Two Reliable Methods for The Solution of Fractional Coupled Burgers’ Equation Arising as a Model of Polydispersive Sedimentation

Ali Kurt
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  • Department of Mathematics, Faculty of Science and Art, Pamukkale University, Denizli, Turkey
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, Mehmet Şenol
  • Department of Mathematics, Faculty of Science and Art, Nevşehir Hacι Bektaş Veli University, Nevşehir, Turkey
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, Orkun Tasbozan and Mehar Chand

Abstract

In this article, we attain new analytical solution sets for nonlinear time-fractional coupled Burgers’ equations which arise in polydispersive sedimentation in shallow water waves using exp-function method. Then we apply a semi-analytical method namely perturbation-iteration algorithm (PIA) to obtain some approximate solutions. These results are compared with obtained exact solutions by tables and surface plots. The fractional derivatives are evaluated in the conformable sense. The findings reveal that both methods are very effective and dependable for solving partial fractional differential equations.

1 Introduction

Fractional calculus, which includes arbitrary order derivatives and integrals, is the generalized form of the classical calculus. In the last decades, it has been frequently researched by many scientists to model real world problems. Therefore, it offered a decent way of implementation for plenty of models in miscellaneous areas of engineering and physics such as, electrical networks [1], fluid flow [11], image and signal processing [17], mathematical physics [30], viscoelasticity [25], biology [20], control [5] and see references therein [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44].

Besides, seeking analytical and approximate solutions of fractional partial differential equations (FPDEs) become more popular. Therefore, achieving the solutions of FPDEs important for these areas and has a distinct place.

Up to now, various powerful numerical techniques have been proposed for solutions of the (FPDEs). Some of them are, Adomian decomposition method (ADM) [21, 23], homotopy perturbation method (HPM) [2, 14], variational iteration method (VIM) [22], Legendre wavelet operational matrix method (LWOMM) [26], homotopy analysis method (HAM) [18, 19] and residual power series method [6, 7].

In this article, we use exp-function method [15] and perturbation-iteration algorithm (PIA) [27, 28, 29] to present new analytical and numerical solutions of fractional coupled Burgers’ equations given as [24]:

Dtαu+uDxu+vDyu1(Dx2u+Dy2u)=0,Dtαv+uDxv+vDyv1(Dx2v+Dy2v)=0,

The exp-function method is a robust technique for obtaining compacton-like, periodic and solitary solutions of FPDEs. It transforms the given system to an ordinary differential equation and yields to solve it efficiently. In addition, perturbation-iteration algorithm is established by using the perturbation expansion. With choosing proper initial and boundary conditions, it can be performed directly to the model without discretization or any other special conversions.

The methodology in the other sections can be described as follows. Some basic definitions are presented in Section 2. Analysis of the implemented methods are given in Section 3. In Section 4, both methods are used to obtain analytical and approximate solutions of coupled Burgers’ equation. Finally, the paper ends with a conclusion in Section 5.

2 Preliminaries

There are different types of arbitrary order differentiation. The most widely used are the Riemann-Liouville(RLFD) and Caputo fractional derivatives (CFD).

Definition 1

The RLFD operatorDαf(x) forα > 0 and q – 1 < α < qdefined as [12, 13]:

Dαf(x)=dqdxq1Γ(qα)αxf(t)xtα+1qdt

Definition 2

The CFD of orderα > 0 forn ∈ ℕ, n – 1 < α < n, Dα, defined as [10]:

Dαf(x)=JnαDnf(x)=1Γ(nα)αx(xt)nα1ddtnf(t)dt

Along with these definitions, a new fractional derivative definition, namely the conformable fractional derivative, has been introduced by Khalil et al. [16].

Definition 3

The conformable fractional derivative of anαth order functionf : [0, ∞) → Ris defined by

Tα(f)(t)=limε0f(t+εt1α)(f)(t)ε

where 0 < α ≤ 1 and t > 0.

Theorem 1

Basic properties of conformable derivative ofα-differantiablefandgfunctions for 0 < α ≤ 1 at pointt > 0 are

  1. Tα(mf + ng) = mTα(f) + nTα(g), m, n ∈ ℝ
  2. Tα(tp) = ptpα for all p
  3. Tα(f.g) = fTα(g) + gTα(f)
  4. Tα(fg)=gTα(f)fTα(g)g2
  5. Let f(t) = c be a constant function. Then Tα(c) = 0.
  6. Tα(f)(t)=t1αdf(t)dt, if f is differentiable.

Definition 4

The conformable partial derivatives of anαth orderffunction withx1, …, xnvariables are [8]

dαdxiαf(x1,...,xn)=limε0f(x1,...,xi1,xi+εxi1α,...,xn)f(x1,...,xn)ε.

where 0 < α ≤ 1.

Definition 5

The conformable integral of anαth orderffunction starting froma ⩾ 0 is defined by [19]

Iαa(f)(s)=saf(t)t1αdt.

3 Descriptions of the Implemented Methods

3.1 Exp-Function Method

Taking account into the following nonlinear time fractional equation in order to explain the basic idea of the implemented method [15]

Fu,αutα,ux,uy,2αut2α,2ux2,2uy2,=0

where the fractional derivatives are in conformable sense. We can introduce the wave variable as

u(x,y,t)=u(η),η=kx+wy+ctαα

where k, w, c are arbitrary constants that can be examined later. With the help of conformable chain rule [1], we have

α(.)tα=cd(.)dη,(.)x=kd(.)dη,(.)y=wd(.)dη,.

Hence Eq.(6) changes into differential equation with integer order as follows.

Qu,uη,uηη,uηηη,uηηηη,...=0.

Due to exp-function method, it is supposed that the wave solution can be regarded in the following form

u(η)=n=djanenηs=qpbmemη=ajejη++adedηbpepη++aqeqη

where p, q, j and d are positive integers that can be examined later, an and bm are unrecognized constants. To calculate the values of j and p, highest order the linear term of Eq. (9) is equalized with the highest order nonlinear term. By using the same procedure, the values for q and d, can be calculated by balancing the lowest order linear term of Eq. (9) with lowest order nonlinear term. As a result we can acquire the traveling wave solutions of the considered Eq. (6)

3.2 Perturbation-Iteration Algorithm (PIA)

Formerly, a perturbation based algorithm has been introduced by Aksoy and Pakdemirli [3]. In the method, an iterative algorithm is proposed using the perturbation expansion. Previously, this method is implemented on ordinary FDEs [27], fractional-integro differential equations [28] and systems of FDEs [29].

In this article, the most basic PIA, PIA(1, 1) is used to attain approximate solutions of FPDEs. For this purpose, one we consider the correction term in the perturbation expansion and correction terms of first derivatives in the Taylor series expansion [3, 4].

To describe the main idea of PIA, take the FPDE

Fuα(x,t),u(x,t),ux(x,t),uxx(x,t),,ε=0,

where ε is assumed as an artificially small parameter. The perturbation expansions with one correction terms are

un+1=un+εucnTα(un+1)=Tα(un)+εucn

Subrogating (12) into (11) and expanding in the Taylor series form for only first order derivatives yields

Funα,un,(un)x,(un)xx,,0+Fuunα,un,(un)x,(un)xx,,0εucn+Fuαunα,un,(un)x,(un)xx,,0εuc(α)n+Fuxunα,un,(un)x,(un)xx,,0ε(uc)xn+Fuxxunα,un,(un)x,(un)xx,,0ε(uc)xxn++Fεunα,un,unx,unxx,,0ε=0

or

uc(α)nFu(α)+ucnFu+(uc)xnFux+(uc)xxnFuxx++Fε+Fε=0.

Rewriting (14) gives the subsequent PIA(1, 1) iteration formula

ut(x,t)+FuFutu(x,t)=Fε+FεFut.

In this expansion, all of the derivatives are evaluated at ε = 0. Using an initial function u0(x, t), firstly the correction term (uc)0(x, t) is computed. Subrogating it into (12) gives the first approximate result u1(x, t). Similar procedure is applied until obtaining the other approximations.

4 Application of the Methods for (2 + 1)-dimensional Time-Fractional Coupled Burgers’ Equation

4.1 Analytical Solution of Coupled Burgers’ Equation

Think of the fractional coupled Burgers’ equation [24] as

Dtαu+uDxu+vDyu1(Dx2u+Dy2u)=0,Dtαv+uDxv+vDyv1(Dx2v+Dy2v)=0,

where 0 < α ≤ 1, 𝔎, is Reynolds number, u = u(x, y, t) and v = v(x, y, t). By the help of the chain rule [1] and the wave transform η=kx+wy+ctαα, we obtain

cU(η)+ku(η)U(η)+wV(η)U(η)1(k2+w2)U(η)=0cV(η)+kuV(η)+wvV(η)1(k2+w2)V(η)=0.

Now assume that the solution of (17) can be described as

U(η)=acecη++adedηbpepη++bqeqη.

V(η)=dsesη++dnenηflelη++frerη.

Using (18), (19) and (17) led to c = p, d = q, s = l and n = r. For convenience lets assume all the coefficients c = p = s = l = n = r = 1. Now rewriting u(η) and v(η) due to above assumptions

U=a1eη+a0+a1eηb1eη+b0+b1eηV=d1eη+d0+d1eηf1eη+f0+f1eη.

Substituting the equations (20) into (17) and equalizing the coefficients of e yields a system of algebraic equations. Solving the system with respect to the constants expressed above we can handle the following solutions

c=wd0b0+k2f0b0+w2f0b0+ka0f0b0f0,a1=f1wd0b0+2k2b0f1f0+2w2b0f1f0+ka0f1f0b0wd1f0kf02,a1=0,b1=b0f1f0,a1=0,b1=0,d1=0,f1=0.

So the solutions can be obtained as

u(x,y,t)=a0+AeBb0+b0f1eBf01kf02

and

v(x,y,t)=d0+d1eBf0+f1eB

where

A=f1wd0b0+2k2b0f1f0+2w2b0f1f0+ka0f1f0b0wd1f0B=kxwy+wd0b0+k2f0b0+w2f0b0+ka0f0tαb0f0α

4.2 Approximate Solution of Coupled Burgers’ Equations

Regard the system (16) with the the conditions u(x,y,0)=1+14+8exy1+12exyandv(x,y,0)=1+exy2+exy. For the values k = 1, w = 1, f0 = 2, d0 = 1, b0 = 1, d–1 = 1, b–1 = 1, f–1 = 1 and a0 = 1, we can acquire the exact solutions as

u(x,y,t)=1+14+8exy+3+4tα2α1+12exy+3+4tα2Rα,v(x,y,t)=1+exy+3+4tα2α2+exy+3+4tα2α.

Now we introduce a small perturbation parameter ε to the system and rewrite the equations as

Dtαu+εuDxu+εvDyu1(Dx2u+Dy2u)=0,Dtαv+εuDxv+εvDyv1(Dx2v+Dy2v)=0,

Therefore, terms in formula (15)turn into

F=t1α(un)t(x,y,t)1(un)xx(x,y,t)+(un)yy(x,y,t),Fu=0,Fut=t1α,Fε=un(x,y,t)unx(x,y,t)+vn(x,y,t)(un)y(x,y,t)

and

F=t1α(vn)t(x,y,t)1(vn)xx(x,y,t)+(vn)yy(x,y,t),Fv=0,Fvt=t1α,Fε=un(x,t)(vn)x(x,t)+vn(x,t)(vn)y(x,t)

Subrogating above terms in the iteration formula (15) gives the subsequent partial differential equations

εtuct=tunt+tα((un)xxε(un(un)x+vn(un)y)+(un)yy)

and

εtvct=tvnt+tα((vn)xxε(un(vn)x+vn(vn)y)+(vn)yy)

Beginning with the initial functions

u(x,y,0)=1+14+8exy1+12exyandv(x,y,0)=1+exy2+exy

and using (15), the numerical results are obtained for n = 0, 1, 2, … respectively.

u1(x,y,t)=4ex+y++822ex+y+1(8)(3+4)tαex+y2α22ex+y+12

v1(x,y,t)=(3+4)tαex+y2α2ex+y+12+ex+y+12ex+y+1

u2(x,y,t)=(8)(3+4)2t2αex+y2ex+y13α2ex+y+12+16tαex+y24α342ex+y+15(8)(3+4)tαex+y2α2ex+y+2+822ex+y+1+1

v2(x,y,t)=2(3+4)2t3αe2x+2y2ex+y13α332ex+y+15+(3+4)2t2αex+y2ex+y18α222ex+y+13+(3+4)tαex+y2α2ex+y+12+ex+y+12ex+y+1

Similarly, the fourth order solutions u4(x, y, t) and v4(x, y, t) are calculated. In Table 1 and 2, the fourth order PIA numerical approximate solutions are compared to exact solutions. Also the absolute errors are calculated for changing values of α and x. The results indicate the reliability of PIA. Besides using Figures 16, figures for the solutions of PIA are illustrated for changing α values. They exhibit that PIA produces highly approximate results. It is also obvious that further iterations would generate convenient solutions.

Fig. 1
Fig. 1

The surface plot of the PIA solution u4(x, y, t) for ℜ = 100 and t = 0.1 when α = 0.75.

Citation: Applied Mathematics and Nonlinear Sciences 4, 2; 10.2478/AMNS.2019.2.00049

Fig. 2
Fig. 2

The surface plot of the PIA solution v4(x, y, t) for ℜ = 100 and t = 0.1 when α = 0.75.

Citation: Applied Mathematics and Nonlinear Sciences 4, 2; 10.2478/AMNS.2019.2.00049

Fig. 3
Fig. 3

The surface plot of the PIA solution u4(x, y, t) for ℜ = 100 and t = 0.1 when α = 0.85.

Citation: Applied Mathematics and Nonlinear Sciences 4, 2; 10.2478/AMNS.2019.2.00049

Fig. 4
Fig. 4

The surface plot of the PIA solution v4(x, y, t) for ℜ = 100 and t = 0.1 when α = 0.85.

Citation: Applied Mathematics and Nonlinear Sciences 4, 2; 10.2478/AMNS.2019.2.00049

Fig. 5
Fig. 5

The surface plot of the PIA solution u4(x, y, t) for ℜ = 100 and t = 0.1 when α = 0.95.

Citation: Applied Mathematics and Nonlinear Sciences 4, 2; 10.2478/AMNS.2019.2.00049

Fig. 6
Fig. 6

The surface plot of the PIA solution v4(x, y, t) for ℜ = 100 and t = 0.1 when α = 0.95.

Citation: Applied Mathematics and Nonlinear Sciences 4, 2; 10.2478/AMNS.2019.2.00049

Table 1

PIA (u4(x, y, t)) and exact solution values with absolute errors for y = 1, t = 0.1 and ℜ = 100.

α = 0.75α = 0.85α = 0.95
xPIAExactErrorPIAExactErrorPIAExactError
0.00.9039930.9039962.72276E-60.9119280.9119284.58229E-70.9170150.9170158.27357E-8
0.10.9113690.9113712.60627E-60.918830.9188314.36394E-70.9236000.9236007.85330E-8
0.20.9183050.9183082.43193E-60.9253000.9253014.05356E-70.9297610.9297617.27317E-8
0.30.9248090.9248112.21778E-60.9313480.9313483.68115E-70.9355080.9355086.58693E-8
0.40.9308890.9308911.97981E-60.9369860.9369863.27310E-70.9408560.9408565.84150E-8
0.50.9365590.9365611.73155E-60.942230.9422302.85150E-70.9458210.9458215.07597E-8
0.60.9418330.94118651.48390E-60.9470950.9470952.43396E-70.9504210.9504214.32130E-8
0.70.9467270.9467291.24520E-60.9516000.9516002.03383E-70.9546740.9546743.60076E-8
0.80.9512600.9512611.02143E-60.9557630.9557631.66055E-70.9586000.9586002.93064E-8
0.90.9554490.955458.16543E-70.9596030.9596031.32019E-70.9622160.9622162.32127E-8
1.00.9593140.9593146.32769E-70.9631390.9631391.01605E-70.9655420.9655421.77807E-8

Table 2

PIA (v4(x, y, t)) and exact solution values with absolute errors for y = 1, t = 0.1 and ℜ = 100.

α = 0.75α = 0.85α = 0.95
xPIAExactErrorPIAExactErrorPIAExactError
0.00.6043550.6043522.95952E-60.5957300.5957304.98075E-70.5902010.5902018.99301E-8
0.10.5963380.5963352.83291E-60.5882280.5882284.74342E-70.5830430.5830438.53620E-8
0.20.5887990.5887962.64341E-60.5811950.5811954.40604E-70.5763470.5763477.90562E-8
0.30.5817290.5817272.41063E-60.5746220.5746224.00125E-70.5701000.5701007.15971E-8
0.40.5751200.5751182.15196E-60.5684930.5684933.55772E-70.5642870.5642876.34946E-8
0.50.5689570.5689561.88212E-60.5627940.5627943.09945E-70.5588900.5588905.51736E-8
0.60.5632250.5632231.61293E-60.5575060.5575062.64561E-70.5538900.5538904.69707E-8
0.70.5579050.5579041.35347E-60.5526090.5526092.21069E-70.5492670.5492673.91387E-8
0.80.5529780.5529771.11025E-60.5480840.5480841.80495E-70.5450000.5450003.18548E-8
0.90.5484250.5484248.87547E-70.5439100.5439101.43499E-70.5410700.5410702.52312E-8
1.00.5442240.5442236.87793E-70.5400660.5400661.10441E-70.5374540.5374541.93268E-8

5 Conclusion

In this study, initially exp-function method is employed to acquire a new exact solution set for fractional coupled Burgers’ system of equations comes with polydispersive sedimentation. Then using PIA, some approximate solutions of the system are presented. It is observed that the exp-function method appears to be a robust and adequate tool for handling of FPDEs. Besides, comparison of the approximate solutions obtained by PIA for α = 0.75, α = 0.85 and α = 0.95 reveals the power and fast convergence rate of the method even after a few approximations. The main advantage of the method is it does not require any special assumptions or transformations. Thus it is obvious that both methods are powerful tools for solution of FPDEs and they are ready to be applied to different types of FPDEs arising in different research areas.

Communicated by Juan Luis García Guirao

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    The surface plot of the PIA solution u4(x, y, t) for ℜ = 100 and t = 0.1 when α = 0.75.

  • View in gallery

    The surface plot of the PIA solution v4(x, y, t) for ℜ = 100 and t = 0.1 when α = 0.75.

  • View in gallery

    The surface plot of the PIA solution u4(x, y, t) for ℜ = 100 and t = 0.1 when α = 0.85.

  • View in gallery

    The surface plot of the PIA solution v4(x, y, t) for ℜ = 100 and t = 0.1 when α = 0.85.

  • View in gallery

    The surface plot of the PIA solution u4(x, y, t) for ℜ = 100 and t = 0.1 when α = 0.95.

  • View in gallery

    The surface plot of the PIA solution v4(x, y, t) for ℜ = 100 and t = 0.1 when α = 0.95.