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]:
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.
There are different types of arbitrary order differentiation. The most widely used are the Riemann-Liouville(RLFD) and Caputo fractional derivatives (CFD).
Along with these definitions, a new fractional derivative definition, namely the conformable fractional derivative, has been introduced by Khalil et al. [16].
where 0 <
Let
where 0 <
Taking account into the following nonlinear time fractional equation in order to explain the basic idea of the implemented method [15]
where the fractional derivatives are in conformable sense. We can introduce the wave variable as
where
Hence Eq.(6) changes into differential equation with integer order as follows.
Due to exp-function method, it is supposed that the wave solution can be regarded in the following form
where
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
where
Subrogating (12) into (11) and expanding in the Taylor series form for only first order derivatives yields
or
Rewriting (14) gives the subsequent PIA(1, 1) iteration formula
In this expansion, all of the derivatives are evaluated at
Think of the fractional coupled Burgers’ equation [24] as
where 0 <
Now assume that the solution of (17) can be described as
Using (18), (19) and (17) led to
Substituting the equations (20) into (17) and equalizing the coefficients of
So the solutions can be obtained as
and
where
Regard the system (16) with the the conditions
Now we introduce a small perturbation parameter
Therefore, terms in formula (15)turn into
and
Subrogating above terms in the iteration formula (15) gives the subsequent partial differential equations
and
Beginning with the initial functions
and using (15), the numerical results are obtained for
Similarly, the fourth order solutions
PIA (
0.0
0.903993
0.903996
2.72276E-6
0.911928
0.911928
4.58229E-7
0.917015
0.917015
8.27357E-8
0.1
0.911369
0.911371
2.60627E-6
0.91883
0.918831
4.36394E-7
0.923600
0.923600
7.85330E-8
0.2
0.918305
0.918308
2.43193E-6
0.925300
0.925301
4.05356E-7
0.929761
0.929761
7.27317E-8
0.3
0.924809
0.924811
2.21778E-6
0.931348
0.931348
3.68115E-7
0.935508
0.935508
6.58693E-8
0.4
0.930889
0.930891
1.97981E-6
0.936986
0.936986
3.27310E-7
0.940856
0.940856
5.84150E-8
0.5
0.936559
0.936561
1.73155E-6
0.94223
0.942230
2.85150E-7
0.945821
0.945821
5.07597E-8
0.6
0.941833
0.9411865
1.48390E-6
0.947095
0.947095
2.43396E-7
0.950421
0.950421
4.32130E-8
0.7
0.946727
0.946729
1.24520E-6
0.951600
0.951600
2.03383E-7
0.954674
0.954674
3.60076E-8
0.8
0.951260
0.951261
1.02143E-6
0.955763
0.955763
1.66055E-7
0.958600
0.958600
2.93064E-8
0.9
0.955449
0.95545
8.16543E-7
0.959603
0.959603
1.32019E-7
0.962216
0.962216
2.32127E-8
1.0
0.959314
0.959314
6.32769E-7
0.963139
0.963139
1.01605E-7
0.965542
0.965542
1.77807E-8
PIA (
0.0
0.604355
0.604352
2.95952E-6
0.595730
0.595730
4.98075E-7
0.590201
0.590201
8.99301E-8
0.1
0.596338
0.596335
2.83291E-6
0.588228
0.588228
4.74342E-7
0.583043
0.583043
8.53620E-8
0.2
0.588799
0.588796
2.64341E-6
0.581195
0.581195
4.40604E-7
0.576347
0.576347
7.90562E-8
0.3
0.581729
0.581727
2.41063E-6
0.574622
0.574622
4.00125E-7
0.570100
0.570100
7.15971E-8
0.4
0.575120
0.575118
2.15196E-6
0.568493
0.568493
3.55772E-7
0.564287
0.564287
6.34946E-8
0.5
0.568957
0.568956
1.88212E-6
0.562794
0.562794
3.09945E-7
0.558890
0.558890
5.51736E-8
0.6
0.563225
0.563223
1.61293E-6
0.557506
0.557506
2.64561E-7
0.553890
0.553890
4.69707E-8
0.7
0.557905
0.557904
1.35347E-6
0.552609
0.552609
2.21069E-7
0.549267
0.549267
3.91387E-8
0.8
0.552978
0.552977
1.11025E-6
0.548084
0.548084
1.80495E-7
0.545000
0.545000
3.18548E-8
0.9
0.548425
0.548424
8.87547E-7
0.543910
0.543910
1.43499E-7
0.541070
0.541070
2.52312E-8
1.0
0.544224
0.544223
6.87793E-7
0.540066
0.540066
1.10441E-7
0.537454
0.537454
1.93268E-8
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