1　Introduction

With the development of computer technology and computational software,analytic approximate solutions with high accuracy for nonlinear problems become feasible by the analytic approximate methods such as the Adomian decomposition method(ADM)[Adomian(1986,1989,1994)],collocation method[Dai,Schnoor,and Atluri(2012)],variational iteration method[Wazwaz(2009)]and perturbation method[Hinch(1991)],etc.

The ADM[Adomian and Rach(1983);Adomian(1983,1986,1989,1994);Wazwaz(2009,2011);Serrano(2011);Lai,Chen,and Hsu(2008);Duan,Rach,Baleanu,and Wazwaz(2012)]is a practical technique for solving nonlinear functional equations,including ordinary differential equations(ODEs),partial differential equations(PDEs),integral equations,integro-differential equations,etc.The ADM provides efficient algorithms for analytic approximate solutions and numeric simulations for real-world applications in the applied sciences and engineering without unphysical restrictive assumptions such as required by linearization and perturbation.It is convenient for implementation in computational software[Duan,Rach,and Wazwaz(2013)]and new algorithms for the Adomian polynom ials increase its computing efficiency[Duan(2011)].The accuracy of the obtained analytic approximate solutions can be verified by direct substitution into the original equation[Fu,Wang,and Duan(2013);Duan,Rach,Wazwaz,Chaolu,and Wang(2013)].

First,we demonstrate the procedure of the ADM by solving a first-order nonlinear differential equation

where the functionsα,gandfare analytic.

We rew rite Eq.(1)in Adomian’s operator-theoretic form

whereand

Applying the integral operatorto both sides of Eq.(2)yields

In the ADM,the solutionu(t)is represented by the Adomian decomposition series

and the nonlinearity comprises the Adomian polynom ials

where the Adomian polynom ialsAn(t)[Adomian and Rach(1983)]are de fined as

Various algorithms for the Adomian polynom ials have been developed by Rach(1984,2008),Wazwaz(2000),Abdelwahid(2003)and several others[Abbaoui,Cherruault,and Seng(1995);Zhu,Chang,and Wu(2005);Biazar,Ilie,and Khoshkenar(2006);Azreg-A?nou(2009)].Recently,new algorithms and subroutines written in MATHEMATICA for fast generation of the Adomian polynomials to high orders have been developed by Duan(2010a,b,2011)and Duan and Guo(2010).

Substituting Eqs.(4)and(5)into Eq.(3)yields

From Eq.(7),the solution components are determined by the Adomian recursion scheme

where then-term approximation is given as

We remark that the convergence of the Adomian decomposition series has previously been proven by several investigators[Abbaoui and Cherruault(1994,1995);Abdelrazec and Pelinovsky(2011);Rach(2008)].For example,Abdelrazec and Pelinovsky(2011)have published a rigorous proof of convergence for the ADM under the aegis of the Cauchy-Kovalevskaya theorem.In point of fact,the Adomian decomposition series is found to be a computationally advantageous rearrangement of the Banach-space analog of the Taylor expansion series about the initial solution component function.

We remark that the domain of convergence for the Adomian decomposition series may not always be sufficiently large for engineering purposes.In order to cope with such occurrences some authors have applied one of several well-know convergence acceleration techniques such as the diagonal Padé approximants[Adomian(1994);Wazwaz(2009)]or the iterated Shanks transform[Adomian(1994);Duan,Chaolu,Rach,and Lu(2013)].

Wazwaz(1999)and Wazwaz and El-Sayed(2001)haveproposed differentmodi fied decomposition methods,where the sum Φ+L?1gwas partitioned or decomposed and then its components were distributed to subsequent solution components in order to suppress the occasional phenomenon of noisy convergence as well as to facilitate the calculation of integrals.

In[Duan(2010a);Duan,Rach,and Wang(2013)],the parametrized recursion scheme,which embeds a convergence parametercinto the recursion scheme,was proposed in order to obtain decomposition solutions with larger effective regions of convergence.

By introducing a parametercwith a specified decompositioninto the recursion scheme(8)and(9),we deduce the parametrized recursion scheme

Similarly,we can introduce the convergence parametercinto the modified recursion schemes[Wazwaz(1999);Wazwaz and El-Sayed(2001)].

For sake of discussion,we list two specific decompositions of the convergence parametercand their corresponding parametrized recursion schemes as and

where then-term approximation as parametrized bycis thusIn this paper,we shall present a method to determine the value of the convergence parametercthrough examination of the curves ofφn(t;c)versuscfor different values ofnandtsuch that the parametrized decomposition series has a larger effective region of convergence.

2　Illustration of the proposed method

For a specifiedt,φn(t;c)denotes the analytic approximations of the solutionu(t)such that the curves ofφn(t;c)versuscbecome horizontal over the effective field ofc.By virtue of this property,we can efficiently determine the value of the convergence parametercsuch that the decomposition series has a larger effective region of convergence.We illustrate the effectiveness of our method through the following four examples.

Example 1.Consider the Riccati equation

where the exact solution isfort＞?1.

Applying the integral operatorto both sides of Eq.(18)yieldsu=

Next,we decompose the solutionand the nonlinearityf(u)=u2=where the Adomian polynomialsAnare

Applying the parametrized recursion scheme(12)–(14)

we calculate the parametrized solution components as

where then-term parametrized approximation is

In Figs.1(a)–1(d),we plot the curves ofφ10(t,c),φ15(t,c),φ20(t,c)versuscfort=1,t=1.5,t=2 andt=2.5,respectively.We observe in each sub figure that there is an interval,where the horizontal segments overlap one another,and the corresponding fields ofccontract as the values oftincrease.Thus we observe thatφn(t;c)converges for a larger field oftwhencequals about 0.6.

In Fig.2,we plot the exact solutionu?(t)and the 16-term approximationsφ16(t;c)forc=0.4,0.6,0.8.We observe by comparing the three values ofcthat the decomposition series forc=0.6 has the largest effective region of convergence.We note that if the convergence parametercwas not introduced,which corresponds to the case ofc=0,then then-term approximation would be the Maclaurin polynomials of the exact solution,i.e.φn(t)=1?t+t2?···+(?t)n?1,which converges only on the interval?1＜t＜1.

Next,we apply the parametrized recursion scheme(15)–(17)to this example,

Figure 1:Curves of φn(t;c)versus c for(a)t=1,(b)t=1.5,(c)t=2,(d)t=2.5 and for n=10(solid line),n=15(dot line),n=20(dash line).

Figure 2:The exact solution u?(t)(solid line)and the 16-term approximations φ16(t;0.4)(dot line),φ16(t;0.6)(dash line)and φ16(t;0.8)(dot-dash line).

Figure 3:Curves of φn(t;c)versus c for(a)t=2,(b)t=3,(c)t=4,(d)t=5 and for n=10(solid line),n=15(dot line),n=20(dash line).

where|c|＜1.We calculate the parametrized solution components as

where then-term parametrized approximation isIn Figs.3(a)–3(d),we plot the curves ofversuscfort=2,t=3,t=4 andt=5,respectively.Astincreases,the horizontal segments contract to a neighborhood ofc=0.85.

In Fig.4,we plot the exact solutionand the 18-term approximationsφ18(t;c)forc=0.4,0.6,0.8.We observe for the three values ofcthat the decomposition series forc=0.8 has the largest effective region of convergence.

Figure 4:The exact solution u?(t)(solid line)and the 18-term approximations φ18(t;0.4)(dot line),φ18(t;0.6)(dash line)and φ18(t;0.8)(dot-dash line).

In fact,then-term approximation is

which converges fortsuch that|c?t+ct|＜1,i.e.the interval

Example 2.Consider the nonlinear differential equation with a negative power nonlinearity

where the exact solution is

By the ADM,we rew rite Eq.(23)as

whereThe first several Adomian polynomials for the nonlinearityf(u)=1/uare

By the parametrized recursion scheme(12)–(14),the solution componentsunare determined as

Then,we calculate the parametrized solution components in succession as

In Figs.5(a)–5(d),we plot the curves ofφ11(t,c),φ13(t,c),φ15(t,c)versuscfort=1,t=5,t=10andt=15,respectively.The plots showas the parametercdecreases from 0 to?6 that the effective region of convergence of the decomposition series gradually increases.

In Fig.6,we plot the exact solutionu?(t)and the 15-term approximationsφ15(t;c)forc=0,?2,?4,where the gradual expansion of effective regions of convergence is obvious.

We note that if the parametercwas not introduced,i.e.the case ofc=0,then then-term approximationφn(t)would be the Maclaurin polynomials of the exact solutionu?(t),which converges only on the interval?1≤t≤1.

Figure 5:Curves of φn(t;c)versus c for(a)t=1,(b)t=5,(c)t=10,(d)t=15 and for n=11(solid line),n=13(dot line),n=15(dash line).

Figure 6:The exact solution u?(t)(solid line)and the 15-term approximations φ15(t;0)(dot line),φ15(t;?2)(dash line)and φ15(t;?4)(dot-dash line).

We have checked by using the specific decomposition(15)of the parametercthat the decomposition series forc=?0.35 has a larger effective region of convergence than that forc=0.

Example 3.Consider the Lane-Emden equation

where the exact solution isLet

then Eq.(25)becomes

Applying the inverse operatorto both sides of Eq.(27)yields

The first several Adomian polynomials for the quintic nonlinearityf(u)=u5are

The components of the solutionare determined by the parametrized recursion scheme(12)–(14)

We obtain the parametrized solution components as

Figure 7:Curves of φn(t;c)versus c for(a)t=1.5,(b)t=2,(c)t=2.5,(d)t=3 and for n=12(solid line),n=14(dot line),n=16(dash line).

where then-term parametrized approximation is

In Figs.7(a)–7(d),we plot the curves ofversuscfort=1.5,t=2,t=2.5 andt=3,respectively.We observe in each of the sub figures that there is an interval,where the horizontal segments overlap one another,and the corresponding fields ofccontract as the values oftincrease.Thus we observe thatφn(t;c)converges for a larger field oftwhencequals about 0.3.

In Fig.8,we plot the exact solutionu?(t)and the 16-term approximationsφ16(t;c)forc=0,0.15,0.3.We observe for the three values ofcthat the series solution has the largest effective region of convergence forc=0.3.

Figure 8:The exact solution u?(t)(solid line)and the 16-term approximations φ16(t;0)(dot line),φ16(t;0.15)(dash line)and φ16(t;0.3)(dot-dash line).

We have checked by using the speci fic decomposition(15)of the parametercthat the decomposition series forc=0.2 has a larger effective region of convergence than that forc=0.

Exam ple 4.Consider the nonlinear differential equation with a logarithmic nonlinearity

where this initial value problem does not have an exact analytic solution.

Applying the integral operatorto both sides of Eq.(30)yields

We compute the components of the solutionby the parametrized modified recursion scheme as

where the first several Adomian polynomialsAnfor the logarithm ic nonlinearityf(u)=ln(u)are

The parametrized solution components are computed as

where then-term parametrized approximation is

In Figs.9(a)–9(d),we plot the curves ofversuscfort=1.5,t=2,t=2.5 andt=3,respectively.The curves display that the effective region of convergence ofφn(t;c)gradually increases ascdecreases from 0 to?6.In Fig.10,we plot the MATHEMATICA numeric solutionu?(t),and the 12-term approximationsφ12(t;c)forc=0,?3 and?6.We observe for the three values ofcthat the decomposition series forc=?6 has a larger effective region of convergence than that forc=0.

We have checked by using the decomposition(15)of the parametercthat the decomposition series forc=?0.7 has a larger effective region of convergence than that forc=0.

In summary,by introducing the convergence parametercand its speci fied decomposition,we adjust each term of the decomposition series.For some field ofc,the decomposition series has a larger effective region of convergence than for other fields ofc.Through the curves ofφn(t;c)versuscfor different values ofnandt,we can determine the optimal field ofc.

From our investigation,we observe that the decomposition of the convergence parametercis nonunique.Here we have demonstrated two simple decompositions of the parametercin Eqs.(12)and(15)and their effects on the solutons by the Adomian decomposition series.

Figure 9:Curves of φn(t;c)versus c for(a)t=1.5,(b)t=2,(c)t=2.5,(d)t=3 and for n=8(solid line),n=10(dot line),n=12(dash line).

Figure 10:The MATHEMATICA numerical solution u?(t)(solid line)and the 12-term approximations φ12(t;0)(dot line),φ12(t;?3)(dash line)and φ12(t;?6)(dotdash line).

3　Conclusions

In this paper,we have presented a new method to optimize the value of the convergence parametercin the ADM.Our proposed method examines the curves ofφn(t;c)versuscfor different values ofnandt.If the curves become horizontal in some region ofc,then that region corresponds to an effective field ofc.We can contract the effective field ofcby increasing the values of the argumentt.We can choosecthrough this method and thus expand the effective region of convergence of the Adomian decomposition series for solutions of nonlinear differential equations.We have investigated four examples of nonlinear differential equations to demonstrate how to practically expand the region of convergence for the solution by the Adomian decomposition series of nonlinear ODEs.

Acknowledgement:This work was supported by the Innovation Program of the Shanghai Municipal Education Commission(14ZZ161).

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