Approximate analytical solutions of fractional gas dynamic equations

$Page 1: Approximate analytical solutions of fractional gas dynamic equations$
Applied Mathematics and Computation 217 (2011) 9905–9915

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate/amc

Approximate analytical solutions of fractional gas dynamic equations

S. Das ⇑, R. KumarDepartment of Applied Mathematics, Institute of Technology, Banaras Hindu University, Varanasi 221 005, India

a r t i c l e i n f o a b s t r a c t

Keywords:Fractional order PDEsCaputo derivativeDifferential transforms methodFractional Brownian motion

0096-3003/$ - see front matter � 2011 Elsevier Incdoi:10.1016/j.amc.2011.03.144

⇑ Corresponding author.E-mail address: [email protected] (S. Da

In this article, differential transform method (DTM) has been successfully applied to obtainthe approximate analytical solutions of the nonlinear homogeneous and non-homoge-neous gas dynamic equations, shock wave equation and shallow water equations with frac-tional order time derivatives. The true beauty of the article is manifested in its emphaticapplication of Caputo fractional order time derivative on the classical equations with theachievement of the highly accurate solutions by the known series solutions and even formore complicated nonlinear fractional partial differential equations (PDEs). The methodis really capable of reducing the size of the computational work besides being effectiveand convenient for solving fractional nonlinear equations. Numerical results for differentparticular cases of the equations are depicted through graphs.

� 2011 Elsevier Inc. All rights reserved.

1. Introduction

The equations of gas dynamics are mathematical expressions based on the physical laws of conservation namely, the lawsof conservation of mass, conservation of momentum, conservation of energy etc. The nonlinear equations of ideal gasdynamics are applicable for three types of nonlinear waves like shock fronts, rarefactions and contact discontinuities. In1981, Steger and Warming [1] addressed that the conservation-law form of the in viscid gas dynamic equations possessesa remarkable property by virtue of which the nonlinear flux vectors are homogeneous functions of degree one which permitsthe splitting of flux vectors into sub vectors by similarity transformations and as a result new explicit and implicit dissipativefinite-difference schemes are developed for solving first-order hyperbolic systems of equations. The different types of gasdynamics equations in physics have been solved by Elizarova [2], Ames [3], Evans and Bulut [4], Polyanin and Zaitsev [5]by applying various kinds of analytical and numerical methods. In 1985, Aziz and Anderson [6] used a pocket computer (Ca-sio model FX-702P) to solve some problems arising in gas dynamics. Liu [7] has studied nonlinear hyperbolic-parabolic par-tial differential equations related to gas dynamics and mechanics. In 2003, Rasulov and Karaguler [8] applied the differencescheme for solving nonlinear system of equations of gas dynamic problems for a class of discontinuous functions.

Shallow water (SW) equations, a set of hyperbolic PDEs that describe the flow below a pressure surface in a fluid whichare derived from Navier–Stokes equations, nowadays is an important area of research. The equations are used to model Ros-shy and Kelvin waves in the atmosphere, rivers, lakes and oceans. The equations are also used to study the physical phenom-ena like Tsunami wave and modeling the transport of chemical species. For shallow water equation, it is considered that thedepth of water bodies is relatively small in comparison with the scale of the wave propagation of the body. The free orbitalmotion of water is disrupted and unable to return to the original motion during their flow in the ocean bottom. As a resultwater becomes shallower which causes higher and steeper swelling ultimately producing familiar sharp-crested wave shape.Various researchers like Whitham [9], Broer [10], Kaup [11] have studied the propagations of shallow water in their

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9906 S. Das, R. Kumar / Applied Mathematics and Computation 217 (2011) 9905–9915

literatures. Crnjaric-zic et al. [12] have made a sincere attempt to extend balanced finite volume WENO and central WENOschemes to the hyperbolic balance laws with geometric source term and flux function for solving shallow water and openchannel flow equations where the source term depends on the channel geometry. The different numerical schemes for solv-ing SW equations can be found in the research articles of Bermudez and Vazquez [13], Burguete and Garcia-Navarro [14],Hubbard and Garcia- Navarro [15], Gallouet et. al. [16], Garcia- Navarro and Vazquez- Cendon [17], etc.

During the last decade, fractional calculus has found diverse applications in various scientific and technological fieldswhich can be found in the monograph of Podlubny [18]. Fractional differential equations are increasingly used to modelproblems in fluid mechanics, acoustics, biology, electromagnetism, diffusion, signal processing and many other physical pro-cesses [19,20,21].

The authors in this article have made a sincere endeavour to solve nonlinear gas dynamics equations, shock wave equa-tion and shallow water equations by using a powerful numerical method called generalized two dimension differentialtransform. The differential transform method was first proposed and applied to solve linear and nonlinear initial value prob-lems in electric circle analysis by Zhou [22]. The DTM gives approximate analytical solution in the form of a polynomial. It isdifferent from the traditional higher order Taylor series method which requires symbolic completion of the necessary deriv-atives of the data functions. The DTM introduces a promising approach for many applications in various domains of Sciences.However, DTM has some drawbacks. Using DTM we obtain a series solution which is actually a truncated series solution thatdoes not exhibit the real behaviour of the problems but gives a good approximation to the true solution in a very small re-gion. But there are a lot of advantages of the method which encourages the researchers to apply the method for getting reli-able results towards obtaining rapid convergence of the solution in comparison with the other existing methods. Thismethod gives approximate analytical solution on each subinterval which is not possible in the purely numerical methods.The absence of small parameter assumption, no requirement of linearization or discretization, very little computational bur-den and the simplistic approach of the method are certain attributes that make it very relevant for solving a large class oflinear and nonlinear differential equations.

In 2001, Jang et al. [23] stated that this method is an iterative procedure for obtaining Taylor series solutions to differen-tial equations and so it requires less computational work for systems of higher order as compared to the Taylor series meth-od. Ayaz [24] added two theorems on it and successfully applied the method to solve linear and nonlinear problems withlesser computational work. Yu and Chen [25,26] applied this method to the optimization of the rectangular tins with variablethermal parameters. Chen and Liu [27] have applied this method to solve two-boundary-value problems. Presently thismethod has been successfully employed to solve different types of nonlinear problems in science and engineering. In2009, Ahmad [28] did a work to evaluate the performance of DTM by comparing the continuous solution of Chaotic Genesiosystem obtained using DTM to that obtained using Runge–Kutta method. Recently, Zou et al. [29] has successfully appliedDTM to obtain analytical solution of solitary waves with and without continuity at crest governed by Cammasa–Holm equa-tion. The article provides a new analytical way to solve solitary wave problem with discontinuity. Jafari et al. [30] usedhomotopy analysis method to solve the nonlinear gas dynamics equations. A similar type of problem was later solved byapplying DTM by Jafari et al. [31]. Allan and Al-Khaled [32] applied Adomian decomposition method (ADM) for gettingthe solution of shock wave equation. The same type of problem was solved by Khatami et al. [33] using the mathematicaltools viz., Homotopy analysis method (HAM) and Variational iteration method (VIM), while Berberler and Yildirim [34] usedHomotopy perturbation method for obtaining analytical approach to the shock wave equation. In 2001, Xie et al. [35] havegiven the exact solutions of coupled Whitham-Broer-Kaup (WBK) equations which describe the propagation of shallowwater waves. Later, the similar type of coupled equations was solved by Ganji et al. [36] using homotopy perturbation meth-od and Rashidi and Erfani [37] using DTM. Recently Liu and Hou [38] have solved time fractional coupled Burger equationsby DTM. But to the best of authors’ knowledge these problems with fractional time derivative have not yet been solved byany researcher. The fractional order system of the gas dynamic equations and shock wave equation are obtained by replacingthe first order time derivative term by fractional order b(0 < b 6 1). The advantage of the fractional order system is that theyallow greater degree of freedom in the model. An integer order differential operator is a local operator. Whereas the frac-tional order differential operator is non local in the sense that it takes into account the fact that the future state not onlydepends upon the present state but also upon all of the history of its previous states. For this realistic property, the fractionalorder systems are becoming popular. Another reason behind using fractional order derivatives is that these are naturally re-lated to the systems with memory which prevails for most of the physical and scientific system models.

In this article, a sincere attempt has been made to explore the analytical expressions of probability density functions ofhomogeneous and non homogeneous nonlinear gas dynamic equations, shock wave equation and shallow water equationsfor different fractional Brownian motions and also for the standard motion. The numerical results of the problems are pre-sented graphically.

2. Generalized two dimensional differential transform method

The generalized two dimensional differential transform method [39,40,41] is applied to solve various nonlinear gas dy-namic equations with fractional time derivatives. The proposed method is based on the combination of the classical differ-ential transform method (DTM) and generalized Taylor’s formula [42]. Considering the function of two variables u(x, t) isanalytic and continuously differentiable with respect to time t in the domain of interest and assuming that it can be repre-

S. Das, R. Kumar / Applied Mathematics and Computation 217 (2011) 9905–9915 9907

sented as the product of two single variable functions i.e., u(x, t) = f(x)g(t), based on the properties of the generalized twodimensional differential transform.

uðx; tÞ ¼X1k¼0

FaðkÞðx� x0ÞkaX1h¼0

GbðhÞðt � t0Þhb ¼X1k¼0

X1h¼0

Ua;bðk; hÞðx� x0Þkaðt � t0Þhb;

where 0 < a, b 6 1 and Ua,b(k,h) = Fa(k)Gb(h) is called the spectrum of u(x, t).

Ua;bðk;hÞ ¼1

Cðakþ 1ÞCðbhþ 1Þ Dax0

� �kDb

t0

� �huðx; tÞ

� �x0 ;t0

;

where Dax0

� �k¼ Da

x0

� �Da

x0

� �Da

x0

� �. . . ; (k-times).

In case of a = 1 and b = 1, the generalized two dimensional differential transform method reduces to the classical twodimensional differential transform method.

Theorems. Let U(k,h), V(k,h) and W(k,h) be the differential transformations of the functions u(x, t), v(x, t) and w(x, t) respectively.Then

(a) If u(x, t) = v(x, t) ± w(x, t) then Ua,b(k,h) = Va,b(k,h) ± Wa,b(k,h)(b) If u(x, t) = a v(x, t), a 2 R then Ua,b(k,h) = a Va,b(k,h)(c) If u(x, t) = v(x, t)w(x, t) then Ua;bðk;hÞ ¼

Pkr¼0

Phs¼0Va;bðr;h� sÞWa;bðk� r; sÞ

(d) If u(x, t) = (x � x0)ma(t � t0)nb then Ua,b(k,h) = d(k �m)d(h � n).

(e) If uðx; tÞ ¼ Dax0

vðx; tÞ then Ua;bðk;hÞ ¼ Cðaðkþ1Þþ1ÞCðakþ1Þ Va;bðkþ 1;hÞ:

(f) If uðx; tÞ ¼ Dcx0

vðx; tÞ;m� 1 < c 6 m and u(x, t) = f(x)g(t) then

Ua;bðk;hÞ ¼Cðaðkþ 1Þ þ 1Þ

Cðakþ 1Þ Va;bðkþ c=a;hÞ:

3. Applications

In this section three examples on nonlinear fractional order homogeneous and non-homogeneous time-fractional gas dy-namic equations and shock wave equation are solved to demonstrate the performance and efficiency of the DTM.

Example 1. Consider the following homogeneous nonlinear time-fractional gas dynamic equation

@bu@tb þ u

@u@x� uð1� uÞ ¼ 0; 0 < b 6 1; ð1Þ

with the initial condition

uðx;0Þ ¼ e�x: ð2Þ

Taking the generalized two dimensional differential transform of Eq. (1) and using related theorem, we obtain

Cðbðhþ 1Þ þ 1ÞCðbhþ 1Þ U1;bðk;hþ 1Þ þ

Xk

r¼0

Xh

s¼0

fðkþ 1� rÞU1;bðkþ 1� r; sÞU1;bðr;h� sÞg � U1;bðk; hÞ

þXk

r¼0

Xh

s¼0

U1;bðk� r; sÞU1;bðr; h� sÞ ¼ 0:

ð3Þ

The related initial condition should also be transformed as follows:

Uðk;0Þ ¼ ð�1Þk

k!; k ¼ 0;1;2; . . . ð4Þ

Substituting Eq. (4) into Eq. (3), we have calculated some components of series solution as given below:

U1;bð0;1Þ ¼1

Cðbþ 1Þ ; U1;bð0;2Þ ¼1

Cð2bþ 1Þ ; U1;bð0;3Þ ¼1

Cð3bþ 1Þ ; . . .

U1;bð1;1Þ ¼ �1

Cðbþ 1Þ ; U1;bð1;2Þ ¼ �1

Cð2bþ 1Þ ; U1;bð1;3Þ ¼ �1

Cð3bþ 1Þ ; . . .

U1;bð2;1Þ ¼1

2!Cðbþ 1Þ ; U1;bð2;2Þ ¼1

2!Cð2bþ 1Þ ; U1;bð2;3Þ ¼1

2!Cð3bþ 1Þ ; . . .

U1;bð3;1Þ ¼ �1

3!Cðbþ 1Þ ; U1;bð3;2Þ ¼ �1

3!Cð2bþ 1Þ ; U1;bð3;3Þ ¼ �1

3!Cð3bþ 1Þ ; . . .


Therefore,

uðx; tÞ ¼X1k¼0

X1h¼0

U1;bðk; hÞxkthb: ð5Þ

Substituting all U1,b(k,h) into Eq. (5), we have series solution as follows

uðx; tÞ ¼ e�x 1þ 1Cðbþ 1Þ t

b þ 1Cð2bþ 1Þ t

2b þ 1Cð3bþ 1Þ t

3b þ � � ��

¼ e�xEbðtbÞ; ð6Þ

which is the exact solution of the time- fractional gas dynamic equation (1), where Eb(z) is the Mittag–Lefller function de-fined by EbðzÞ ¼

P1n¼0

zn

Cðbnþ1Þ.

Now for the standard case i.e. forb = 1, u(x, t) becomes u(x, t) = et�x, the result is same as the result of Jafari et al. [31].

Example 2. Consider the following non-homogeneous nonlinear time-fractional gas dynamic equation

@bu@tb þ u

@u@x� uð1� uÞ ¼ �et�x; 0 < b 6 1; ð7Þ


uðx;0Þ ¼ 1� e�x: ð8Þ

Taking the generalized two dimensional differential transform of Eq. (7), we have

U1;bðk;hþ 1Þ ¼ Cðbhþ 1ÞCðbðhþ 1Þ þ 1Þ �

Xk

r¼0

Xh

s¼0

fðkþ 1� rÞU1;bðkþ 1� r; sÞ"

U1;bðr; h� sÞg þ U1;bðk;hÞ

�Xk

r¼0

Xh

s¼0

U1;bðk� r; sÞU1;bðr;h� sÞ � ð�1Þk

k!

1h!

� ð9Þ

The related initial condition should be also transformed as follows:

Uð0;0Þ ¼ 0;

Uðk;0Þ ¼ ð�1Þkþ1

k!; k ¼ 1;2; . . . ð10Þ

Substituting Eq. (10) into Eq. (9), we have calculated some components of series solution as given below:

U1;bð0;1Þ ¼ �1

0!0!Cðbþ 1Þ ; U1;bð0;2Þ ¼ �Cðbþ 1Þ

0!1!Cð2bþ 1Þ ; U1;bð0;3Þ ¼ �Cð2bþ 1Þ

0!2!Cð3bþ 1Þ ; . . .

U1;bð1;1Þ ¼1

1!0!Cðbþ 1Þ ; U1;bð1;2Þ ¼Cðbþ 1Þ

1!1!Cð2bþ 1Þ ; U1;bð1;3Þ ¼Cð2bþ 1Þ

1!2!Cð3bþ 1Þ ; . . .

U1;bð2;1Þ ¼ �1

2!0!Cðbþ 1Þ ; U1;bð2;2Þ ¼ �Cðbþ 1Þ

2!1!Cð2bþ 1Þ ; U1;bð2;3Þ ¼ �Cð2bþ 1Þ

2!2!Cð3bþ 1Þ ; . . .

U1;bð3;1Þ ¼1

3!0!Cðbþ 1Þ ; U1;bð3;2Þ ¼Cðbþ 1Þ

3!1!Cð2bþ 1Þ ; U1;bð3;3Þ ¼Cð2bþ 1Þ

3!Cð3bþ 1Þ ; . . .

Therefore,

uðx; tÞ ¼X1k¼0

X1h¼0

U1;bðk; hÞxkthb ð11Þ

Substituting all U1,b(k,h) into Eq. (11), we have series solution as follows

uðx; tÞ ¼ 1� e�x 1þ 10!Cðbþ 1Þ t

b þ Cðbþ 1Þ1!Cð2bþ 1Þ t

2b þ Cð2bþ 1Þ2!Cð3bþ 1Þ t

3b þ � � ��

¼ 1� e�xEbðKtbÞ ð12Þ

where, Kn ¼1; n ¼ 0Cððn�1Þbþ1Þðn�1Þ! ; n > 0

�Again at b = 1 i.e., for standard motion,u(x, t) = 1 � et�x, which is similar to the result of Jafari et al. [31].

Example 3. Consider the following nonlinear time-fractional Shock wave equation

@buðx; tÞ@tb þ 1

c0

@u@x� cþ 1

2c20

u@u@x¼ 0; ðx; tÞ 2 R� 0; T½ �; 0 < b 6 1; ð13Þ



uðx;0Þ ¼ e�x2=2; ð14Þ

where c0, c are constants andc is the specific heat.Taking, c0 = 2 and c ¼ 3

2 , the Eq. (13) can be re-written in the following Shock Wave equation as

@buðx; tÞ@tb þ 1

2@u@x� 5

16u@u@x¼ 0; ðx; tÞ 2 R� 0; T½ �: ð15Þ

Taking the generalized two dimensional differential transform of Eq. (15), we have

U1;bðk;hþ 1Þ ¼ Cðbhþ 1ÞCðbðhþ 1Þ þ 1Þ �

12ðkþ 1ÞUðkþ 1; hÞ þ 5

16

Xk

r¼0

Xh

s¼0

fðkþ 1� rÞU1;bðkþ 1� r; sÞ"

U1;bðr;h� sÞg ð16Þ

The related initial condition should be also transformed as follows:

Uðk;0Þ ¼ ð�1Þk=2

ðk=2Þ!2k==2 ; k ¼ 0;2;4; . . .

¼ 0 k ¼ 1;3;5; . . .

(ð17Þ

Substituting Eq. (17) into Eq. (16), we get some components of series solution as follows

U1;bð0;1Þ ¼ 0; U1;bð1;1Þ ¼3

161

Cðbþ 1Þ ; U1;bð2;1Þ ¼ 0; U1;bð3;1Þ ¼1

16; U1;bð4;1Þ ¼ 0;

U1;bð5;1Þ ¼ �3

32; . . .

U1;bð0;2Þ ¼ �9

2561

Cð2bþ 1Þ ; U1;bð1;2Þ ¼ 0; U1;bð2;2Þ ¼ �63

5121

Cð2bþ 1Þ ; U1;bð3;2Þ ¼ 0;

U1;bð4;2Þ ¼155

20481

Cð2bþ 1Þ ; . . .

U1;bð0;3Þ ¼ 0; U1;bð1;3Þ ¼ �819

212 þ45

212

Cð2bþ 1ÞðCðbþ 1ÞÞ2

� 405

213

1Cð2bþ 1Þ

!1

Cð3bþ 1Þ ; U1;bð2;3Þ ¼ 0;

U1;bð3;3Þ ¼ �1140

213 þ15

210


!1

Cð3bþ 1Þ ; U1;bð4;3Þ ¼ 0; . . .

Therefore,

uðx; tÞ ¼X1k¼0

X1h¼0

U1;bðk; hÞxkthb: ð18Þ

Substituting all U1,b(k,h) into Eq. (18), we finally obtain the series solution as

uðx; tÞ ¼ 1� x2

2þ x4

8þ 3x

16þ x3

16� 3x5

32

� �tb

Cðbþ 1Þ þ � 9256� 63x2

512þ 155x4

2048

� �t2b

Cð2bþ 1Þ

þ �819212 þ

45212


� 405213

1Cð2bþ 1Þ

!xþ �1140

213 þ15210


!x3

!t3b

Cð3bþ 1Þ þ � � � ð19Þ

Example 4. Consider the following nonlinear time-fractional Whitham–Broer–Kaup (WBK) shallow water equations

@au@ta þ u

@u@xþ @v@xþ b

@2u@x2 ¼ 0; ð20Þ

@bu@tb þ

@

@xðuvÞ þ a

@3u@x3 � b

@2v@x2 ¼ 0; 0 < a;b 6 1; ð21Þ

with the initial conditions

uðx;0Þ ¼ k� 2kðaþ b2Þ1=2 cothðkðxþ x0ÞÞ; ð22Þvðx;0Þ ¼ �2k2ðaþ b2 þ bðaþ b2Þ1=2Þ cos ech2ðkðxþ x0ÞÞ: ð23Þ

where u(x, t) is the field of horizontal velocity and v(x, t) is the height that deviate from equilibrium position of liquid, and k, a,b, k and x0 are arbitrary constants.


Taking the generalized two dimensional differential transform of Eqs. (20) and (21), we have

Cðaðjþ 1Þ þ 1ÞCðajþ 1Þ U1;aði; jþ 1Þ þ

Xi

r¼0

Xj

s¼0

fU1;aðr; j� sÞðiþ 1� rÞU1;aðiþ 1� r; sÞg þ ðiþ 1ÞV1;bðiþ 1; jÞ

þ bðiþ 1Þðiþ 2ÞU1;aðiþ 2; jÞ ¼ 0;

Cðbðjþ 1Þ þ 1ÞCðbjþ 1Þ V1:bði; jþ 1Þ þ ðiþ 1Þ

Xiþ1

r¼0

Xj

s¼0

fU1;aðr; j� sÞV1;bðiþ 1� r; sÞg þ aðiþ 1Þðiþ 2Þðiþ 3ÞU1;aðiþ 3; jÞ

� bðiþ 1Þðiþ 2ÞV1;bðiþ 2; jÞ ¼ 0;

The related initial conditions are also be transformed as follows:

Uð0;0Þ ¼ k� 2kðaþ b2Þ1=2 cothðkx0Þ;

Uð1;0Þ ¼ 2k2ðaþ b2Þ1=2 cos ech2ðkx0Þ;

Uð2;0Þ ¼ �2k3ðaþ b2Þ1=2 cothðkx0Þ cos ech2ðkx0Þ;

Uð3;0Þ ¼ 23

k4ðaþ b2Þ1=2 2þ coshð2kx0Þð Þ cos ech4ðkx0Þ;

Uð4;0Þ ¼ �16

k5ðaþ b2Þ1=2 11 coshðkx0Þ þ coshð3kx0Þð Þ cos ech5ðkx0Þ;

Vð0;0Þ ¼ �2k2 aþ b2 þ bðaþ b2Þ1=2� �

cos ech2ðkx0Þ;

Vð1;0Þ ¼ 4k3 aþ b2 þ bðaþ b2Þ1=2� �

cothðkx0Þ cos ech2ðkx0Þ;

Vð2;0Þ ¼ �2k4 aþ b2 þ bðaþ b2Þ1=2� �

2þ coshð2kx0Þð Þ cos ech4ðkx0Þ;

Vð3;0Þ ¼ 23

k5 aþ b2 þ bðaþ b2Þ1=2� �

11 coshð2kx0Þ þ coshð3kx0Þð Þ cos ech5ðkx0Þ;

Vð4;0Þ ¼ �16

k6 aþ b2 þ bðaþ b2Þ1=2� �

33þ 26 coshð2kx0Þ þ coshð4kx0Þð Þ cos ech6ðkx0Þ;

Substituting transformed initial conditions into transformed equations, we have calculated some components of series solu-tion as given below:

U1;að0;1Þ ¼ �2

Cðaþ 1Þ k2ðaþ b2Þ1=2k cos ech2ðkx0Þ;

V1;bð0;1Þ ¼ �4

Cðbþ 1Þ k3k aþ b2 þ bðaþ b2Þ1=2� �

cothðkx0Þ cos ech2ðkx0Þ;

U1;að1;1Þ ¼1

Cðaþ 1Þ4k3ðaþ b2Þ1=2k cothðkx0Þ cos ech2ðkx0Þ;

V1;bð1;1Þ ¼1

Cðbþ 1Þ4k4k aþ b2 þ bðaþ b2Þ1=2� �

2þ coshð2kx0Þð Þ cos ech4ðkx0Þ;

U1;að2;1Þ ¼ �1

Cðaþ 1Þ2k4kðaþ b2Þ1=2 2þ coshð2kx0Þð Þ cos ech4ðkx0Þ;

V1;bð2;1Þ ¼ �1

Cðbþ 1Þ2k5k aþ b2 þ bðaþ b2Þ1=2� �

11 coshð2kx0Þ þ coshð3kx0Þð Þ cos ech5ðkx0Þ;

U1;að3;1Þ ¼1

Cðaþ 1Þ23

k5kðaþ b2Þ1=2 11 coshðkx0Þ þ coshð3kx0Þð Þ cos ech5ðkx0Þ;

V1;bð3;1Þ ¼1

Cðbþ 1Þ23

k6k aþ b2 þ bðaþ b2Þ1=2� �

33þ 26 coshð2kx0Þ þ coshð4kx0Þð Þ cos ech6ðkx0Þ;

U1;að4;1Þ ¼ �1

Cðaþ 1Þ16

k6kðaþ b2Þ1=2 33þ 26 coshð2kx0Þ þ coshð4kx0Þð Þ cos ech6ðkx0Þ;

V1;bð4;1Þ ¼ �1

Cðbþ 1Þ1

12k7 2aðaþ b2Þ1=2k�

1208þ 1191 coshð2kx0Þ þ 120 coshð4kx0Þ þ coshð6kx0Þð Þ

� �aðaþ b2Þ1=2�

k 2416þ 2382k coshð2kx0Þ þ 240 coshð4kx0Þ þ 2 coshð6kx0Þð Þþ

� aþ b2 þ bðaþ b2Þ1=2� �

k 245 sinhð2kx0Þ þ 56 sinhð4kx0Þ þ sinhð6kx0Þð ÞÞ cos ech8ðkx0Þ;


U1;að0;2Þ ¼ �1

Cð2aþ 1Þ2k3k cos ech4ðkx0Þð2ðaþ b2 þ bðaþ b2Þ1=2Þkð2þ coshð2kx0ÞÞCðaþ 1ÞCðbþ 1Þ

� ð2ðaþ b2 þ bðaþ b2Þ1=2Þkð2þ coshð2kx0ÞÞ � ðaþ b2Þ1=2k sinhð2kx0ÞÞÞ;

V1;bð0;2Þ ¼ �1

Cð2bþ 1Þ2k4k cos ech5ðkx0Þð4k coshðkx0Þð�2b2ðbþ ðaþ b2Þ1=2Þ þ að�2bþ 3ðaþ b2Þ1=2Þ

þ aðaþ b2Þ1=2 coshð2kx0ÞÞCðbþ 1ÞCðaþ 1Þ þ ð2ðað4b� 7ðaþ b2Þ1=2Þ þ 4b2ðbþ ðaþ b2Þ1=2Þk coshðkx0Þ

� 2aðaþ b2Þ1=2k coshð3kx0Þ þ ðaþ b2 þ bðaþ b2Þ1=2Þkð3 sinhðkx0Þ þ sinhð3kx0ÞÞÞ;

Therefore,

uðx; tÞ ¼X1i¼0

X1j¼0

U1;aði; jÞxitja; vðx; tÞ ¼X1i¼0

X1j¼0

V1;bði; jÞxitjb ð24Þ

Substituting all U1,a(i, j) andV1,b(i, j), we get the series solutions of u(x, t) and v(x, t).

4. Numerical results and discussion

In this section, the numerical results of the probability density function u(x, t) are calculated for different fractionalBrownian motions b ¼ 1

3 ;12 ;

23 and also the standard motion b = 1. The numerical values of u(x, t) vs. time at x = 1 and also those

vs. x and t at b = 1/2 for all the three examples are calculated for different particular cases and are depicted through Figs. 1–6.During numerical computations only four iterations are considered. It is evident that by using more terms, the accuracies ofthe results can be improved and the errors converge to zero.

It is seen from Example 1 which is described by Figs. 1 and 2 that u(x, t) increases with the increase in t and decreases withthe increase in b. Also for a particular case b = 1/2, u (x, t) increases with increase in t and decreases with the increase in x. Butfor Example 2, the nature of figures (Figs. 3 and 4) are remarkably opposite. Fig. 3 depicts that u(x, t) decreases with increasein t but increases with the increase in b. Again Fig. 4 depicts that forb = 1/2, u(x, t) decreases with increase in t and increaseswith increase in x. Thus the presence of source term makes a rapid changes of the results of u(x, t) for the two considered

0.2 0.4 0.6 0.8 1

0.4

0.6

0.8

1.2

1.4

1.6

1.8 3/1=β

2/1=β

3/2=β

1=β

),( txu

t

Fig. 1. Plot of u(x, t) vs. t at different values of b for Example 1.

00.2

0.4

0.6

0.8

1 0

0.2

0.4

0.6

0.8

1

01

2

3

4

00.2

0.4

0.6

0.8

t

x

,( t)xu

Fig. 2. Plot of u(x, t) w.r.to x and t at b = 1/2 for Example 1.

0.2 0.4 0.6 0.8 1

-0.2

0.2

0.4

0.6

1=β

3/2=β2/1=β3/1=β

t

),( txu


00.2

0.4

0.6

0.8

1 0

0.2

0.4

0.6

0.8

1

-2

-1

0

00.2

0.4

0.6

0.8

),( txu

x

t


0.2 0.4 0.6 0.8 1.0

0.40

0.45

0.50

0.55

0.60

0.65),( txu

1=β

3/2=β

2/1=β

3/1=β

t





problems. The shock wave problem considered in Example 3 is graphically represented through Figs. 5, 6. It is observed fromFig. 5 that u(x, t) initially increases with the increase in t for every b but afterwards it decreases. It is seen that the tendency ofdecrease of u(x, t) is higher for lower values of b. It is also observed from Fig. 6 that forb = 1/2, u(x, t) decreases with the in-creases in x and t.

The SW equations with fractional time derivative considered in Example 4 are described through Figs. 7–10 for k = 0.5,a = 1.5, b = 1.5, k = 0.1, x0 = 12; The Figs. 7 and 9 show that probability density functions u(x, t) and v(x, t) decrease withthe increase in t. It is also seen that u(x, t) increases with the increase in a and v(x, t) increases with the increase in b. Figs. 8and 10 are the 3-D representations of u(x, t) and v (x, t) with t and x at a = b = 1/2. For the standard motion i.e., at a = b = 1,Figs. 11 and 12 respectively depict the comparison of numerical values of u(x, t) and v(x, t) with the exact results obtained by[35]. It is clear from the figures that in small range of time the results obtained by DTM with only four iterations are almostaccurate.

0.2 0.4 0.6 0.8 1.0

0.044

0.046

0.048

0.050 1=α=2/3α

2/1=α3/1=α),( txu

t

Fig. 7. Plot of u(x, t) vs. t at different values of a for Example 4.

Fig. 8. Plot of u(x, t) w.r.to x and t at a = 1/2 Example 4.

0.2 0.4 0.6 0.8 1.0

0.052

0.051

0.050

0.049

0.048

0.047

0.046

1=β3/2=β

2/1=β3/1=β

),( txv

t

Fig. 9. Plot of v(x, t) vs. t at different values of b for Example 4.

Fig. 10. Plot of v(x, t) w.r.to x and t at b = 1/2 for Example 4.

0.2 0.4 0.6 0.8 1.0

0.045

0.046

0.047

0.048

0.049

0.050),( txu

t

Fig. 11. Plot of comparison of u(x, t) vs. t at a = 1 with exact solution for Example 4.

0.2 0.4 0.6 0.8 1.0

0.051

0.050

0.049

0.048

0.047),( txv

t

Fig. 12. Plot of comparison of v(x, t) vs. t at b = 1 with exact solution for Example 4.


5. Conclusion

The major achievement of the article is the demonstration of the successful application of the DTM to obtain analyticsolutions of the nonlinear gas dynamic equations, shock wave equation and shallow water equations with fractional timederivatives. Results confirm that the method is really effective for handling solutions of a class of nonlinear PDEs of fractionalorder system.

Acknowledgements

The authors of this article express their sincere thanks to the reviewers for their valuable suggestions for the improve-ment of the article.


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Approximate analytical solutions of fractional gas dynamic equations