Zakharov-Kuznetsov and modified Zakharov-Kuznetsov

Abstraction

In the present paper, we construct the analytical exact solutions of some nonlinear development equations in mathematical natural philosophies ; viz. the space–time fractional Zakharov-Kuznetsov ( ZK ) and fractional modified Zakharov-Kuznetsov ( mZK ) equations by utilizing fractional sub equation method. As a consequence, new types of exact analytical solutions are obtained. The obtained consequences are shown diagrammatically. Here the fractional derived function is described in the Jumarie’s modified Riemann-Liouville sense.

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Cardinal words:Fractional sub-equation method, fractional Zakharov-Kuznetsov ( ZK ) equation ; fractional modified Zakharov-Kuznetsov ( mZK ) equation ; modified Riemann–Liouville derivative ; Mittag-leffler map.

  1. Introduction

See the space-time fractional Zakharov-Kuznetsov ( ZK ) [ 1, 2 ] equation

""( 1.1 )

and the space-time fractional modified Zakharov-Kuznetsov ( mZK ) [ 3, 4 ] equation

""( 1.2 )

where""and"","","",""are arbitrary invariables.

The Zakharov-Kuznetsov ( ZK ) equation was foremost derived for analyzing decrepit nonlinear ion acoustic moving ridges in to a great extent magnetized lossless plasma and geophysical flows in two dimensions [ 5 ] . The ZK equation is one of the two well-established canonical planar extensions of the Korteweg-de Vries equation [ 6 ] . There are some analytical methods like variational loop method ( VIM ) [ 7, 8 ] by which fractional ZK equation solved. The fractional Zakharov-Kuznetsov ( ZK ) has been examined once by utilizing VIM [ 9 ] and HPM [ 10 ] . The ZK equation determines the behavior of infirmly nonlinear ion-acoustic moving ridges in a integrated of hot isothermal negatrons and cold ions in the being of a unvarying magnetic field [ 11, 12 ] . A going moving ridge analysis is disposed in [ 13 ] for the ZK equation.

When the negatron plasma or ion does non carry through the Boltzmann distribution, Munro and Parkes obtain modified ZK ( mZK ) equation and they besides examined the stableness of going wave solutions in isolation and two-dimensional periodic planar long moving ridge disturbance wave solutions [ 14, 15 ] . The modified ZK equation interprets an anisotropic planar generalisation of the KdV equation and can be analysed in magnetic plasma for a bantam amplitude Alfven wave at a critical angle to the uninterrupted magnetic field. The mZK equation is effectively applied to place the development of several lone moving ridges in isothermal multicomponent magnetized plasma, likewise the analysis of stability of lone moving ridges of mZK equation has described in [ 16 ] . The mZK equation has enticed the attending for research workers in the past few old ages.

In this paper, we will use the fractional sub-equation method [ 17-21 ] for work outing fractional space-time fractional ZK equation and the space-time fractional mZK equation.

The remainder of this paper is organized as follows. In Section 2, we introduce some definitions and belongingss of Jumarie’s modified Riemann-Liouville derivative and we give the description of the fractional sub-equation method for work outing Fractional Partial Differential equations ( FPDEs ) . Then in Section 3 and 4, we employ the present method to set up exact solutions for the fractional space-time fractional ZK equation and the space-time fractional mZK equation severally. In Section 5, we discuss the numerical simulations for present method. Some of import decisions are presented at the terminal of the paper.

  1. Descriptions of Modified Riemann-Liouville Derivative and the Proposed Method

The Jumarie’s modified Riemann–Liouville derivative [ 22 ] of order""is defined by the look

""( 2.1 )

Some belongingss for the modified Riemann–Liouville derived function listed in [ 22 ] are as follows:

"",""( 2.2 )

""( 2.3 )

""( 2.4 )

The above belongingss play an of import function in the fractional sub-equation method [ 15-18 ] . The chief stairss of this method are described as follows:

Measure 1:Suppose that a nonlinear FPDE, say in three independent variables"",""and""is given by

"",""( 2.5 )

where"",""and""are Jumarie’s modified Riemann–Liouville derived functions ofU, where""is an unknown map,""is a multinomial inUand its assorted partial derived functions in which the highest order derived functions and nonlinear footings are involved.

Measure 2:By utilizing the going wave transmutation:

"",""( 2.6 )

where"",""andVare invariables to be determined subsequently, the FPDE ( 2.5 ) is reduced to the undermentioned nonlinear fractional ordinary differential equation ( ODE ) for"":

""( 2.7 )

Measure 3:We suppose that combining weight. ( 2.7 ) has the undermentioned solution:

""( 2.8 )

where""are invariables to be determined subsequently,""is a positive whole number determined by equilibrating the highest order derivative term and nonlinear term in combining weight. ( 2.7 ) and""satisfies the undermentioned fractional Riccati equation:

""( 2.9 )

where""is a changeless. By utilizing the generalised Exp-function method via Mittag-Leffler maps, Zhang et Al. [ 23 ] foremost obtained the undermentioned solutions of fractional Riccati equation ( 2.9 )

""( 2.10 )

where the generalized hyperbolic and trigonometric maps are defined as

"","",""

"","","",""

where""and""denotes the Mittag-Leffler map, given as

""( 2.11 )

Measure 4:Substituting combining weight. ( 2.8 ) along with combining weight. ( 2.9 ) into combining weight. ( 2.7 ) and utilizing the belongingss of Jumarie’s modified Riemann–Liouville derivative combining weight. ( 2.2 ) to eq. ( 2.4 ) , we can acquire a multinomial in"""". Comparing each coefficients of""""to zero, outputs a set of nonlinear algebraic equations for"""","","",""andV.

Measure 5:Solving the algebraic equations system in Step 4, replacing these invariables""""and"", solutions of combining weight. ( 2.9 ) given in combining weight. ( 2.10 ) into combining weight. ( 2.8 ) , we can obtain the expressed solutions of combining weight. ( 2.5 ) instantly.

Remark:If""the Riccati equation becomes"". So the method in this illustration can be used to work out integer-order differential equations. In this sense, we would wish to reason that our method includes the bing tanh-function method as particular instance.

  1. Application of Fractional Sub-equation to the space-time fractional Zakharov-Kuznetsov ( ZK ) equation

In this subdivision, we apply the fractional sub-equation method to find the exact solutions for space-time fractional Zakharov-Kuznetsov ( ZK ) equation ( 1.1 ) .

By sing the going moving ridge transmutation ( 2.6 ) , eq. ( 1.1 ) can be reduced to the undermentioned nonlinear fractional Ode:

""( 3.1 )

By equilibrating the highest order derivative term and nonlinear term in combining weight. ( 3.1 ) , the value of""can be determined, which is""in this job.

We suppose that combining weight. ( 3.1 ) has the following formal solution:

""( 3.2 )

where""satisfies eq. ( 2.9 ) .

Substituting combining weight. ( 3.2 ) along with combining weight. ( 2.9 ) into combining weight. ( 3.1 ) and so comparing each coefficients of""""to zero, we can obtain a set of algebraic equations for""as follows:

""

""

""

""( 3.3 )

""

""

Solving the above algebraic equations ( 3.3 ) , we have

Case1:

In this instance,a,Banddegree Celsiussare arbitrary. The set of coefficients for the solution of combining weight. ( 3.2 ) are given as

"","",""( 3.4 )

We, hence, obtain from combining weight. ( 2.10 ) , ( 3.2 ) and ( 3.4 ) three types of exact solutions of combining weight. ( 3.1 ) , viz. , two generalized inflated map solutions, two generalized trigonometric map solutions and one rational solution as follows:

""

""

""( 3.5 )

""

""

Case 2:

In the 2nd instance,""are arbitrary and"". The set of coefficients of the solution of combining weight. ( 3.2 ) are given as

"","",""( 3.6 )

We, hence, obtain from combining weight. ( 2.10 ) , ( 3.2 ) and ( 3.6 ) three types of exact solutions of combining weight. ( 3.1 ) , viz. , two generalized inflated map solutions, two generalized trigonometric map solutions and one rational solution as follows:

""

""

""( 3.7 )

""

""

Case 3:

In the 3rd instance,""are arbitrary and"". The set of coefficients of the solution of combining weight. ( 3.2 ) are given as

"","",""( 3.8 )

We, hence, obtain from combining weight. ( 2.10 ) , ( 3.2 ) and ( 3.8 ) three types of exact solutions of combining weight. ( 3.1 ) , viz. , two generalized inflated map solutions, two generalized trigonometric map solutions and one rational solution as follows:

""

""

""( 3.9 )

""

""

  1. Execution of Fractional Sub-equation to the Space-time fractional Modified Zakharov-Kuznetsov( mZK ) equation

In this subdivision, we apply the fractional sub-equation method to build the exact solutions for space-time fractional modified Zakharov-Kuznetsov ( mZK ) equation ( 1.2 ) .

By sing the going moving ridge transmutation ( 2.6 ) , eq. ( 1.2 ) can be reduced to the undermentioned nonlinear fractional Ode:

""( 4.1 )

By equilibrating the highest order derivative term and nonlinear term in combining weight. ( 4.1 ) , we have

""

This implies"".

We suppose that combining weight. ( 4.1 ) has the following formal solution:

""( 4.2 )

where""satisfies eq. ( 2.9 ) .

Substituting combining weight. ( 4.2 ) along with combining weight. ( 2.9 ) into combining weight. ( 4.1 ) and so comparing each coefficients of""""to zero, we can obtain a set of algebraic equations for""as follows:

""

""

""

""( 4.3 )

""

Solving the above algebraic equations ( 4.3 ) , we have:

Case 1:

  1. "","",""( 4.4 )

We, hence, obtain from combining weight. ( 2.10 ) , ( 4.2 ) and ( 4.4 ) two types of exact solutions of combining weight. ( 4.1 ) , viz. , two generalized inflated map solutions, two generalized trigonometric map solutions as follows:

""

""( 4.5 )

""

""

  1. "","",""( 4.6 )

We, hence, obtain from combining weight. ( 2.10 ) , ( 4.2 ) and ( 4.6 ) two types of exact solutions of combining weight. ( 4.1 ) , viz. , two generalized inflated map solutions, two generalized trigonometric map solutions as follows:

""

""( 4.7 )

""

""

  1. Numeric Simulations of space- clip fractional ZK and mZK equations

In this present numerical experiment, the solutions obtained by fractional sub-equation method have been used to pull the solution graphs for space- clip fractional Zakharov-Kuznetsov ( ZK ) and modified Zakharov-Kuznetsov ( mZK ) equations.

  1. Numeric Simulations of space- clip fractional ZK equation

The numerical solutions of space- clip fractional Zakharov-Kuznetsov ( ZK ) equation in combining weight. ( 1.1 ) have been shown in Figs.1-4 with the aid of fractional sub-equation method, when"","","","","","",""and"".

""

Fig. 1.The bell shaped lone moving ridge solution for""obtained from combining weight. ( 3.5 ) , when"".

""

Fig. 2.The lone moving ridge solution for""obtained from combining weight. ( 3.5 ) , when"".

""

Fig. 3.The lone moving ridge solution for""obtained from combining weight. ( 3.5 ) , when"".

""

Fig. 4.The lone moving ridge solution for""obtained from combining weight. ( 3.5 ) , when"".

In this numerical simulation, figs. 1-4 show lone wave solutions for"","",""and""from combining weight. ( 3.5 ) in the three-dimensional figures severally. These solution surfaces have been drawn when"".

  1. Numeric Simulations of mZK equation

The numerical solutions of fractional modified Zakharov-Kuznetsov ( mZK ) equation in combining weight. ( 1.2 ) have been shown in Figs.5 and 6 with the aid of fractional sub-equation method, when"","","","",""and"".

""

Fig. 5.The lone moving ridge solution for""obtained from combining weight. ( 4.5 ) .

""

Fig. 6.The lone moving ridge solution for""obtained from combining weight. ( 4.5 ) .

In this numerical simulation, fig.5 and 6 show lone wave solution for""and""from combining weight. ( 4.5 ) in the three-dimensional figures severally. These solution surfaces have been drawn when"".

5.3Influence of fractional order derived function on lone moving ridge solutions

In this present analysis, we have examined the lone moving ridge solutions by plotting 2-D figures of fractional Zakharov-Kuznetsov and fractional modified Zakharov-Kuznetsov for different values of"".

""

Fig. 7.The lone moving ridge solution for""obtained from combining weight. ( 3.5 ) for the ZK equation for different values of"".

""

Fig. 8.The lone moving ridge solution for""obtained from combining weight. ( 4.5 ) for the mZK equation for different values of"".

In this numerical simulation, fig.7 and 8 show lone wave solution for the fractional ZK and mZK equation for different values of"". It may be observed from fig.7 and 8 that when the fractional derivative parametric quantity""increases the extremum of the lone moving ridges nature additions.

  1. Decision

In this paper, utilizing fractional sub-equation method, we successfully obtained three types of exact analytical solutions including the generalised inflated map solutions, generalized trigonometric map solutions and rational solution for the space-time fractional ZK equation and two types exact analytical solutions including the generalised inflated map solutions, generalized trigonometric map solutions for the space-time fractional mZK equation. From our consequences obtained in this paper, we conclude that the fractional sub-equation method is powerful, effectual and convenient for nonlinear FPDEs. The present work demonstrates the pertinence of fractional sub-equation method to nonlinear FPDEs like fractional ZK equation and fractional mZK equation. Finally, this method provides a powerful mathematical tool to obtain more general explicit exact analytical solutions for nonlinear FPDEs arising in mathematical natural philosophies.

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