G.Bengochea,L.Verde-Star,and M.Ortigueira

1Departamento de Matem′aticas,Universidad Aut′onoma Metropolitana,Iztapalapa,Apartado 55–534,Ciudad de M′exico,M′exico

2CTS–UNINOVA/Department of Electrical Engineering,Faculdade de Ci′encias e Tecnologia da Universidade Nova de Lisboa,Campus da FCT da UNL,Quinta da Torre 2825–149 Monte da Caparica,Portugal

1 Introduction

The nonlinear Klein-Gordon equation plays an important role in several fields of physics,such as quantum mechanics,general relativity,and nonlinear optics,[1?2]In the literature,there are numerous papers dealing with diverse methods to solve nonlinear Klein-Gordon equations such as the collocation method,[3]the decomposition method,[4]the homotopy method[5]and other methods.[6?7]The nonlinear Klein-Gordon equation that we will study in this paper has the form

with t≥0,and initial conditions

where

x=(x1,x2,...,xm)∈Rm,t∈(0,T],b is a real number,g is a given nonlinear function,assumed to be analytic at zero,and f is a known function.We study the one-dimensional case(m=1),but it is not difficult to see that the method can be applied in the general case.

The operational calculus that we will use in this paper is a particular instance of a general algebraic operational calculus introduced in Ref.[8].Our operational calculus uses basic linear algebra tools to solve equations involving operators that generalize differential and difference operators.The algebraic setting is a field of generalized Laurent series with a multiplication,called algebraic convolution,and a modified shift operator that can be considered as a generalization of differentiation.Taking suitable concrete realizations of our operational calculus we can solve several kinds of differential and difference equations using the same procedure for all kinds of equations.

In this paper,we will use an algebraic convolution product that extends the multiplication formula(tn/n!)×(tm/m!)=tn+m/(n+m)!and coincides with the usual convolution defined by means of integrals.The concrete realization of the operational calculus that we will use allows us to solve linear and nonlinear differential equations,without using integral transforms or computing integrals.Since our operational methods are based on simple linear algebra they are simpler than those based on the Mikusi′nski operational calculus,[9]and can be used to provide a rigorous foundation of the main results of Heaviside’s operational calculus.Several applications of our methods can be found in Refs.[10–13],and some variations in Refs.[14–15].

The nonlinear term in Eq.(1)will be handled by means of the Adomian polynomials,[16]which we calculate using expansions in Taylor series.Combining our methods with the Adomian polynomial expansions we solve Eq.(1)with a purely operational approach.

The paper is organized as follows.Section 2 contains an introduction to the operational calculus introduced in Ref.[8],and the procedure to calculate the Adomian polynomials.In Sec.3 we solve Eq.(1)using the tools introduced in Sec.2.In Sec.4 we show how the method is applied to several equations that have been considered in research papers related with the subject.Finally,some conclusions are presented in Sec.5.

2 Algebraic Setting and Preliminary Results

2.1 Operational Calculus

In this section we present a brief description of the algebraic setting and the basic properties of the general operational calculus introduced in Ref.[8],where the reader can find a detailed presentation and the proofs of the statements that we summarize next.

Let{pk:k∈Z}be a group with the multiplication defined by pkpn=pk+n,for k,n∈Z.Let F be the set of all the formal series of the form

where akis a complex number for each k∈Z and,either,all the akare equal to zero,or there exists an integer v(a)such that ak=0 whenever k

This multiplication in F is associative and commutative and p0is its unit element.With this multiplication F is a field.We define the series

These elements of F are called generalized geometric series and satisfy ex,k(p?k(p0?xp1)k+1)=p0.In particular,when k=0 we have

If xy then

In particular,if m=n=0 then we have

The linear operator L on F is defined by Lpk=pk?1for k0,and Lp0=0,and it is called the modified left shift.As an example,if a=∑i≥0yipi,then La= ∑

i≥0yi+1pi.

One important property of the operator L is

where I is the identity operator and Pnis the projection on the subspace generated by pn,that is Pna=ynpn.See Ref.[8]page 333.

In order to apply our operational method to nonlinear Klein-Gordon differential equations we will use the concrete realization of the field F that has pk=tk/k!,for k∈Z,where t is a real or complex variable and k!is defined for negative values of k by

The modified left shift in this concrete realization is L=?/?t=Dt,and the generalized geometric series become the basic exponential polynomials

Let us note that the generalized geometric series are convergent for all x and t in C.

If instead of taking pk=tk/k!we take pkas a normalized Hermite polynomial of degree k we obtain another concrete realization of the operational calculus that can be used to solve differential equations of the kind that we consider in this paper.

2.2 Adomian Polynomials

The Adomian polynomials and decomposition methods[16]are a very important tool for the solution of many kinds of equations,including nonlinear differential equations.The general theory and convergence properties of the Adomian decomposition methods have been studied in Refs.[17–19],but,in many applications the convergence properties depend on the particular functions involved in the equations.In the examples that we consider in this paper the domain of convergence of the series that represent the solutions is easily determined.

Using the Adomian polynomials we can write a nonlinear term g(y)as a power series.In this paper,we calculate the Adomian polynomials by expanding the nonlinear term g(y)in a Taylor series about the point y0,that is

where y=∑yktk/k!.After some algebraic manipulation and grouping terms that correspond to the same monomial we obtain a series

where the Akare the Adomian polynomials.

3 Operational Solution of Nonlinear Klein-Gordon Equations

In this section we will solve Eq.(1)using the theory introduced in Sec.2.We will take? = ?2/?x2and suppose that f(x,t)is of the form

where the functions f1,kand f2,khave a convergent Taylor series representation in a neighborhood of zero and

Suppose that the solution of Eq.(1)can be expressed in the formfor some sequence of functions yk(x).Using the concrete realization of the operational method with pk=tk/k!and L= ?/?t=Dtwe can rewrite Eq.(1)as

From Eq.(4)we have that

and therefore the previous equation becomes

From Eq.(5)we write g(y)in terms of the Adomian polynomials asAlso observe that we have the factorizationand that,by Eq.(2),the multiplicative inverse of(p?2+bp0)in the fieldTherefore

Performing the operations in the right-hand side of Eq.(6)we obtain a series in terms of the pkwhich begins with p0.Consider that y0(x)and y1(x)are the initial conditions a0(x)and a1(x),respectively.Then we equate the coefficients of pkin both sides of the equation and obtain recursively the solution yk(x).Finally,we replace pkby tk/k!and obtain the solution as a function of t and x.In the next section we will solve in detail several examples.Remark 1 It is worth mentioning that the operations in Eq.(6)can be simplified using the multiplication formula(3).

4 Implementation of the Operational Method

Our first example is a linear non-homogeneous equation.

Example2 Consider the non-homogeneous Klein-Gordon equation of the form

with initial conditions

Equation(7)can be written as

where Dtand Dxdenote the derivatives with respect to t and x,respectively,and I is the identity operator.Suppose that the solution can be written in the form

for some sequence of functions yk(x).Observe that

and using the Taylor series expansion at zero of sin(t)we obtain

Let pk=tk/k!,for k∈ Z,and L=Dt.Since,by Eq.(4),we have L2=p?2(p0I?P0?P1),Eq.(8)can be expressed in the form

After some algebraic manipulations and using the fact that Pky(x,t)=yk(x)pk,we get

From Eq.(2)we see that eis the multiplicative inverse ofin F.Hence

From Eq.(3)it is clear that

In order to simplify the notation we will write ykinstead of yk(x)and y instead of y(x,t).Therefore we have

Equating corresponding coefficients of pkin both sides of the last equation we get

The initial conditions give us y0=e?axand y1=0.Then the previous equations become y2=(a2? 2)e?ax,y3= ?(31?1)sinx,y4=(a2?2)2e?ax,y5=(32?1)sinx,...Therefore the solution is given by

and then,writing the pkin terms of t we obtain

If we define A2=a2?2 and compute the sums we get

Remark 3 If we change the initial conditions in Example 2 and consider instead y(x,0)=0,(?y/?t)(x,0)=sin(x).We obtain y0=0 and y1=sin(x),and recursively y2=0,y3=?sin(x),y4=0,y5=sin(x),...The solution in terms of the concrete realization is given by

Our solution coincides with the solution obtained in Ref.[4].The series t?t3/3!+t5/5!?···is convergent for all real values of t since it is the expansion in Taylor series of sin(t).Therefore our solution converges for all real values of t.Table 1 shows the absolute error between our truncated series solution(15 terms)and the closed form solution.

Example 4 Now,consider the nonlinear Klein-Gordon equation of the form

Equation(9)can be written as

with initial conditions

Let us suppose that the solution has the form

where the ykare functions of x.Then we have

The nonlinear term y2can be expanded as a Taylor series about the point y0as follows

and the function?xcos(t)+x2cos2(t)is expressed as the convergent series

Following the same procedure used in Example 2,we write Eq.(9)in terms of the pk’s as

Multiplying both sides of the equation by p2,which is the multiplicative inverse of p?2,we obtain

Equating corresponding coefficients of the pkin both side of the equation we get

The initial conditions give us y0=x and y1=0,and by forward substitutions we obtain y2=?x,y3=0,y4=x,y5=0,...Therefore the solution is given by∑k≥0(?1)kxp2k,and this is equivalent to

Our solution coincides with the solution presented in Ref.[20].The series 1 ? t2/2!+t4/4!? ···is convergent for all real values of t since it is the expansion in Taylor series of cos(t).Therefore our solution converges for all real values of t.Table 2 shows the absolute error between our approximation to the solution by a truncated series and the closed form solution.

Example 5 Consider now the nonlinear Klein-Gordon equation

with initial conditions

where L0≤ x≤ L1,t0≤ t,and c,α,β,γ are constants,andEquation(10)can be written as

As in the previous examples we suppose that

Expanding the nonlinear term y3as a Taylor series about the point y0,we get

Following the same procedure that we used in Examples 2 and 3,we write Eq.(10)in terms of the pk’s as

where d=Since the product e?di,0edi,0is the multiplicative inverse of(p0+dip1)(p0? dip1)in the field F,we have

Equating coefficients we get

The given initial conditions yield y0=B tan(Kx)and y1=BcKsec2(Kx),and from the previous equations we obtain

Our series∑yktk/k!with the coefficients described in Eq.(11)is the Taylor series,around t=0,of y=B tan(K(x+ct)),which is the exact solution of Eq.(10),see Ref.[21].The series obtained by our method is convergent for all real values of t between ?π/2 and π/2 since it is the expansion in Taylor series of y=B tan(K(x+ct)).Table 3 shows the absolute error between our solution and the exact solution when we take B=0.816 497,K=0.426 401 and c=0.5.

5 Conclusions

In the examples presented in the previous section we have shown how our operational method is applied to obtain solutions of Klein-Gordon equations.In all the examples our solutions coincide with the ones obtained by other authors using diverse methods.The main difference between our method and the usual ones is that we do not require the computation of integrals,and we do not use any integral transforms.In addition,since the computational parts of our method are mainly operations with power series and differentiation,they can be easily done using symbolic computational software packages,such as Maple and Mathematica.The numerical results presented in the solved examples show that the values computed by evaluation of truncations of the series representations produce good approximations of the exact solutions.This fact indicates that we can expect to obtain good approximations to the solutions of problems for which the exact solution is not known.

Our operational methods can also be used to solve other kinds of differential and difference equations,including fractional differential equations,with several different definitions of the fractional differentiation operators.The procedures used for all kinds of equations are essentially the same and do not depend on particular properties of the elements pkand the operator L of the concrete realization used in each case.

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