ 0.

Stability of solution

The solution of the problem (12)(14) depends continuously on g. In fact, let uε, τ and vε, τ be two solutions of (12)(14) corresponding to the final data gε, τ and hε, τ, and uε, τ and vε, τ are represented by





Direct computation leads us to


Applying Lemma 1.1 directly, we get




The proof is complete. □

We have shown that the regularization problem (12)(14) is a well-posed problem in the sense of Hadamard. Now, the main goal of the coming theorem is to provide an error estimation between the regularization solution and the exact solution.

Theorem 2.2

Let g, f, uε, τ as in Theorem 2.1 and assume that problem (6)(8) has a solution uC2, 1( (0, a)×(0, T); L2(Ω))and n=10Tknsα1fn(s)eknsααds2 < ∞, where kn = (n π/a)2. Suppose that the problem (6)(8) has uniquely a solution u such thatu(⋅, 0) ‖ < ∞. Then the following estimate holds for all 0 < tT,




and uε, τ is the unique solution of problem (12)(14).


The exact solution satisfies


On the other hand, in terms of un(0), we have




It follows that


Combining (18) and (27), we get


From the inequality (a + b)2 ≤ 2(a2 + b2), we have


From (31) and Parseval’s identity u(,t)uε,τ(,t)2=a2n=1un(t)unε,τ(t)2, we get




This completes the proof of Theorem 2.2. □

Theorem 2.3

(Error estimates in case of non-exact data) Let f, g as in Theorem 2.1. Let τ ≥ 0 and ε ∈ 𝔻 be given. Assume u is the unique solution of problem (6)(8) corresponding to the exact data g. Suppose that gε, τ is measured data such that


Then there exists an approximate solution Uε, τ, which links to the noisy data gε, τ, satisfying



Let Uε, τ be the solution of the regularized problem (12)(14) corresponding to data gε, τ and let uε, τ be the solution of the problem (12)(14) corresponding to the data g. Let u(x, t) be the exact solution, and in view of the triangle inequality, one has


Combining the results from Theorem 2.1 (see the proof) and Theorem 2.2, for every t ∈ [0, T], we get


The proof is complete. □

3 Numerical illustration

In this section, we illustrate the theoretical results in Section 2 through an example. Consider the space domain Ω = [0, a] in association with the final time T, and our problem is




where g(x) = eTααsinπax. Under the above assumptions, the exact solution of the problem is


Now, due to the error in the measuring process, the measured data is perturbed by a “noise” with level ε, i.e


where P0 is a natural number and cp is a finite sequence of random normal numbers with mean 0 and variance A2. It follows that the error in the measurement process is bounded by ε, ‖gnoiseg ‖ ≤ R ε where R is some positive number. The error between the measured data and the exact data will tend to 0 as ε tends to 0 . Regarding (18), the regularized solution corresponding to the measured data takes the following form


Let a = 5, P0 = 1000, A2 = 100. Consider the following situation:

  • Situation 1

    In this situation, the regularization parameter ε will be discussed. Fix α = 0.3, τ = 0.5. Consider ε1 = 10–1, ε2 = 10–3, ε3 = 10–5. We have the following figures:

For each point of time we evaluate the “Relative error” between the exact solution and the regularized solution which is defined by


The relative error is a better representation of the difference between the exact and the approximate solution. When the value of the exact solution is large, the difference between the exact and the approximate solution does not tell us much information about the accuracy of the approximation. In this case, the relative error is a better measurement. Figure 3 shows errors for a comparison between the exact solution and the regularized solution at the initial time t0 = 0 and τ = 1 with various values of ε. In Table 1, we have the error table at time time t* = 0.1.

Table 1

The error and Relative error at time t* = 0.1.

Remark 3.1

From Figure 1, Figure 2, Figure 3, Figure 4 and Table 1, it is clear that as the mearsuring error ε gets smaller, the regularized solution gets closer and closer to the exact one. It is also noted that in this situation, the noise parameter cp varies from -249.689 to 242.4461.

The exact solution with α = 0.3 (a) and regularized solution with ε1 = 10–1 (b)
Fig. 1

The exact solution with α = 0.3 (a) and regularized solution with ε1 = 10–1 (b)

The regularized solution with ε2 = 10–3 (a) and with ε3 = 10–5 (b)
Fig. 2

The regularized solution with ε2 = 10–3 (a) and with ε3 = 10–5 (b)

At time t0 = 0 and τ = 1: Exact solution with α = 0.9 (black) and Regularized solution with ε1 = 10–1 (blue), ε3 = 10–3 (green), ε5 = 10–5 (red)
Fig. 3

At time t0 = 0 and τ = 1: Exact solution with α = 0.9 (black) and Regularized solution with ε1 = 10–1 (blue), ε3 = 10–3 (green), ε5 = 10–5 (red)

The exact solution with α = 0.1 (a) and regularized solution with ε1 = 10–1 (b)
Fig. 4

The exact solution with α = 0.1 (a) and regularized solution with ε1 = 10–1 (b)

The regularized solution with ε2 = 10–3 (a) and with ε3 = 10–5 (b)
Fig. 5

The regularized solution with ε2 = 10–3 (a) and with ε3 = 10–5 (b)

  • Situation 2

    In this situation, the focusing parameter is τ. Let α = 0.1 and fix ε = 10–3. Consider the series of τ: τ1 = 0.3, τ2 = 0.5, τ3 = 1. We have Figure 6 to illustrate our theoretical results. It is also noted that in this situation, the noise parameter cp varies from -220.5152 to 352.6678.

Case α = 0.5, At time t = 0 and ε1 = 10–3: Exact solution (black) and Regularized solution with τ1 = 0 (blue), τ2 = 0.3 (green), τ3 = 1 (red)
Fig. 6

Case α = 0.5, At time t = 0 and ε1 = 10–3: Exact solution (black) and Regularized solution with τ1 = 0 (blue), τ2 = 0.3 (green), τ3 = 1 (red)

Remark 3.2

Figure 6 agrees with the theoretical result: the regularized solution with a higher value of τ is closer to the exact one. The parameter τ is very useful if we want to get a more accurate approximation if the measuring process cannot be improved or if the cost of measuring better is very expensive. In this case, with the appearance of τ, the error can be improved without any extra cost on measuring (as we can see in Figure 6).

4 Conclusion

In this paper, we have stated and discussed the quasi-boundary value regularization method for the inverse problem in the heat equation under the conformable derivative. In addition, we have also established an error estimate between exact and regularized solutions. These estimates are supported by several numerical examples. The estimate is a Hölder-type estimate (εt/T) for all values of t in the interval (0, T]. However, at the initial time t = 0, the error estimate is of logarithm type only. In the future, we hope to improve the error estimate as well as to consider the nonlinear case of f.


The authors are very grateful to the referees for their valuable suggestions, which helped to improve the paper significantly. The third author would like to thank Professor Dang Duc Trong, Associate Professor Nguyen Huy Tuan and Tra Quoc Khanh for their support.


  • [1]

    Tikhonov A.N., Arsenin V.Y., Solutions of ill-posed problems, Vh Winston, 1977. Google Scholar

  • [2]

    Glasko V. B., Inverse problems of mathematical physics, New York: American Institute of Physics, 1984. Google Scholar

  • [3]

    Denche M., Bessila K., A modified quasi-boundary value method for ill-posed problems, J. of Math. Anal. and Appl., 2005, 301 419-426. CrossrefGoogle Scholar

  • [4]

    Hao D.N., Duc N.V., Stability results for the heat equation backward in time, J. of Math. Anal. and Appl., 2009, 353, 627-641. CrossrefWeb of ScienceGoogle Scholar

  • [5]

    Hao D.N., Duc N.V., D. Lesnic, Regularization of parabolic equations backward in time by a non-local boundary value problem method, IMA J. of Appl. Math., 2010, 75, 291-315. CrossrefGoogle Scholar

  • [6]

    Hao D.N., Duc N.V., Regularization of backward parabolic equations in Banach spaces, J. of Inverse. and Ill-posed Probs., 2012, 20, 745-763. Google Scholar

  • [7]

    Fu C.L., Qian Z., Shi R., A modified method for a backward heat conduction problem, Appl. Math. and Comput., 2007, 185, 564-573. Web of ScienceGoogle Scholar

  • [8]

    Fu C.L., Xiong X.T., Qian Z., Fourier regularization for a backward heat equation, J. Math. Anal. Appl., 2007, 331, 472-480. CrossrefWeb of ScienceGoogle Scholar

  • [9]

    Trong D.D., Quan P.H., Khanh T.V., Tuan N.H., A nonlinear case of the 1-D backward heat problem: Regularization and error estimate, Zeitschrift Analysis und ihre Anwendungen, 2007, 26, 231-245. Google Scholar

  • [10]

    Trong D.D., Tuan N.H., Regularization and error estimate for the nonlinear backward heat problem using a method of integral equation, Nonlin. Anal.: Theo., Meth. and Appl., 2009, 71, 4167-4176. CrossrefGoogle Scholar

  • [11]

    Trong D.D., Tuan N.H., Regularization and error estimate for the nonlinear backward heat problem using a method of integral equation, Nonlin. Anal.: Theo., Meth. and Appl., 2009, 71, 4167-4176. CrossrefGoogle Scholar

  • [12]

    Magin R., Fractional calculus models of complex dynamics in biological tissues, Computers & Mathematics with Applications, 2010, 59, 1586-1593. Web of ScienceCrossrefGoogle Scholar

  • [13]

    Magin R., Ortigueira M., Podlubny I., Trujillo J.J., On the fractional signals and systems, Signal Processing, 2011, 91, 350-371. Web of ScienceCrossrefGoogle Scholar

  • [14]

    Merala F.C., Roystona T.J., Magin R., Fractional calculus in viscoelasticity: An experimental study, Communications in Nonlinear Science and Numerical Simulation, 2010, 15, 939-945. Web of ScienceCrossrefGoogle Scholar

  • [15]

    Samko S.G., Kilbas A.A., Marichev O.I., Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Switzerland, 1993. Google Scholar

  • [16]

    Podlubny I., Fractional differential equation, San Diego: Academic Press, 1999. Google Scholar

  • [17]

    Kilbas A.A., Srivastava H.M., Trujillo J.J., Theory and applications of fractional differential equations, Amesterdam: Elsevier Science B.V, 2006. Google Scholar

  • [18]

    Atangana A., Baleanu D., New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model, Thermal Science, 2016, 20, 763-769. CrossrefGoogle Scholar

  • [19]

    Atangana A., Koca I., Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, Chaos, Solitons & Fractals, 2016, 89, 447-454. Web of ScienceCrossrefGoogle Scholar

  • [20]

    Atangana A., Alkahtani B.S.T, Modeling the spread of Rubella disease using the concept of with local derivative with fractional parameter, Complexity, 21, 2015, 442-451. Google Scholar

  • [21]

    Atangana A., Alqahtani R.T., Delling the spread of river blindness disease via the caputo fractional derivative and the beta-derivative, Entropy, 18, 2016, 40.  CrossrefGoogle Scholar

  • [22]

    Liu J.J., Yamamoto M., A backward problem for the time-fractional diffusion equation, Appl. Anal., 2010, 89, 1769-1788. CrossrefWeb of ScienceGoogle Scholar

  • [23]

    Kenichi S., Yamamoto M., Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problem, Journal of Mathematical Analysis and Applications, 2011, 382 426-447. CrossrefGoogle Scholar

  • [24]

    Luchko Y., Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation., Comput. Math. Appl., 2010, 59 1766-1772. CrossrefWeb of ScienceGoogle Scholar

  • [25]

    Jun-Gang W., Wei T., Zhou Y., Optimal error bound and simplified Tikhonov regularization method for a backward problem for the time-fractional diffusion equation, Journal of Computational and Applied Mathematics, 2015, 279, 277-292. Web of ScienceCrossrefGoogle Scholar

  • [26]

    Tuan N.H., Nane E., Inverse source problem for time-fractional diffusion with discrete random noise, Statistics and Probability Letters, 2017, 120, 126-134. CrossrefGoogle Scholar

  • [27]

    Khalil R., Al Horani M., Yousef A., Sababheh M., A new Definition Of Fractional Derivative, J. Comput. Appl. Math., 2014, 264, 65-70 CrossrefWeb of ScienceGoogle Scholar

  • [28]

    Atangana A., Baleanu D., Alsaedi A., New properties of conformable derivative, Open Mathematics, 2015, 13, 889-898. Web of ScienceGoogle Scholar

  • [29]

    Abdeljawad T., On conformable fractional calculus, J. Comput. Appl. Math., 2014, 279, 57-66. Web of ScienceGoogle Scholar

  • [30]

    Anderson D.R., Ulness D.J., Properties of the Katugampola fractional derivative with potential application in quantum mechanics, Journal of Mathematical Physics, 2015, 56, 063-502 Web of ScienceGoogle Scholar

  • [31]

    Abu Hammad I., Khalil R.. Fractional Fourier Series with Applications, American Journal of Computational and Applied Mathematics, 2014, 4, 187-191. Google Scholar

  • [32]

    Chung W.S., Fractional Newton mechanics with conformable fractional derivative, Journal of Computational and Applied Mathematics, 2015, 290, 150-158. Web of ScienceCrossrefGoogle Scholar

  • [33]

    Eslami M., Exact traveling wave solutions to the fractional coupled nonlinear Schrodinger equations, Applied Mathematics and Computation, 2016, 285, 141-148. Web of ScienceCrossrefGoogle Scholar

  • [34]

    Çenesiz Y., Baleanu D., Kurt A., Tasbozan O., New exact solutions of Burgers’ type equations with conformable derivative, Waves in Random and Complex Media, 2016, 1-14. Web of ScienceGoogle Scholar

  • [35]

    Çenesiz Y., Kurt A., The solutions of time and space conformable fractional heat equations with conformable Fourier transform, Acta Universitatis Sapientiae, Mathematica, 2015, 7, 130-140. CrossrefWeb of ScienceGoogle Scholar

  • [36]

    Çenesiz Y., Kurt A., Nane E., Stochastic solutions of conformable fractional Cauchy problems, . arXiv preprint arXiv:1606.07010. Google Scholar

About the article

Received: 2017-08-18

Accepted: 2018-07-03

Published Online: 2018-08-24

Competing interests: The authors declare that they have no competing interests.

Authors’ contributions: All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 999–1011, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2018-0084.

Export Citation

© 2018 Vu et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

Comments (0)

Please log in or register to comment.
Log in