Numerical Solutions of Fractional Burgers’ Type Equations with Conformable Derivative

Chinese Journal of Physics 58 (2019) 75–84

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Numerical Solutions of Fractional Burgers’ Type Equations with Conformable Derivative Mehmet Senola, Orkun Tasbozanb, Ali Kurt a b

T

⁎,b

Department of Mathematics, Faculty of Science and Art, Nevsehir Haci Bektas Veli University, Nevsehir, TURKEY Department of Mathematics, Faculty of Science and Art, Mustafa Kemal University, Hatay, TURKEY

A R T IC LE I N F O

ABS TRA CT

Keywords: Residual Power Series Method Conformable Fractional Derivative Numerical Solutions Burgers’ Type Equations

In this article, we introduce the residual power series method (RPSM) for finding approximate solutions of the time-fractional Burgers type equations using the conformable fractional derivative definition. This definition is simple and effective in the solution procedure of the fractional differential equations that have complicated solutions with classical fractional derivative definitions like Caputo and Riemann-Liouville. The results indicate the proposed method gives significant and reliable solutions.

1. Introduction As the generalization of the integer order calculus, fractional calculus helped scientist to define and model numerous phenomena in many areas of physics and engineering branches. These studies include, fractional differential equations (FDEs) [1,2], fractional kinetic equations [3,4], optics [5,6], signal processing [7], thermodynamics [8] and fluid flow [9] in recent years. There are some common methods used in the literature to obtain approximate or analytical solutions of nonlinear fractional ordinary and partial differential equations. For instance, Adomian decomposition method (ADM) [10] for linear and nonlinear fractional diffusion and wave equations, differential transformation method (DTM) [11] for the convergence of fractional power series, Variational iteration Method (VIM) [12] for the space and time-fractional Burgers equations, homotopy analysis method (HAM) [13] for the conformable fractional Nizhnik-Novikov-Veselov system, finite difference method [14] perturbation-iteration algoritm (PIA) [15] for ordinary fractional differential equations and Elzaki projected differential transform method [16] for system of linear and nonlinear fractional partial differential equations. Many different analytical approaches are used to obtain the exact solutions of fractional partial differential equations arising in different branches of science. For instance exp (−ϕ (ξ )) -expansion method and the novel exponential rational function technique are used to obtain the exact solutions of time fractional Zoomeron equation [17]. Modified Kudryashov method is used to construct new exact solutions for some conformable fractional differential equations [18]. Feng [19] used a method based on the Jacobi elliptic equation to establish the exact solutions of conformable fractional partial differential equations. Many scientists made further studies and explanations on the physical meaning and physical applications of Burgers equation in the past few years. A variety of analytical and numerical solutions constructed for the solution of the problem. Modified Burgers and Burgers-KDV equations which are different forms of the Burgers' equation are regarded in this paper can be used in various scientific areas such as, plasma physics, solid-state physics, optical fibers, biology, fluid dynamics, chemical kinetics, number theory, gas dynamics, heat conduction, turbulence theory etc. [20–22] In this article, the residual power series method [23–30] is used to obtain new approximate solutions of time-fractional Burgers’

⁎

Corresponding author. E-mail addresses: [email protected] (M. Senol), [email protected] (O. Tasbozan), [email protected] (A. Kurt).

https://doi.org/10.1016/j.cjph.2019.01.001 Received 5 November 2018; Received in revised form 4 January 2019; Accepted 15 January 2019 Available online 29 January 2019 0577-9073/ © 2019 The Physical Society of the Republic of China (Taiwan). Published by Elsevier B.V. All rights reserved.

Chinese Journal of Physics 58 (2019) 75–84

M. Senol, et al.

equation

∂αu ∂u ∂ 2u +u − υ 2 = 0, α ∂t ∂x ∂x

(1)

modified Burgers’ equation

∂αu ∂u ∂ 2u + u2 − υ 2 = 0, ∂t α ∂x ∂x

(2)

and Burgers–Korteweg–de Vries equation

∂αu ∂u ∂ 2u ∂3u + ϖu + β 2 + s 3 = 0, α ∂t ∂x ∂x ∂x

(3)

where u = u (x , t ) . In the RPSM, the coefficients of the power series are calculated by means of the concept of residual error with the help of one or more variable algebraic equation chains, and finally, a so-called truncated series solution is obtained [30]. The major improvement of the RPSM is that it can be implemented to the problem directly without linearization, perturbation or discretization and without any transformation by selecting appropriate initial conditions [29]. After giving a few preliminary definitions and brief description of the RSPM, we have presented the solution procedure of two nonlinear fractional partial differential equations that shows the reliability and efficiency of the method. Also figures and a tables are presented in order to compare their numerical results. At last, we discussed about obtained results as a section for conclusion. 2. Preliminaries There are a few definition of fractional derivative of order α > 0. The most widely used are the Riemann-Liouville and Caputo fractional derivatives. Definition 2.1. The Riemann-Liouville fractional derivative operator Dαf(x) defined as [23,31,32]:

dq ⎡ 1 dx q ⎢ Γ(q − α ) ⎣

D α f (x ) =

x

f (t )

∫ (x − t )α+1−q dt ⎤⎥

(4)

⎦

α

where α > 0 and q − 1 < α < q . Definition 2.2. The Caputo fractional derivative of order α defined as [33]:

1 D α f (x ) = J n − α Dnf (x ) = * Γ(n − α )

x

n

∫ (x − t )n−α−1 ⎛⎝ dtd ⎞⎠ f (t ) dt

(5)

α

where α > 0 for n ∈ , n − 1 < α < n, D (.) means Caputo fractional derivative operator and * integral operator. α

J n − α (.)

means Caputo fractional

Recently, a new definition, so-called ”conformable fractional derivative” has been proposed by R. Khalil et al. [34]. Definition 2.3. An α − th order “conformable fractional derivative” of a function defined by

f (t + εt 1 − α ) − (f )(t ) ε→0 ε

Tα (f )(t ) = lim

(6)

for f: [0, ∞) → R and for all t > 0, α ∈ (0, 1). The properties of this new definition are given in the following theorem [34] Theorem 2.4. Let α ∈ (0, 1] and f, g are α-differantiable functions at point t > 0, then 1. 2. 3. 4.

Tα (mf + ng ) = mTα (f ) + nTα (g ) for all m , n ∈  Tα (t p) = pt p − α for all p Tα (f . g ) = fTα (g ) + gTα (f ) f gT (f ) − fT (g ) Tα ( g ) = α 2 α g

5. Tα (c ) = 0 for all constant functions f (t ) = c df (t ) 6. In addition, if f is differentiable, then Tα (f )(t ) = t 1 − α dt Definition 2.5. Let f is a function with n variables x1, ..., xn, and the conformable partial derivatives of f of order α ∈ (0, 1] in xi is defined as follows [35]

f (x1, ...,x i − 1, x i + εx i1 − α , ...,x n ) − f (x1, ...,x n ) dα f (x1, ...,x n) = lim . α ε→0 dx i ε Definition 2.6. The conformable integral of a function f starting from a ≥ 0 is defined as [36] 76

(7)

Chinese Journal of Physics 58 (2019) 75–84

M. Senol, et al. s

Iαa (f )(s ) =

f (t )

∫ t1−α dt.

(8)

a

3. Residual power series method In this section, some important definitions and theorems about residual power series will be given. Theorem 3.1. Suppose that f is an infinitely α−differentiable function at a neigborhood of a point t0 for some 0 < α ≤ 1, then f has the fractional power series expansion of the form: ∞

f (t ) =

∑ k=0

Here

(Tαt0 f )(k ) (t0)

(Tαt0 f )(k ) (t0)(t − t0) kα 1 , t0 < t < t0 + R α , R > 0. α k k!

(9)

represents the application of the fractional derivative k−times [37]. ∞

Definition 3.2. A multiple fractional power series about t0 = 0 is defined by ∑n = 0 fn (x ) t nα for 0 ≤ m − 1 < α < m , where t is a variable and fn(x) are functions called the coefficients of the series. [28,38]. Theorem 3.3. Assume that u(x, t) has a multiple fractional power series representation at t0 = 0 of the form [38] ∞

u (x , t ) =

∑ fn (x ) t nα,

1

0 ≤ m − 1 < α < m, x ∈ I , 0 ≤ t ≤ R α . (10)

n=0 1

If ut(nα ) (x , t ), n = 0, 1, 2, … are continuous on I × (0, R α ), then fn (x ) =

ut(nα ) (x , 0) . αnn !

To clarify the basic concept of RPSM, let’s take a nonlinear fractional differential equation of the form:

Tα u (x , t ) + N [x ] u (x , t ) + R [x ] u (x , t ) = c (x , t ), x ∈ , n − 1 < nα ≤ n, t > 0

(11)

given with the initial condition

f0 (x ) = u (x , 0) = f (x )

(12)

Here, R[x] is a linear, N[x] is a non-linear operator and c(x, t) are continuous functions. The RPSM method made up of stating the solution of the equation (11) subject to (12) as a fractional power series expansion around t = 0 .

f(n − 1) (x ) = Tt(n − 1) α u (x , 0) = h (x )

(13)

The expansion form of the solution is given by ∞

t nα

∑ fn (x ) αnn!

u (x , t ) = f (x ) +

(14)

n=1

In the next step, the k.truncted series of u(x, t), namely uk(x, t) can be written as: k

uk (x , t ) = f (x ) +

t nα

∑ fn (x ) αnn!

(15)

n=1

If the 1. RPS approximate solution u1(x, t) is

u1 (x , t ) = f (x ) + f1 (x )

tα αn

(16)

then uk(x, t) could be reformulated as

uk (x , t ) = f (x ) + f1 (x )

tα + αn

k

t nα

∑ fn (x ) αnn!

(17)

n=2

1 v,

for 0 < α ≤ 1, 0 ≤ t < x ∈ I and k = 2, 3, 4, ... First we express the residual function as

Res (x , t ) = Tα u (x , t ) + N [x ] u (x , t ) + R [x ] u (x , t ) − c (x , t )

(18)

and the k. residual function as

Resk (x , t ) = Tα uk (x , t ) + N [x ] uk (x , t ) + R [x ] uk (x , t ) − g (x , t ), k = 1, 2, 3, ... It is clear that Res (x , t ) = 0 and lim Resk (x , t ) = Res (x , t ) for each x ∈ I and 0 ≤ t. In fact this lead to k →∞

(19) ∂(n − 1) α ∂t (n − 1) α

Resk (x , t ) for

n = 1, 2, 3, ...,k because in the conformable sense, the fractional derivative of a constant is zero [24,29,30]. Solving the equation ∂(n − 1) α ∂t (n − 1) α

Resk (x , 0) = 0 gives us the desired fn(x) coefficients. Thus the un(x, t) approximate solutions can be obtained respectively. 77

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We can express the following theorem for the convergence analysis for the method. Theorem 3.4. If there exists a fixed constant 0 < K < 1 such that un + 1 (x , t ) ≤ K un (x , t ) for all n ∈  and 0 < t < R < 1, then the sequence of approximate solution converges to an exact solution [39]. Proof. For all 0 < t < R < 1, we have ∞

∞

∑

u (x , t ) − un (x , t ) =

ui (x , t )

∑

≤

i=n+1

∞

ui (x , t ) ≤ f (x )

i=n+1

∑

Ki =

i=n+1

K n+1 1−K

f (x )

n →∞ 0. →

(20)

□ 4. Application of Residual Power Series Method 4.1. Solution of time-Fractional Burgers’ Equation Consider the nonlinear time-fractional Burgers’ equation [20]

∂αu ∂u ∂ 2u − υ 2 = 0, +u ∂t α ∂x ∂x

(21)

with the initial condition obtained from the exact solution

c 2 + 2bυ tanh ⎜⎛ ⎝

u (x , 0) = f (x ) = c −

c 2 + 2bυ (x − 2γυ) ⎞ ⎟. 2υ ⎠

(22)

where u = u (x , t ) and υ, γ, b and c are constants. For residual power series ∞

u (x , t ) = f (x ) +

t nα

∑ fn (x ) αnn!

(23)

n=1

and k-th truncated series of u(x, t) k

uk (x , t ) = f (x ) +

t nα

∑ fn (x ) αnn! ,

k = 1, 2, 3, ... (24)

n=1

The k-th residual function of time-fractional Burgers’ equation is:

Resuk (x , t ) =

∂αuk ∂u ∂ 2u + uk k − υ 2k . ∂t α ∂x ∂x

(25)

To determine the coefficients f1(x), in u1(x, t), we should substitute the 1st truncated series u1 (x , t ) = f (x ) + truncated residual function

tα f1 (x ) α

into the 1st

t αf1′ (x ) ⎞ t αf1″ (x ) ⎞ t αf (x ) ⎞ ⎛ ⎛ Resu1 (x , t ) = f1 (x ) + ⎛f (x ) + 1 ⎜f ′ (x ) + ⎟ − υ ⎜f ″ (x ) + ⎟. α ⎠⎝ α ⎠ α ⎠ ⎝ ⎝ ⎜

⎟

(26)

Now for the substitution of t = 0 through equation Resu1(x, t) and to obtain

Resu1 (x , 0) = f1 (x ) + f (x ) f ′ (x ) − υf ″ (x ).

(27)

Thus for Res1 (x , 0) = 0

f1 (x ) = −f (x ) f ′ (x ) + υf ″ (x ).

(28)

Therefore, we obtain the 1st RPS approximate solutions of time-fractional Burgers’ equation as

u1 (x , t ) = f (x ) + Again,

to

t α (−f (x ) f ′ (x ) + υf ″ (x )) . α

determine tα

the

u2 (x , t ) = f (x ) + f1 (x ) α + f2 (x ) Resu2 (x , t ) = f1 (x ) +

t 2α 2α2

t αf2 (x ) α

second

unknown

(29) coefficient

f2(x),

we

substitute

the

2nd

truncated

series

solution

into the 2nd truncated residual function and obtain

+ ⎜⎛f (x ) + ⎝

t αf1′ (x ) t 2αf 2′ (x ) ⎞ t αf1 (x ) t 2αf2 (x ) ⎞ ⎛ + + ⎟ ⎜f ′ (x ) + ⎟ 2 α α 2α ⎠ ⎝ 2α 2 ⎠

t αf1″ (x ) t 2αf 2″ (x ) ⎞ − υ ⎛⎜f ″ (x ) + + ⎟. α 2α 2 ⎠ ⎝ Now, applying Tα on both sides of Resu2(x, t) and equating to 0 for t = 0 gives: 78

(30)

Chinese Journal of Physics 58 (2019) 75–84

M. Senol, et al.

f2 (x ) = −f1 (x ) f ′ (x ) − f (x ) f1′ (x ) + υf1″ (x ).

(31)

Therefore the 2nd RPS approximate solutions of time-fractional Burgers’ equation is obtained as:

u2 (x , t ) = f (x ) +

t 2α (−f1 (x ) f ′ (x ) − f (x ) f1′ (x ) + υf1″ (x )) t α (−f (x ) f ′ (x ) + υf ″ (x )) + . α 2α 2

(32)

In the same manner, we apply the same procedure for n = 3 to obtain the following results.

f3 (x ) = −f2 (x ) f ′ (x ) − 2f1 (x ) f1′ (x ) − f (x ) f2′ (x ) + υf2″ (x ),

(33)

t 2α (−f1 (x ) f ′ (x ) − f (x ) f1′ (x ) + υf1″ (x )) t α (−f (x ) f ′ (x ) + υf ″ (x )) + α 2α 2 t 3α (−f2 (x ) f ′ (x ) − 2f1 (x ) f1′ (x ) − f (x ) f 2′ (x ) + υf 2″ (x )) + . 6α3

u3 (x , t ) = f (x ) +

f4 (x ) =

−f3 (x ) f ′ (x ) − 3f2 (x ) f1′ (x ) − 3f1 (x ) f2′ (x ) − f (x ) f3′ (x ) + υf3″ (x ) α

(34)

,

(35)

t 2α (−f1 (x ) f ′ (x ) − f (x ) f1′ (x ) + υf1″ (x )) t α (−f (x ) f ′ (x ) + υf ″ (x )) + α 2α 2 t 3α (−f2 (x ) f ′ (x ) − 2f1 (x ) f1′ (x ) − f (x ) f 2′ (x ) + υf 2″ (x ))

u4 (x , t ) = f (x ) + + +

t 4α (−f3 (x ) f ′ (x )

6α3 − 3f2 (x ) f1′ (x ) − 3f1 (x ) f 2′ (x ) − f (x ) f3′ (x ) + υf3″ (x )) 24α 4

.

(36)

In Table 1, the fourth-order approximate RPSM solutions of time-fractional Burgers’ equation are compared numerically with the exact solution tα

u (x , t ) = c −

⎛ c 2 + 2bυ (x − c α − 2γυ ) ⎞ c 2 + 2bυ tanh ⎜ ⎟⎟ ⎜ 2υ ⎠ ⎝

(37)

where B is an arbitrary integral constant. Absolute errors are presented for α = 0.50, α = 0.75 and α = 0.95 and. The results indicate that as the x values increase the absolute errors decrease. Besides, as the α values increase, the absolute errors decrease. Also the Table 1 show competitive solutions of the RPSM with highly approximate results. Moreover, in Fig. 1, the surface plots of the approximate and analytical solutions are illustrated for α = 0.50, α = 0.75 and α = 0.95. 4.2. Solution of time-Fractional modified Burgers’ Equation Now consider the time-fractional modified Burgers’ eqution of the form [20]

∂αu ∂u ∂ 2u + u2 − υ 2 = 0, ∂t α ∂x ∂x

(38)

with the initial condition Table 1 Comparison of RPSM approximate (u3(x, t)) and exact solutions with absolute errors for c = 1, b = 1, υ = 2, γ = 1 and t = 0.1. α = 0.75

α = 0.50

α = 0.95

x

RPSM

Exact

Abs. Error

RPSM

Exact

Abs. Error

RPSM

Exact

Abs. Error

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

3.21097 3.20803 3.20474 3.20106 3.19696 3.19239 3.18728 3.18159 3.17524 3.16817 3.16028

3.21102 3.20808 3.20479 3.20112 3.19702 3.19244 3.18734 3.18165 3.17530 3.16822 3.16033

4.67431E-5 4.98003E-5 5.26619E-5 5.51885E-5 5.72035E-5 5.84864E-5 5.87660E-5 5.77131E-5 5.49340E-5 4.99649E-5 4.22688E-5

3.19722 3.19267 3.18759 3.18192 3.17561 3.16856 3.16071 3.15197 3.14223 3.13138 3.11932

3.19722 3.19267 3.18759 3.18193 3.17561 3.16856 3.16071 3.15197 3.14223 3.13138 3.11932

3.61456E-7 3.83535E-7 4.03596E-7 4.20446E-7 4.32587E-7 4.38162E-7 4.34903E-7 4.20080E-7 3.90452E-7 3.42235E-7 2.71092E-7

3.19175 3.18656 3.18078 3.17433 3.16714 3.15912 3.15019 3.14025 3.12919 3.11689 3.10321

3.19175 3.18656 3.18078 3.17433 3.16714 3.15912 3.15019 3.14025 3.12919 3.11689 3.10321

1.12244E-8 1.18929E-8 1.24934E-8 1.29874E-8 1.33271E-8 1.34532E-8 1.32936E-8 1.27616E-8 1.17546E-8 1.01531E-8 7.82087E-9

79

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Fig. 1. The surface plots of u4(x, t) for c = 0.1, b = 0.9, υ = 1.5, γ = 1.5, and for (a) α = 0.50, (b) α = 0.75, (c) α = 0.95 and their corresponding exact solutions (d), (e) and (f) respectively.

3c

u (x , 0) = f (x ) = − 1 − cosh

(

2c (x + 3rυ) υ

) − sinh (

2c (x + 3rυ) υ

)

(39a)

where u = u (x , t ) and υ, c and r are constants. Now repeating the above residual power series proedure we obtained the following results.

f1 (x ) = −f (x ) f ′ (x ) + υf ″ (x ), u1 (x , t ) = f (x ) +

(40)

t α (−f 2 (x ) f ′ (x ) + υf ″ (x )) , α

(41)

f2 (x ) = −2f (x ) f1 (x ) f ′ (x ) − f 2 (x ) f1′ (x ) + υf1″ (x ), u2 (x , t ) = f (x ) +

(42)

t 2α (−2f (x ) f1 (x ) f ′ (x ) − f 2 (x ) f1′ (x ) + υf1″ (x )) t α (−f (x ) f ′ (x ) + υf ″ (x )) + 2α 2 α

f3 (x ) = −2f12 (x ) f ′ (x ) − 2f (x ) f2 (x ) f ′ (x ) − 4f (x ) f1 (x ) f1′ (x ) − f 2 (x ) f2′ (x ) + υf2″ (x ),

t 2α (−2f (x ) f1 (x ) f ′ (x ) − f 2 (x ) f1′ (x ) + υf1″ (x )) t α (−f (x ) f ′ (x ) + υf ″ (x )) + α 2α 2 2 3 α ′ ′ ′ t (−2f1 (x ) f (x ) − 2f (x ) f2 (x ) f (x ) − 4f (x ) f1 (x ) f1 (x ) − f 2 (x ) f 2′ (x ) + υf 2″ (x )) + . 6α3

(43) (44)

u3 (x , t ) = f (x ) +

f4 (x ) = − 6f1 (x ) f2 (x ) f ′ (x ) − − 6f (x ) f2 (x ) f1′ (x ) −

2f (x ) f3 (x ) f ′ (x ) − 6f12 (x ) f1′ (x ) 6f (x ) f1 (x ) f2′ (x ) − f 2 (x ) f3′ (x ) +

υf3″ (x ),

(45)

(46)

t 2α (−2f (x ) f1 (x ) f ′ (x ) − f 2 (x ) f1′ (x ) + υf1″ (x )) t α (−f (x ) f ′ (x ) + υf ″ (x )) + 2α 2 α t 3α (−2f12 (x ) f ′ (x ) − 2f (x ) f2 (x ) f ′ (x ) − 4f (x ) f1 (x ) f1′ (x ) − f 2 (x ) f 2′ (x ) + υf 2″ (x ))

u4 (x , t ) = f (x ) + + + +

6α3 t 4α (−6f1 (x ) f2 (x ) f ′ (x ) − 2f (x ) f3 (x ) f ′ (x ) − 6f12 (x ) f1′ (x )) 24α 4 t 4α (−6f (x ) f2 (x ) f1′ (x ) − 6f (x ) f1 (x ) f 2′ (x ) − f 2 (x ) f3′ (x ) + υf3″ (x )) 24α 4

(47) 80

Chinese Journal of Physics 58 (2019) 75–84

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Table 2 Comparison of RPSM approximate (u4(x, t)) and exact solutions with absolute errors for c = 0.1, r = 0.01, υ = 1 and x = −1. α = 0.75

α = 0.50

α = 0.95

t

RPSM

Exact

Abs. Error

RPSM

Exact

Abs. Error

RPSM

Exact

Abs. Error

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

−1.28821 −1.25302 −1.23939 −1.22927 −1.22097 −1.21381 −1.20747 −1.20175 −1.19650 −1.19166 −1.18714

−1.28821 −1.25302 −1.23939 −1.22927 −1.22096 −1.21380 −1.20745 −1.20171 −1.19646 −1.19159 −1.18706

0.00000000 2.91417E-7 1.61184E-6 4.36721E-6 8.84005E-6 1.52557E-5 2.38040E-5 3.46504E-5 4.79422E-5 6.38121E-5 8.23816E-5

−1.28821 −1.27462 −1.26564 −1.25793 −1.25100 −1.24462 −1.23865 −1.23302 −1.22767 −1.22256 −1.21767

−1.28821 −1.27462 −1.26564 −1.25793 −1.25100 −1.24462 −1.23865 −1.23301 −1.22766 −1.22255 −1.21766

0.00000000 2.23481E-9 2.96367E-8 1.33890E-7 3.89356E-7 8.89592E-7 1.74511E-6 3.08170E-6 5.03917E-6 7.77023E-6 1.14396E-5

−1.28821 −1.28138 −1.27513 −1.26913 −1.26333 −1.25770 −1.25220 −1.24684 −1.24158 −1.23644 −1.23140

−1.28821 −1.28138 −1.27513 −1.26913 −1.26333 −1.25770 −1.25220 −1.24684 −1.24158 −1.23644 −1.23140

0.00000000 6.9280E-11 1.84574E-9 1.25438E-8 4.87331E-8 1.39371E-7 3.28385E-7 6.76937E-7 1.26543E-6 2.19533E-6 3.59076E-6

In Table 2, the fourth-order approximate RPSM solutions of time-fractional modified Burgers’ equation are compared numerically with the exact solution

3c

u (x , t ) = −

α

1−

t ⎛ 2c ⎛x − c α + 3rυ ⎞⎠ ⎞ cosh ⎜ ⎝ υ ⎟⎟ ⎜

⎝

tα

⎛ 2c ⎛x − c α − sinh ⎜ ⎝ υ ⎜ ⎠ ⎝

+ 3rυ ⎞ ⎞ ⎠

⎟⎟ ⎠

(48)

for α = 0.50, α = 0.75 and α = 0.95 and absolute errors are presented. The results show that as the absolute value of the t values increase the absolute errors also increase. Besides, as the α values increase, the absolute errors decrease. Also in Fig. 2, the surface plots of the approximate and analytical solutions are compared for α = 0.50, α = 0.75 and α = 0.95. 4.3. Solution of time-Fractional Burgers’–Kdv Equation Now consider the time-fractional modified Burgers’ eqution of the form [20]

∂αu ∂u ∂ 2u ∂3u + β 2 + s 3 = 0, + ϖu α ∂t ∂x ∂x ∂x

(49)

with the initial condition

Fig. 2. The surface plots of u4(x, t) for c = 0.1, r = 0.01, υ = 1, and for (a) α = 0.50, (b) α = 0.75, (c) α = 0.95 and their corresponding exact solutions (d), (e) and (f) respectively. 81

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( ( ) − sinh ( ) ) 2βx

u (x , 0) = f (x ) =

12β 2 cosh 5s 12β 2 − 25sϖ 25sϖ 1 + cosh

(

2βx 5s

( ) − sinh ( ) ) βx 5s

βx 5s

2

(50)

where u = u (x , t ) and ϖ, β and s are constants. Now repeating the above residual power series proedure we obtained the following results.

f1 (x ) = −ϖf (x ) f ′ (x ) − βf ″ (x ) − sf (3) (x ),

(51)

t α (−ϖf (x ) f ′ (x ) − βf ″ (x ) − sf (3) (x )) , α

(52)

f2 (x ) = −ϖf1 (x ) f ′ (x ) − ϖf (x ) f1′ (x ) − βf1″ (x ) − sf1(3) (x ),

(53)

u1 (x , t ) = f (x ) +

t α (−ϖf (x ) f ′ (x ) − βf ″ (x ) − sf (3) (x )) α t 2α (−ϖf1 (x ) f ′ (x ) − ϖf (x ) f1′ (x ) − βf1″ (x ) − sf1(3) (x ))

u2 (x , t ) = f (x ) + +

2α 2

(54)

f3 (x ) = −ϖf2 (x ) f ′ (x ) − 2ϖf1 (x ) f1′ (x ) − ϖf (x ) f2′ (x ) − βf2″ (x ) − sf 2(3) (x ),

(55)

t α (−ϖf (x ) f ′ (x ) − βf ″ (x ) − sf (3) (x )) α t 2α (−ϖf1 (x ) f ′ (x ) − ϖf (x ) f1′ (x ) − βf1″ (x ) − sf1(3) (x ))

u3 (x , t ) = f (x ) + + +

f4 (x ) =

t 3α (−ϖf2 (x ) f ′ (x )

2α 2 − 2ϖf1 (x ) f1′ (x ) − ϖf (x ) f 2′ (x ) − βf 2″ (x ) − sf 2(3) (x )) 6α3

.

−ϖf3 (x ) f ′ (x ) − 3ϖf2 (x ) f1′ (x ) − 3ϖf1 (x ) f2′ (x ) − ϖf (x ) f3′ (x ) − βf3″ (x ) − sf 3(3) (x ) α

(56)

,

(57)

t α (−ϖf (x ) f ′ (x ) − βf ″ (x ) − sf (3) (x )) α t 2α (−ϖf1 (x ) f ′ (x ) − ϖf (x ) f1′ (x ) − βf1″ (x ) − sf1(3) (x ))

u4 (x , t ) = f (x ) + + + +

2α 2 t 3α (−ϖf2 (x ) f ′ (x ) − 2ϖf1 (x ) f1′ (x ) − ϖf (x ) f 2′ (x ) − βf 2″ (x ) − sf 2(3) (x )) 6α3 t 4α (−ϖf3 (x ) f ′ (x ) − 3ϖf2 (x ) f1′ (x ) − 3ϖf1 (x ) f 2′ (x ) − ϖf (x ) f3′ (x ) − βf3″ (x ) − sf 3(3) (x )) 24α 4

.

(58)

In Table 3, the fourth-order approximate RPSM solutions of time-fractional Burgers–Korteweg–de Vries equation are compared numerically with the exact solution Table 3 Comparison of RPSM approximate (u3(x, t)) and exact solutions with absolute errors for s = 1, β = 1, ϖ = 1 and t = 0.1. α = 0.75

α = 0.50

α = 0.95

x

RPSM

Exact

Abs. Error

RPSM

Exact

Abs. Error

RPSM

Exact

Abs. Error

− 1.0 − 0.9 − 0.8 − 0.7 − 0.6 − 0.5 − 0.4 − 0.3 − 0.2 − 0.1 0.0

0.330901 0.333532 0.336145 0.338740 0.341315 0.343871 0.346406 0.348920 0.351412 0.353882 0.356330

0.330901 0.333532 0.336145 0.338740 0.341315 0.343871 0.346406 0.348920 0.351412 0.353882 0.356330

4.48850E-11 4.33953E-11 4.18117E-11 4.01384E-11 3.83798E-11 3.65412E-11 3.46276E-11 3.26443E-11 3.05975E-11 2.84928E-11 2.63364E-11

0.333398 0.336012 0.338608 0.341184 0.343741 0.346277 0.348792 0.351285 0.353756 0.356205 0.358630

0.333398 0.336012 0.338608 0.341184 0.343741 0.346277 0.348792 0.351285 0.353756 0.356205 0.358630

3.30624E-13 3.19578E-13 3.07809E-13 2.95375E-13 2.82163E-13 2.68341E-13 2.54186E-13 2.39364E-13 2.24099E-13 2.08389E-13 1.92457E-13

0.334146 0.336755 0.339345 0.341916 0.344467 0.346997 0.349506 0.351993 0.354458 0.356900 0.359319

0.334146 0.336755 0.339345 0.341916 0.344467 0.346997 0.349506 0.351993 0.354458 0.356900 0.359319

1.01030E-14 9.71445E-15 9.49241E-15 9.04832E-15 8.60423E-15 8.21565E-15 7.77156E-15 7.32747E-15 6.82787E-15 6.32827E-15 5.88418E-15

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Fig. 3. The surface plots of u4(x, t) for s = 1, β = 1, ϖ = 1, and for (a) α = 0.50, (b) α = 0.75, (c) α = 0.95 and their corresponding exact solutions (d), (e) and (f) respectively. 2α

⎜

u (x , t ) =

12β 2 − 25sϖ

2α

⎛ ⎛ 2β ⎛x − 625β sαt ⎞ ⎞ ⎛ 2β ⎛x − 625β sαt ⎞ ⎞ ⎞ ⎠ ⎠ 12β 2 ⎜cosh ⎜ ⎝ 5s − sinh ⎟ ⎜ ⎝ 5s ⎟⎟ ⎜ ⎟ ⎜ ⎟⎟ ⎜ ⎝ ⎠ ⎝ ⎠⎠ ⎝ ⎟

⎜

⎟

2α

2

2α

⎛ ⎛ β ⎛x − 625β sαt ⎞ ⎞ ⎛ β ⎛x − 625β sαt ⎞ ⎞ ⎞ 25sϖ ⎜1 + cosh ⎜ ⎝ 5s ⎠ ⎟ − sinh ⎜ ⎝ 5s ⎠ ⎟ ⎟ ⎜ ⎟ ⎜ ⎟⎟ ⎜ ⎝ ⎠ ⎝ ⎠⎠ ⎝ ⎜

⎟

⎜

⎟

(59)

for α = 0.50, α = 0.75 and α = 0.95 and absolute errors are presented. The results show that as the x values increase the absolute errors decrease. Besides, as the α values increase, the absolute errors decrease. Also in Fig. 3, comparison of the surface plots of the approximate and exact solutions are given for α = 0.50, α = 0.75 and α = 0.95. 5. Conclusion In this paper, approximate solutions of the nonlinear time-fractional Burgers’, modified Burgers’ and Burgers-Korteweg-de Vries equations are obtained by the residual power series method (RPSM). One can easily transform fractional differential equations to the known classical differential equations by using conformable fractional derivative definition. By the proposed method and conformable fractional derivative definition, it is shown a very simple way of obtaining approximate solutions for nonlinear fractional partial differential equations. Approximate solutions are compared with the exact solutions to show the reliability of the method. Absolute errors are given with approximate and exact solutions with the help of figures and tables. Therefore, we can conclude that the method is a very effective and reliable tool for FDEs arising in different branches of physics and engineering. Supplementary material Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.cjph.2019.01.001. References [1] C. Celik, M. Duman, Finite element method for a symmetric tempered fractional diffusion equation, Appl. Num. Math. 120 (2017) 270–286. [2] A. Yakar, H. Kutlay, Monotone iterative technique via initial time different coupled lower and upper solutions for fractional differential equations, Filomat 31 (4) (2017) 1031–1039. [3] P. Agarwal, M. Chand, G. Singh, Certain fractional kinetic equations involving the product of generalized k-bessel function, Alexandria Engineering Journal 55 (4) (2016) 3053–3059. [4] M. Chand, J.C. Prajapati, E. Bonyah, Fractional integrals and solution of fractional kinetic equations involving k-mittag-leffler function, Transactions of A. Razmadze Mathematical Institute 171 (2) (2017) 144–166.

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