A New Jacobi Elliptic Function Expansion Method for Solving a Nonlinear PDE Describing Pulse Narrowing Nonlinear Transmission Lines

ZAYED E.M.E.and ALURRFI K.A.E.

Department of Mathematics,Faculty of Science,Zagazig University,P.O.Box 44519, Zagazig,Egypt.

A New Jacobi Elliptic Function Expansion Method for Solving a Nonlinear PDE Describing Pulse Narrowing Nonlinear Transmission Lines

ZAYED E.M.E.?and ALURRFI K.A.E.

Department of Mathematics,Faculty of Science,Zagazig University,P.O.Box 44519, Zagazig,Egypt.

.In this article,we apply the first elliptic function equation to find a new kind of solutions of nonlinear partial differential equations(PDEs)based on the homogeneous balance method,the Jacobi elliptic expansion method and the auxiliary equation method.New exact solutions to the Jacobi elliptic functions of a nonlinear PDE describing pulse narrowing nonlinear transmission lines are given with the aid of computer program,e.g.Maple or Mathematica.Based on Kirchhoff’s current law and Kirchhoff’s voltage law,the given nonlinear PDE has been derived and can be reduced to a nonlinear ordinary differentialequation(ODE)using a simple transformation.The given method in this article is straightforward and concise,and can be applied to other nonlinear PDEs in mathematical physics.Further results may be obtained.

New Jacobi elliptic function expansion method;pulse narrowing nonlinear transmission lines;exact solutions;Kirchhoff’s current law;Kirchhoff’s voltage law.

1 Introduction

The nonlinear PDEs in mathematical physics are major subjects in physical science[1]. Exactsolutionsfor these equations play an importantrole in many phenomena in physics, such as fluid mechanics,hydrodynamics,optics,plasma physics and so on.Recently, many methods for finding these solutions have been presented,for example,tanh-sechmethod[2-4],extended tanh-method[5-7],sine-cosine method[8-10],homogeneous balance method[11,12],Jacobi elliptic function method[13-16],F-expansion method[17-19], exp-function method[20,21],trigonometric function series method[22],expansion method[23-27],the modified simple equation method[28-33],the modified mapping method[34],the firstintegralmethod[35-38],the multiple exp-function algorithm method [39,40],the transformed rationalfunction method[41],the Frobeniusdecomposition technique[42],the local fractional variation iteration method[43],the local fractional series expansion method[44]and so on.

The objective of this article is to use a new Jacobi elliptic function expansion method [45]to find the exactsolutions ofthe following nonlinear PDE describing pulse narrowing nonlinear transmission lines[46]:

whereV(x,t)is the voltage of the pulse andC0,L,δandb1are constants.The physical details of the derivation of Eq.(1.1)is elaborated in[46]using the Kirchhoff’s current law and Kirchhoff’s voltage law,which are omitted here for simplicity.It is well-known[46] that Eq.(1.1)has the solution:

wherevis the propagation velocity of the pulse andprovided thatv＞v0.

This paper is organized as follows:In Sec.2,the description of a new Jacobi elliptic function expansion method is given.In Sec.3,we use the given method described in Sec. 2,to find exact solutions of Eq.(1.1).In Sec.4,the physical explanations of some results are presented.In Sec.5,some conclusions are obtained.

2 Description of a new Jacobielliptic function expansion method

Consider a nonlinear PDE in the form

whereV=V(x,t)is an unknown function,Pis a polynomial inV(x,t)and its partial derivatives in which the highest order derivatives and nonlinear terms are involved.Let us now give the main steps of the Jacobi elliptic function expansion method[45]:

Step 1.We look for the voltageV(x,t)of the pulse in the traveling form:wherevis the propagation velocity of the pulse,to reduce Eq.(2.1)to the following nonlinear(ODE):

whereHis a polynomial ofV(ξ)and its total derivativesV′(ξ),V′′(ξ),...and′=d/dξ.

Step 2.We suppose that the solution of Eq.(2.3)has the form:

wherez(ξ)satisfies the Jacobi elliptic equation:

Step 3.We determine the positive integerNin(2.4)by balancing the highest-order derivatives and the nonlinear terms in Eq.(2.3).

Step 4.Substituting(2.4)along with Eq.(2.5)into Eq.(2.3)and collecting all the coefficients ofthen setting them to zero,yield a set of algebraic equations.

Step 5.Solving the algebraic equations in Step 4,using the Maple or Mathematica to findg0,gi,fi,v,a,b,c.

Step 6.It is well-known[45]that Eq.(2.5)has many families of solutions as follows:

?

In this table,snξ=sn(ξ,m),cnξ=cn(ξ,m),dnξ=dn(ξ,m),nsξ=ns(ξ,m),csξ=cs(ξ,m),dsξ=ds(ξ,m),scξ=sc(ξ,m),sdξ=sd(ξ,m)are the Jacobi elliptic function with modulusm,where 0＜m＜1.These functions degenerate into hyperbolic functions whenm→1 as follows:

snξ→tanhξ,cnξ→sechξ,dnξ→sechξ,nsξ=cothξ,csξ=cschξ,dsξ=cschξ,scξ=sinhξ,sdξ=sinhξ,ncξ=coshξ,and into trigonometric functions whenm→0 as follows:

snξ→sinξ,cnξ→cosξ,dnξ→1,nsξ→cscξ,csξ→cotξ,dsξ→cscξ,scξ→tanξ,sdξ→sinξ,ncξ→secξ.

Also,these functions satisfy the following formulas:

and

sn′ξ=cnξdnξ,cn′ξ=?snξdnξ,dn′ξ=?m2snξcnξ,cd′ξ=?(1?m2)sdξndξ,ns′ξ=?csξdsξ,dc′ξ=(1?m2)ncξscξ,cn′ξ=scξdcξ,nd′ξ=m2cdξsdξ,sc′ξ=dcξncξ,cs′ξ=?nsξdsξ,ds′ξ=?csξnsξ,sd′ξ=ndξcdξ,where′=d/dξ.

Step 7.Substituting the solutions of Step 6,into(2.4)we have the exact solutions of Eq.(2.1).

3 Exact solutions of Eq.(1.1)using the given method in Sec.2

In this section,we apply the Jacobi elliptic function expansion method of Sec.2,to find the exact solutions of Eq.(1.1).To this end,we use the transformation(2.2)to reduce Eq. (1.1)to the following nonlinear ODE:

BalancingV′′withV2givesN=2.Therefore,(2.4)reduces to

whereg0,g1,f1,g2andf2are constants to be determined such that

Substituting(3.3)along with Eq.(2.5)into Eq.(3.1)and collecting all the coefficients ofzi(ξ),(i=0,1,...,8)and setting them to zero,we have the following algebraic equations:

On solving the above algebric equations(3.4)by Maple or Mathematica,we have the following results:

From(3.3)and(3.5),we get the exact solutions of Eq.(3.1)as follows:

whereb/=2c.

Sincea=c,we deduce from the table of Sec.2,the two cases:

Case 1.

Ifa=1/4,b=(1?2m2)/2,c=1/4 andz(ξ)=nsξ±csξorz(ξ)=snξ/(1±cnξ),then we get the Jacobi elliptic function solutions

respectively.

Ifm→1,thena=1/4,b=?1/2,c=1/4 andz(ξ)=coth(ξ)±csch(ξ)orz(ξ)=tanh(ξ)/ (1±sech(ξ))wherek1=?4.In this case,(3.7)and(3.8)reduce to the hyperbolic solutions

respectively.

Case 2.Ifa=(1?m2)/4,b=(1+m2)/2,c=(1?m2)/4 andz(ξ)=ncξ±scξorz(ξ)=cnξ/(1±snξ),then we get the Jacobi elliptic function solutions

respectively.

Ifm→1 thena=0,b=1,c=0 andz(ξ)=cosh(ξ)±sinh(ξ)orz(ξ)=sech(ξ)/(1±tanh(ξ)) wherek1=?4..In this case,(3.13)and(3.14)reduce to the hyperbolic solutions

respectively.

Remark 1.From Cases 1,2 we have shown that whenk1=?4,then we have the solution

With the aid of(3.2),we deduce that

which is equivalent to the well known(1.2)obtained in[46].

Remark 2.Eq.(3.1)can be solved using a direct method as follows:

Multiply Eq.(3.1)byV′(ξ)and integrate with zero constant of integration,we get

whereα=?k1,β=?2k2/3.

It is easy to get the solution

With the aid of(3.2),we deduce that

which is equivalent to the well known(1.2)obtained in[46].

3.1Physical explanations of some results

In this section,we have presented some graphs of the exact solutions(3.9),(3.12),(3.15) and(3.17)constructed by taking suitable values of involved unknown parameters to visualize the mechanism of the original Eq.(1.1).These solutions are kink,singular kinkshaped soliton solution,bell-shaped soliton solutions,singular bell-shaped soliton solutions and hyperbolic solutions.For more convenience the graphical representations of these solutions are shown in the following figures:

3.2Conclusions

In this article,we have solved the nonlinear PDE describing the pulse narrowing nonlinear transmission lines(1.1)using a new Jacobi elliptic function expansion method described in Sec.2.Families of exact solutions including Jacobi elliptic solutions,thedegenerated hyperbolic function solutions(whenm→1)of Eq.(1.1)have been found. Comparing our results obtained in this paper with the well-known results obtained in [46],we deduce that our results(3.20)and(3.23)are equivalent to the well known(1.2) obtained in[46]and the other solutions obtained in article are new and not found elsewhere.Using the Maple,we have shown that all solutions obtained in this article satisfy the original Eq.(1.1).A new Jacobi elliptic function expansion method used in this paper is effective in getting solutions and can be applied to explore the exact solutions of other nonlinear evolution equations in mathematical physics,which will be done in forthcoming papers.

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Received 19 January 2015;Accepted 24 March 2015

?Corresponding author.Email addresses:e.m.e.zayed@hotmail.com(E.M.E.Zayed),alurrfi@yahoo.com (K.A.E.Alurrf i)

AMS Subject Classifications:35K99,35P05,35P99,35C05

Chinese Library Classifications:O175.2