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

Open Physics

formerly Central European Journal of Physics

Editor-in-Chief: Seidel, Sally

Managing Editor: Lesna-Szreter, Paulina


IMPACT FACTOR 2018: 1.005

CiteScore 2018: 1.01

SCImago Journal Rank (SJR) 2018: 0.237
Source Normalized Impact per Paper (SNIP) 2018: 0.541

ICV 2017: 162.45

Open Access
Online
ISSN
2391-5471
See all formats and pricing
More options …
Volume 15, Issue 1

Issues

Volume 13 (2015)

Analysis of a New Fractional Model for Damped Bergers’ Equation

Jagdev Singh / Devendra Kumar / Maysaa Al Qurashi / Dumitru Baleanu
  • Department of Mathematics, Faculty of Arts and Sciences, Cankaya University, Eskisehir Yolu 29. Km, Yukariyurtcu Mahallesi Mimar Sinan Caddesi No: 4 06790, Etimesgut, Turkey
  • Institute of Space Sciences, Magurele-Bucharest, Romania
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2017-03-12 | DOI: https://doi.org/10.1515/phys-2017-0005

Abstract

In this article, we present a fractional model of the damped Bergers’ equation associated with the Caputo-Fabrizio fractional derivative. The numerical solution is derived by using the concept of an iterative method. The stability of the applied method is proved by employing the postulate of fixed point. To demonstrate the effectiveness of the used fractional derivative and the iterative method, numerical results are given for distinct values of the order of the fractional derivative.

Keywords: Time-fractional damped Bergers’ equation; Nonlinear equation; Caputo-Fabrizio fractional derivative; Iterative method; Fixed-point theorem

PACS: 02.30.Jr; 02.30.Mv; 02.30.Uu; 05.45.-a

1 Introduction, Motivation and Preliminaries

Fractional calculus has been gaining more and more importance due to its extensive uses in the multiple aspects of science and engineering. Fractional derivatives and fractional integrals are key topics in fractional calculus and recently many related studies have been published by researchers and scientists [18]. Tarasov [9] investigated the 3 D lattice equations pertaining to long-range interactions of Grünwald-Letnikov kind for fractional extension of gradient elasticity. Choudhary et al. [10] studied the fractional order differential equations occurring in fluid dynamics. Bulut et al. [11] reported the analytical study of differential equations of arbitrary order. Razminia et al. [12] analyzed the fractional diffusivity equation considering wellbore storage and skin effects. Atangana and Koca [13] studied the new fractional Baggs and Freedman models. Atangana [14] presented a fractional model of a nonlinear Fisher’s reaction-diffusion equation. Singh et al. [15] discussed a fractional biological populations model. Singh et al. [16] reported the solution of fractional reaction-diffusion equations. Baleanu et al. [17] studied the fractional finite difference inclusion. In a recent work, Atangana et al. [18] analyzed the fractional Hunter-Saxton equation. Sequentially, Kurt et al. [19] derived the solution of Burgers’ equation, Tasbozan et al. [20] presented new solutions for conformable fractional Boussinesq and combined KdV-mKdV equations, Atangana and Baleanu [21] introduced a novel fractional derivative having nonlocal and non-singular kernel, Alsaedi et al. [22] studied the coupled systems of time-fractional differential problems, Coronel-Escamilla et al. [23] investigated the formulation of Euler-Lagrange and Hamilton equations, Gomez-Aguilar et al. [24] obtained the analytical solutions of the electrical RLC circuit, Gomez-Aguilar et al. [25] examined a fractional Lienard type model of a pipeline, Doungmo Goufo [26] studied the Korteweg-de Vries-Burgers equation involving the Caputo-Fabrizio fractional derivative without singular kernel and many others. Consequently, numerous distinct definitions of fractional derivatives and fractional integrals have been used in literature for example the Riemann-Liouville definition and the Caputo definition. In a recent work, Caputo and Fabrizio proposed a novel derivative of arbitrary order without singular kernel [1, 2]. This new Caputo is an improvement over the old version because the full outcome of the memory can be predicted. In this paper, it is asserted that the Caputo-Fabrizio fractional derivative has additional stimulus effects in comparison to the older derivative.

Definition 1

Assume that ϕH1 (a, b), b > a, β ∈ [0, 1] then the new fractional derivative discovered by Caputo and Fabrizio is explained and presented as: Dtβϕ(t)=M(β)1βatϕ(x)expβtx1βdx,(1)

In the above equation M(β) denotes the normalization function, which satisfies the property M(0) = M(1) = 1 [1]. The derivative can be reformulated as given below when ϕH1 (a, b) Dtβϕ(t)=βM(β)1βat(ϕ(t)ϕ(x))expβtx1βdx.(2)

The Eq. (2) takes the below form Dtβϕ(t)=N(σ)σatϕ(x)exptxσdx,N(0)=N()=1,(3) if σ=1ββ[0,),β=11+σ [0,1] as stated by the authors [14].

Moreover, Limσ01σexptxσ=δ(xt).(4)

The authors of [2] proposed the associated integral of the new Caputo derivative of arbitrary order as described in the following manner.

Definition 2

The fractional order integral of the function ϕ(t) of order β, 0 < β < 1 is expressed as [2] Iβtϕ(t)=2(1β)(2β)M(β)ϕ(t)+2β(2β)M(β)0tϕ(s)ds,t0.(5)

Eq. (5) further yields 2(1β)(2β)M(β)+2β(2β)M(β)=1.(6)

The above result gives an explicit formula for M(β)=2(2β),0β1.(7)

Further, the authors of [2] suggested that the new fractional derivative of order 0 < β < 1 defined by Caputo and Fabrizio can be redefined in view of the above discussed relation as Dtβϕ(t)=11βatϕ(x)expβtx1βdx.(8)

Here we give some important theorems and important properties of new fractional derivative that will be employed in the present article.

Theorem 1

[1] Consider the function ϕ(t) given by ϕs (a) = 0, s = 1, 2, …, n then for the new Caputo-Fabrizio derivative of fractional order we have, 0CFDtβ0CFDtn[ϕ(t)]=0CFDtn0CFDtβ[ϕ(t)].

For proof see [1].

Theorem 2

[14] The fractional derivative defined by Caputo-Fabrizio with the following ordinary differential equation 0CFDtβ[ϕ(t)]=ϕ(t),ϕ(0)0,(9) gives a non-trivial solution for 0 < β < 1. For proof see [14].

Theorem 3

[1] The Laplace transform of the new fractional derivative of a function ϕ (t) defined by Caputo and Fabrizio is expressed as L0CFDtβ[ϕ(t)]=M(β)sϕ¯(s)ϕ(0)s+β(1s), where ϕ¯ (s) indicates the Laplace transform of the function ϕ (t). For proof of this theorem see [1].

2 Fractional model of the damped Bergers’ equation with Caputo-Fabrizio derivative

The damped Bergers’ equation arises in fluid mechanics, gas dynamics, traffic flow and nonlinear acoustics among other fields. Along with the initial condition, it is presented as ut+uuxuxx+λu=0,(10) u(x,0)=ψ(x)(11) where u(x, t) is a function of two variables x and t indicating the displacement; x is a space variable, t is a time variable and λ is a constant. The damped Burgers’ equations has been studied by several researchers such as Babolian and Saeidian [27], Fakhari et al. [28], Inc [29], Song and Zhang [30], Peng and Chen [31] and others. The fractional generalization of the damped Bergers’ equations is investigated by Esen et al. [32]. A recent study carried out by Hristov [33] showed that the Cattaneo constitutive equation with Jeffrey’s fading memory naturally results in a heat conduction equation having a relaxation term interpreted by the Caputo-Fabrizio time derivative of fractional order. This paved the way to see the physical background of the newly defined Caputo-Fabrizio derivative having non-singular kernel in scientific fields. Therefore, we replace the first order time-derivative by the newly introduced derivative of fractional order in Eq. (10), this converts it into the nonlinear fractional damped Burgers’ equation written as 0CFDtβu(x,t)=u(x,t)u(x,t)x+2u(x,t)x2λu(x,t),(12) with the initial condition u(x,0)=ψ(x).(13)

3 Solution of the fractional model of the nonlinear damped Burgers’ equation by an iterative scheme

In this section, we find the solution of Eq. (12), with the help of an iterative approach. Employing the Laplace transform (LT) on Eq. (12) yields M(β)su¯(x,s)u(x,0)s+β(1s)=Luux+2ux2λu.(14)

On simplifying, we get u¯(x,s)=u(x,0)s+s+β(1s)M(β)sLuux+2ux2λu.(15)

Next, employing the inverse LT on Eq. (15), it yields u(x,t)=u(x,0)+L1s+β(1s)M(β)sLuux+2ux2λu.(16)

Then, the recursive formula is expressed as u0(x,t)=u(x,0),(17) and un+1(x,t)=un(x,t)+L1s+β(1s)M(β)sLununx+2unx2λun.(18)

Thus, the solution of Eq. (12) is presented in the following manner u(x,t)=Limnun(x,t).(19)

4 Application of fixed-point theorem for stability analysis of the iterative scheme

Let H be a self-map of X where (X, ‖⋅‖) stands for a Banach space and a particular recursive procedure assumed by yn+1 = f(H, yn). Suppose at least one element and that converges to a point pF(H) where F(H) is the fixed-point set of H. Consider the relation en = ‖ xn+1f(H, xn)‖ where {xn} ⊆ X is to be assumed. The iteration scheme yn+1 = f(H, yn) is called H-stable if Limn en = O implies that Limn xn = p. It must be assumed that the sequence {xn} has an upper boundary without any loss of generality otherwise the possibility of convergence cannot be expected. If all these restrictions are fulfilled for yn+1 = f(H, yn)this is called Picard’s iteration, hence the iteration is known as H-stable.

Theorem 4

[34] Suppose that H is a self-map of X where (X, ‖⋅‖) is a Banach space verifying HxHyCxHx+cxy, For all values of x, y in X, where the values of C and c are O ≤ C and, 0 ≤ c ≤ 1. Assume that H satisfies Picard’s H-stability.

For the fractional model of the damped Burgers’ equation, consider the following succession as un+1(x,t)=un(x,t)+L1s+β(1s)M(β)sLu~nunx+2unx2λun,(20) where u~n indicates a restricted variation satisfying δu~n=0 and s+β(1s)M(β)s denotes the fractional Lagrange’s multiplier.

Next, we establish the following result.

Theorem 5

Suppose T be a self-map described in the following way Tun(x,t)=un+1(x,t)=un(x,t)+L1s+β(1s)M(β)sLununx+2unx2λun,(21) then the iteration is T-Stable in L1 (a, b) if the following condition holds 1+β12(A+B)f(γ)+β22g(γ)+λh(γ)<1, where f, g and h are functions arising from L1s+β(1s)M(β)sL.

Proof

Firstly we show that T has a fixed point. In order to derive this result, we assess the succession for (n, m)∈ N × N. T(un(x,t))T(um(x,t))=un(x,t)um(x,t)+L1s+β(1s)M(β)sLununxumumx+2x2(unum)λ(unum).(22) =un(x,t)um(x,t)+L1s+β(1s)M(β)sL12(un2)x(um2)x+2x2(unum)λ(unum).(23)

Now applying the norm on both sides of Eq. (23) and without loss of generality, we get T(un(x,t))T(um(x,t))un(x,t)um(x,t)+L1s+β(1s)M(β)sL12(un2)x(um2)x+2x2(unum)λ(unum).(24)

Using the property of the norm in particular, the triangular inequality, the R.H.S. of equation (24) is converted to T(un(x,t))T(um(x,t))un(x,t)um(x,t)+L1[s+β(1s)M(β)sL(12((un2)x(um2)x))]+L1[s+β(1s)M(β)sL(2x2(unum))]+L1[s+β(1s)M(β)sL(λ(unum))]un(x,t)um(x,t)+L1[s+β(1s)M(β)sL(12β1(un2um2))]+L1[s+β(1s)M(β)sL(β22(unum))]+L1[s+β(1s)M(β)sL(λ(unum))](25)

Since un (x, t), um (x, t) are bounded functions, two distinct constants can be obtained A, B > 0 such that ∀ t un(x,t)A,um(x,t)B.(26)

Next, using Eq. (26) in Eq. (25), we arrive at the following result T(un(x,t))T(um(x,t))1+12β1(A+B)f(γ)+β22g(γ)+λh(γ).un(x,t)um(x,t),(27) where f, g and h are functions of L1s+β(1s)M(β)sL.

For 1+12β1(A+B)f(γ)+β22g(γ)+λh(γ)<1.

Therefore, the nonlinear T-self mapping attains a fixed point. Now, we verify that T fulfills the conditions in Theorem 4. Suppose Eq. (27) be held, hence substituting c=0,C=1+12β1(A+B)f(γ)+β22g(γ)+λh(γ).(28)

Then Eq. (28) demonstrates that for the nonlinear mapping T the inequality of Theorem 4 holds. Thus, considering the nonlinear mapping T along with the satisfaction of all the conditions in Theorem 4, T is Picard’s T-stable. Hence, we proved the Theorem 5.

The plots of the solution u(x, t) w.r. to space x and time t are found at β = 1 and λ = 1.
Figure 1

The plots of the solution u(x, t) w.r. to space x and time t are found at β = 1 and λ = 1.

The plots of the solution u(x, t) w.r. to space x and time t are derived if β = 0.85 and λ = 1.
Figure 2

The plots of the solution u(x, t) w.r. to space x and time t are derived if β = 0.85 and λ = 1.

The plots of the solution u(x, t) w.r. to space x and time t are found at β = 0.75 and λ = 1.
Figure 3

The plots of the solution u(x, t) w.r. to space x and time t are found at β = 0.75 and λ = 1.

The plots of the solution u(x, t) w.r. to space x and time t are derived at β = 0.65 and λ = 1.
Figure 4

The plots of the solution u(x, t) w.r. to space x and time t are derived at β = 0.65 and λ = 1.

The plots of the solution u(x, t) vs. t for distinct values of β at x = 0.05 and λ = 1.
Figure 5

The plots of the solution u(x, t) vs. t for distinct values of β at x = 0.05 and λ = 1.

5 Numerical simulations

In this section, we investigate the numerical simulations of the special solution of Eq. (12) as function of time and space for u(x, 0) = λx, λ = 1 and distinct values of β. Figs. 15 represent simulations of the solution. The graphical representations demonstrate that the model depends notably on the fractional order. Figures 14 show the clear difference at β = 1, β = 0.85, β = 0.75 and β = 0.65. The model narrates a new characteristic at β = 0.85 β = 0.75 and β = 0.65 that was invisible when modeling at β = 1. From Figs. 14 we can see that the displacement u(x, t) increases with increasing the value of x whereas the displacement u(x, t) decreases with increasing the value of t. Fig. 5 describes the displacement u(x, t) for distinct values of β. It can be noticed from Figs. 15, that the order of derivative significantly affects the displacement.

6 Conclusions

The Caputo-Fabrizio fractional derivative has many important qualities. For instance, at distinct scales it can illustrate matter diversities and configurations, where in local theories these clearly cannot be controlled. We exerted this new derivative to adapt the damped Burgers’ equation. By using an iterative scheme we have derived the solution of the equation. In order to show the stability of the iterative technique we applied the theory of T-stable mapping and the postulate of fixed-point. We presented some interesting numerical simulations for distinct values of β and λ =1. The outcomes demonstrate that the new fractional order derivative can be used to model various scientific and engineering problems.

Acknowledgement

The authors extend their appreciation to the International Scientific Partnership Program ISPP at King Saud University for funding this research work through ISPP# 63.

References

  • [1]

    Caputo M., Fabrizio M., A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 2015, 1, 73-85. Google Scholar

  • [2]

    Losada J., Nieto J.J., Properties of the new fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 2015, 1, 87-92. Google Scholar

  • [3]

    Odibat Z.M., Momani, S., Application of variational iteration method to nonlinear differential equation of fractional order, Int. J. Nonlinear Sci. Numer. Simul., 2006, 7(1), 27-34.Google Scholar

  • [4]

    Kilbas A.A., Srivastava H.M., Trujillo J.J., Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006.Google Scholar

  • [5]

    Caputo M., Linear models of dissipation whose Q is almost frequency independent, part II, Geophys. J. Int., 1967, 13(5), 529-539.CrossrefGoogle Scholar

  • [6]

    Podlubny L., Fractional Differential Equations, Academic Press, London, 1999.Google Scholar

  • [7]

    Magin R.L., Fractional Calculus in Bioengineering, Begell House, Connecticut, 2006.Google Scholar

  • [8]

    Baleanu D., Guvenc Z.B., Machado J.A.T.(Ed.), New Trends in Nanotechnology and Fractional Calculus Applications, Springer Dordrecht Heidelberg, London New York, 2010. Google Scholar

  • [9]

    Tarasov V.E., Three-dimensional lattice models with long-range interactions of Grünwald-Letnikov type for fractional generalization of gradient elasticity, Meccanica, 2016, 51(1), 125-138. CrossrefGoogle Scholar

  • [10]

    Choudhary A., Kumar D., Singh J., Analytical solution of fractional differential equations arising in fluid mechanics by using sumudu transform method, Nonlinear Eng., 2014, 3(3), 133-139. Google Scholar

  • [11]

    Bulut, H., Baskonus H.M., Belgacem F.B.M., The analytical solutions of some fractional ordinary differential equations by sumudu transform method, Abst. Appl. Anal., 2013, Article ID 203875, 6 pages. Google Scholar

  • [12]

    Razminia K., Razminia A., Machado J.A.T., Analytical solution of fractional order diffusivity equation with wellbore storage and skin effects, J. Comput. Nonlinear Dyn., 2016, 11(1), . CrossrefWeb of ScienceGoogle Scholar

  • [13]

    Atangana A., Koca I., On the new fractional derivative and application to nonlinear Baggs and Freedman Model, J. Nonlinear Sci. Appl., 2016, 9, 2467-2480. CrossrefGoogle Scholar

  • [14]

    Atangana A., On the new fractional derivative and application to nonlinear Fisher’s reaction-diffusion equation, Appl. Math. Comput., 2016, 273 948-956. Web of ScienceGoogle Scholar

  • [15]

    Singh J., Kumar D., Kilichman A., Numerical solutions of nonlinear fractional partial differential equations arising in spatial diffusion of biological populations, Abst. Appl. Anal., 2014, Article ID 535793, 12 pages. Google Scholar

  • [16]

    Singh J., Kumar D., Rathore S., On the solutions of fractional reaction-diffusion equations, Le Matematiche, 2013, 68(1), 23-32. Google Scholar

  • [17]

    Baleanu D., Rezapour S., Salehi S., A fractional finite difference inclusion, J. Comput. Anal. Appl., 2016, 20(5), 834-842. Google Scholar

  • [18]

    Atangana A., Baleanu D., Alsaedi A., Analysis of time-fractional Hunter-Saxton equation: a model of neumatic liquid crystal, Open Phys., 2016, 14, 145-149. Web of ScienceGoogle Scholar

  • [19]

    Kurt A., Çenesiz Y., Tasbozan O., On the Solution of Burgers’ Equation with the new fractional derivative, Open Phys., 2015, 13 (1), 355-360. Web of ScienceGoogle Scholar

  • [20]

    Tasbozan O., Çenesiz Y., Kurt A., New solutions for conformable fractional Boussinesq and combined KdV-mKdV equations using Jacobi elliptic function expansion method, Eur. Phys. J. Plus, 2016 131 (7), 244. Web of ScienceCrossrefGoogle Scholar

  • [21]

    Atangana A., Baleanu D., New fractional derivatives with nonlocal and non-singular kernel, Theory and application to heat transfer model, Thermal Science, 2016, 20(2), 763-769. CrossrefGoogle Scholar

  • [22]

    Alsaedi A., Baleanu D., Etemad S., Rezapour, S., On coupled systems of time-fractional differential problems by using a new fractional derivative, Journal of Function Spaces, 2016, Article Number: 4626940,  CrossrefWeb of ScienceGoogle Scholar

  • [23]

    Coronel-Escamilla A., Gomez-Aguilar J.F., Baleanu D., Escobar-Jiménez R.F., Olivares-Peregrino V.H., Abundez-Pliego A., Formulation of Euler-Lagrange and Hamilton equations involving fractional operators with regular kernel, Adv. Difference Equ., 2016, Article Number: 283, . CrossrefWeb of ScienceGoogle Scholar

  • [24]

    Gomez-Aguilar J.F., Morales-Delgado V.F., Taneco-Hernandez M.A., Baleanu D., Escobar-Jiménez, R.F., Al Qurashi M.M., Analytical solutions of the electrical RLC circuit via Liouville-Caputo operators with local and non-local kernels, Entropy, 2016, 18(8), Article Number: 402. Web of ScienceGoogle Scholar

  • [25]

    Gomez-Aguilar J. F., Torres L., Yepez-Martinez H., Baleanu D., Reyes J. M., Sosa, I. O., Fractional Lienard type model of a pipeline within the fractional derivative without singular kernel. 2016, Adv. Difference Equ., Article Number: 173, . CrossrefWeb of ScienceGoogle Scholar

  • [26]

    Doungmo Goufo E.F., Application of the Caputo-Fabrizio fractional derivative without singular kernel to Korteweg-de Vries-Burgers equation, Mathematical Modelling and Analysis, 2016, 21(2), 188-198. CrossrefGoogle Scholar

  • [27]

    Babolian E., Saeidian J., Analytic approximate solutions to Burgers, Fisher, Huxley equations and two combined forms of these eqautions, Commun. Nonlinear. Sci. Numer. Simulat., 2009, 14, 1984-1992. CrossrefGoogle Scholar

  • [28]

    Fakhari A., Domairry G., Ebrahimpour, Approximate explicit solutions of nonlinear BBMB equations by homotopy analysis method and comparison with the exact solution, Phys. Lett. A, 2007, 368, 64-68. CrossrefGoogle Scholar

  • [29]

    Inc M., On numerical solution of Burgers equation by homotopy analysis method, Phys. Lett. A, 2008, 372, 356-360. Web of ScienceCrossrefGoogle Scholar

  • [30]

    Song L., Zhang H.Q., Application of homotopy analysis method to fractional KdV-Burgers-Kuramoto equation, Phys. Lett. A, 2007, 367, 88-94. CrossrefWeb of ScienceGoogle Scholar

  • [31]

    Peng Y., Chen W., A new similarity solution of the Burgers equation with linear damping, Czech. J, Phys., 2008, 56, 317-428. Google Scholar

  • [32]

    Esen A., Yagmurlu N.M., Tasbozan O., Approximate Analytical Solution to Time-Fractional Damped Burger and Cahn-Allen Equations, Appl. Math. Inf. Sci., 2013, 7(5), 1951-1956. CrossrefWeb of ScienceGoogle Scholar

  • [33]

    Hristov J., Transient heat diffusion with a non-singular fading memory: from the Cattaneo constitutive equation with Jeffrey’s kernel to the Caputo-Fabrizio time-fractional derivative, Thermal Science, 2016, 20, 757-762. CrossrefWeb of ScienceGoogle Scholar

  • [34]

    Qing Y., Rhoades, B.E., T-stability of Picard iteration in metric spaces, Fixed Point Theory and Applications, 2008, Article ID 418971, 4 pages. Web of ScienceGoogle Scholar

About the article

Received: 2016-05-30

Accepted: 2017-01-10

Published Online: 2017-03-12


Citation Information: Open Physics, Volume 15, Issue 1, Pages 35–41, ISSN (Online) 2391-5471, DOI: https://doi.org/10.1515/phys-2017-0005.

Export Citation

© 2017 J. Singh et al.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
Zeliha Korpinar, Mustafa Inc, and Mustafa Bayram
Applied Mathematics and Computation, 2020, Volume 367, Page 124781
[2]
Devendra Kumar, Jagdev Singh, Kumud Tanwar, and Dumitru Baleanu
International Journal of Heat and Mass Transfer, 2019, Volume 138, Page 1222
[3]
Abdullahi Yusuf, Mustafa Inc, Aliyu Isa Aliyu, and Dumitru Baleanu
Chaos, Solitons & Fractals, 2018, Volume 116, Page 220
[4]
Devendra Kumar, Jagdev Singh, Dumitru Baleanu, and Sushila Rathore
The European Physical Journal Plus, 2018, Volume 133, Number 7
[5]
Devendra Kumar, Jagdev Singh, Dumitru Baleanu, and Sushila
Physica A: Statistical Mechanics and its Applications, 2018, Volume 492, Page 155
[6]
Mustafa Inc, Abdullahi Yusuf, Aliyu Isa Aliyu, and Dumitru Baleanu
Physica A: Statistical Mechanics and its Applications, 2018
[7]
Leonardo Martínez Jiménez, J. Juan Rosales García, Abraham Ortega Contreras, and Dumitru Baleanu
Open Physics, 2017, Volume 15, Number 1, Page 627
[8]
B. Cuahutenango-Barro, M. A. Taneco-Hernández, and J. F. Gómez-Aguilar
The European Physical Journal Plus, 2017, Volume 132, Number 12
[10]
A. Coronel-Escamilla, J.F. Gómez-Aguilar, L. Torres, and R.F. Escobar-Jiménez
Physica A: Statistical Mechanics and its Applications, 2017
[11]
Chen Wan, Tao Li, and Zhicheng Sun
Advances in Difference Equations, 2017, Volume 2017, Number 1
[12]
A. Coronel-Escamilla, J.F. Gómez-Aguilar, L. Torres, R.F. Escobar-Jiménez, and M. Valtierra-Rodríguez
Physica A: Statistical Mechanics and its Applications, 2017, Volume 487, Page 1

Comments (0)

Please log in or register to comment.
Log in