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Analysis of a New Fractional Model for Damped Bergers’ Equation

  • Jagdev Singh EMAIL logo , Devendra Kumar , Maysaa Al Qurashi and Dumitru Baleanu
Published/Copyright: March 12, 2017
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Open Physics
From the journal Open Physics Volume 15 Issue 1

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 [1–8]. 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 ϕ ∈ H1 (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)1−β∫atϕ′(x)exp−βt−x1−βdx,(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 ϕ ∉ H1 (a, b)

Dtβϕ(t)=βM(β)1−β∫at(ϕ(t)−ϕ(x))exp−βt−x1−βdx.(2)

The Eq. (2) takes the below form

Dtβϕ(t)=N(σ)σ∫atϕ′(x)exp−t−xσdx,N(0)=N(∞)=1,(3)

if σ=1−ββ∈[0,∞),β=11+σ∈ [0,1] as stated by the authors [14].

Moreover,

Limσ→01σexp−t−xσ=δ(x−t).(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(1−β)(2−β)M(β)ϕ(t)+2β(2−β)M(β)∫0tϕ(s)ds,t≥0.(5)

Eq. (5) further yields

2(1−β)(2−β)M(β)+2β(2−β)M(β)=1.(6)

The above result gives an explicit formula for

M(β)=2(2−β),0≤β≤1.(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)=11−β∫atϕ′(x)exp−βt−x1−βdx.(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(1−s),

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

∂u∂t+uux−uxx+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)∂x2−λu(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(1−s)=L−u∂u∂x+∂2u∂x2−λu.(14)

On simplifying, we get

u¯(x,s)=u(x,0)s+s+β(1−s)M(β)sL−u∂u∂x+∂2u∂x2−λu.(15)

Next, employing the inverse LT on Eq. (15), it yields

u(x,t)=u(x,0)+L−1s+β(1−s)M(β)sL−u∂u∂x+∂2u∂x2−λu.(16)

Then, the recursive formula is expressed as

u0(x,t)=u(x,0),(17)

and

un+1(x,t)=un(x,t)+L−1s+β(1−s)M(β)sL−un∂un∂x+∂2un∂x2−λun.(18)

Thus, the solution of Eq. (12) is presented in the following manner

u(x,t)=Limn→∞un(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 p ∈ F(H) where F(H) is the fixed-point set of H. Consider the relation en = ‖ xn+1 −f(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

Hx−Hy≤Cx−Hx+cx−y,

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)+L−1s+β(1−s)M(β)sL−u~n∂un∂x+∂2un∂x2−λun,(20)

where u~n indicates a restricted variation satisfying δu~n=0 and s+β(1−s)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)+L−1s+β(1−s)M(β)sL−un∂un∂x+∂2un∂x2−λun,(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 L−1s+β(1−s)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)+L−1s+β(1−s)M(β)sL−un∂un∂x−um∂um∂x+∂2∂x2(un−um)−λ(un−um).(22)
=un(x,t)−um(x,t)+L−1s+β(1−s)M(β)s⋅L−12∂(un2)∂x−∂(um2)∂x+∂2∂x2(un−um)−λ(un−um).(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)∥+∥L−1s+β(1−s)M(β)s⋅L−12∂(un2)∂x−∂(um2)∂x+∂2∂x2(un−um)−λ(un−um).(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)∥+L−1[s+β(1−s)M(β)sL(−12(∂(un2)∂x−∂(um2)∂x))]+L−1[s+β(1−s)M(β)sL(∂2∂x2(un−um))]+L−1[s+β(1−s)M(β)sL(∥−λ(un−um)∥)]≤∥un(x,t)−um(x,t)∥+L−1[s+β(1−s)M(β)sL(12β1(un2−um2))]+L−1[s+β(1−s)M(β)sL(∥β22(un−um)∥)]+L−1[s+β(1−s)M(β)sL(∥−λ(un−um)∥)](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 L−1s+β(1−s)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.

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.
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.

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.
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.

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.
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.

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.
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.

Figure 5 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. 1–5 represent simulations of the solution. The graphical representations demonstrate that the model depends notably on the fractional order. Figures 1–4 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. 1–4 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. 1–5, 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.

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Received: 2016-5-30
Accepted: 2017-1-10
Published Online: 2017-3-12

© 2017 J. Singh et al.

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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https://doi.org/10.1515/phys-2017-0005
Keywords for this article
Time-fractional damped Bergers’ equation; Nonlinear equation; Caputo-Fabrizio fractional derivative; Iterative method; Fixed-point theorem
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  197. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  198. What If Quantum Theory Violates All Mathematics?
  199. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  200. Relativity, anomalies and objectivity loophole in recent tests of local realism
  201. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  202. The photon identification loophole in EPRB experiments: computer models with single-wing selection
  203. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  204. Bohr against Bell: complementarity versus nonlocality
  205. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  206. Is Einsteinian no-signalling violated in Bell tests?
  207. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  208. Bell’s “Theorem”: loopholes vs. conceptual flaws
  209. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  210. Nonrecurrence and Bell-like inequalities
  211. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  212. Three-dimensional computer models of electrospinning systems
  213. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  214. Electric field computation and measurements in the electroporation of inhomogeneous samples
  215. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  216. Modelling of magnetostriction of transformer magnetic core for vibration analysis
  217. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  218. Comparison of the fractional power motor with cores made of various magnetic materials
  219. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  220. Dynamics of the line-start reluctance motor with rotor made of SMC material
  221. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  222. Inhomogeneous dielectrics: conformal mapping and finite-element models
  223. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  224. Topology optimization of induction heating model using sequential linear programming based on move limit with adaptive relaxation
  225. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  226. Detection of inter-turn short-circuit at start-up of induction machine based on torque analysis
  227. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  228. Current superimposition variable flux reluctance motor with 8 salient poles
  229. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  230. Modelling axial vibration in windings of power transformers
  231. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  232. Field analysis & eddy current losses calculation in five-phase tubular actuator
  233. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  234. Hybrid excited claw pole generator with skewed and non-skewed permanent magnets
  235. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  236. Electromagnetic phenomena analysis in brushless DC motor with speed control using PWM method
  237. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  238. Field-circuit analysis and measurements of a single-phase self-excited induction generator
  239. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  240. A comparative analysis between classical and modified approach of description of the electrical machine windings by means of T0 method
  241. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  242. Field-based optimal-design of an electric motor: a new sensitivity formulation
  243. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  244. Application of the parametric proper generalized decomposition to the frequency-dependent calculation of the impedance of an AC line with rectangular conductors
  245. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  246. Virtual reality as a new trend in mechanical and electrical engineering education
  247. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  248. Holonomicity analysis of electromechanical systems
  249. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  250. An accurate reactive power control study in virtual flux droop control
  251. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  252. Localized probability of improvement for kriging based multi-objective optimization
  253. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  254. Research of influence of open-winding faults on properties of brushless permanent magnets motor
  255. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  256. Optimal design of the rotor geometry of line-start permanent magnet synchronous motor using the bat algorithm
  257. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  258. Model of depositing layer on cylindrical surface produced by induction-assisted laser cladding process
  259. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  260. Detection of inter-turn faults in transformer winding using the capacitor discharge method
  261. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  262. A novel hybrid genetic algorithm for optimal design of IPM machines for electric vehicle
  263. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  264. Lamination effects on a 3D model of the magnetic core of power transformers
  265. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  266. Detection of vertical disparity in three-dimensional visualizations
  267. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  268. Calculations of magnetic field in dynamo sheets taking into account their texture
  269. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  270. 3-dimensional computer model of electrospinning multicapillary unit used for electrostatic field analysis
  271. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  272. Optimization of wearable microwave antenna with simplified electromagnetic model of the human body
  273. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  274. Induction heating process of ferromagnetic filled carbon nanotubes based on 3-D model
  275. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  276. Speed control of an induction motor by 6-switched 3-level inverter
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