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【Abstract】In this paper, the homotopy analytic method (HAM) is employed to solve a porous medium equation with a source term, and the analytic approximate solutions are derived. In addition, the analytic approximate solutions obtained by the HAM reveals that the present method works efficiently.
【Key words】Homotopy analytic method (HAM); porous medium equation; analytic approximate solution
一類渗透媒介方程的解析解
张晓莉 赵小山
(天津职业技术师范大学理学院 中国 天津 300222)
【摘 要】本文应用同伦分析法求解一类渗透媒介方程， 得到方程的较高精度的近似解析解。从而证明同论分析法在求解非线性动力系统的有效性。
【关键词】同伦分析；渗透媒介方程；近似解析解
1.Introduction
Scientific fields,such as plasma physics,fluid dynamics,solid state physics,mathematical biology and chemical kinetics are usually governed by nonlinear equations.A broad range of analytical solution methods have been propsed,such as Bucklund transformation method[1],Hirota’s bilinear scheme[2],inverse scattering method[3],the homotopy analytic method[4-5], the homotopy perturbation method[6],Adomian Pade approximation[7],the extan
ded tanh method[8],variational method[9]and others.
In this paper, the homotopy analytic method(HAM)is applied to give analytic solutions of a porous medium equation[10]with a source term, which is proved to be very powerful,effective, and convenient to solve nonlinear equations with boundary value.
2.The solution of the porous medium equation
In this section,we consider the Cauchy problem for a porous medium equation with a source term.This equation has been widely used as a simple model for a non-linear heat propagation in a reactive medium. The porous medium equation with a source term［10］is
=aw+bw (2.1)
For simplicity we take ｆ（ω）＝ω,ｇ（ω）＝bω.Here ｍ，ｋ are rational numbers,a,b are parameters.The case when a=1,b=0 is studied in[11]. Also a=1,b≠0,m=0 is studied in[12].Here we consider the porous equation with m=-2,k=1.Exact solution of this equation is given w(x,t)=exp(bt) as with the initial conditions w(x,0)=w=[12].
Then equation can be rewritten as
=aw+bw (2.2)
Firstly we make the transformation z=x-ct,c≠0.Substituting z=x-ct,c≠0 into (2.2), we have
bw+cw'-a(-+)=0 (2.3)
Now we employ the HAM to solve it.We at first construct a following continue mapping namely the zero-order deformation equation:
(1-p)(bW(z,p,h)+c-bW-c)-ph[bW(z,p,h)+c-a(+)]=0 (2.4)
Then,the approximate solution of Eq.(2.4) can be written as:
w(z)=W+ (2.5) where h is a non-zero real number.Assuming that W(z,p,h)=,then we take first order differential with respect to p in Eq.(2.4),we finally have the following first-order deformation equation:
-+-b(1+h)W(z,p,h)-b(hp+p-1)-c(1+h)-+c(1-p+hp)+[2ah(()+2p+2p+p)]-=0 (2.6)
Let p=0,Eq.(2.6) can be rewritten as
+b+c=0 （2.7)
Solving it, we have
=h(-e) (2.8)
Differentiating about the first-order deformation equation (2.6) with respect to p,we obtain the second-order deformation equation
-{-24ahp+W(z,p,h)[2b(1+h)+b(ph+p-1)+c(2(1+h)+(ph+p-1)]+6ahW(z,p,h)[p()+2(+2p)+p()]-2ahW(z,p,h)[2p()+（2+p+2+p+2p]+ahW(z,p,h)3[2+p]}=0 (2.9)
Let p=0,Eq.(2.9) can be rewritten as
4aeh++---++b+c=0 (2.10)
Solving it, we have
=[2eh(-6aceh(x-1)x+3c(h+1)(e-xe) +6abcehx(x-1)-abehx(x-1))] (2.11)
So,from Eq.(2.8),Eq.(2.11),we have
w(x,t)≈W++= (2.12)
By using mathematic software mathematica,we obtain other terms of series solution.Assuming that a=0,b=1,c=1,h=1.5,we compare exact solution with the solution by HAM at x=2.5,t∈[4,5] by in Fig.1.When x=2,t=4.5, the series solution is convergent quickly to the exact one, as shown in Table. 1.
Fig. 1. Comparison of the exact solution with analytic approximation solution of Eq.(2.2)
Table.1. Comparison between the exact solution and approximation solutions of Eq.(2.2)
3.Conclusions
In order to verify numerically whether the proposed methodology leads to higher accuracy, we study the porous medium equation by the method.We demonstrate the higher efficiency of the corresponding analytical approximate solutions of Eq.(2.2) in the Fig.1 and Table.1.It is to be noted that 13 terms approximate solutions can achieved a very good approximation with the actual solution of the equations by the HAM. It is evident that the solutions are very rapidly convergent to the exact one by the HAM.
【References】
［１］Wadati M.Introduction to solitons [J].Pramana:J Phys.,2001,57: 841－7.
［２］Hirota R. Exact solution of the Korteweg-de Vries equation for multiple collisions of soliton[J].Phys Rev Lett．1971,7：192-201.
［３］Blowitz M J,Clarkson PA.Nonlinear evolution and inverse scattering[M].Cambridge: Cambridge University Press．1991.
［４］S.J. Liao,The proposed homotopy analysis technique for the solution of nonlinear problems[D].PhD thesis, Shanghai Jiao Tong University,1992.
［５］S.J. Liao,An explicit, totally analytic approximate solution for Blasius’ viscous flow problems[J].Int. J. Non-Linear Mech.,1999,34：759-778.
［６］He JH. Homotopy perturbation technique [J]. Comput. Meth .Appl .Mech .Eng,1999,178:257–62.
［７］G. Adomian,A review of the decomposition method in applied mathematics [J].J Math.Anal. Appl.,1998,135:501-504.
［８］De-Sheng L,Feng G,Hong-Qing Z. Solving the(2+1)-dimensional higher order Broer–Kaup system via a transformation and tanh-function method [J].Chaos,Solitons & Fractals,２004,20(5):1021－5.
［９］He JH.Variational principles for some nonlinear partial differential equations with variable coefficients [J].Chaos,Solitons &Fractals,2004,19(4): 847–51.
［１０］Nevin Pamuk.Series solution for porous medium equation with a source term by Adomian”s decomposition method,Appl.Math.Comput[J].2006, 178:480-485.
［１１］S.Pamuk.Solution of the porous media equation by Adomian’s decomposition method[J].Phys.Lett.A,2005,344:184–188.
［１２］A.D. Polyanin,V.F.Zaitsev,Handbook of Nonlinear Partial Differential Equations[M].Chapman and Hall/CRC Press,Boca Raton,2004.
【Key words】Homotopy analytic method (HAM); porous medium equation; analytic approximate solution
一類渗透媒介方程的解析解
张晓莉 赵小山
(天津职业技术师范大学理学院 中国 天津 300222)
【摘 要】本文应用同伦分析法求解一类渗透媒介方程， 得到方程的较高精度的近似解析解。从而证明同论分析法在求解非线性动力系统的有效性。
【关键词】同伦分析；渗透媒介方程；近似解析解
1.Introduction
Scientific fields,such as plasma physics,fluid dynamics,solid state physics,mathematical biology and chemical kinetics are usually governed by nonlinear equations.A broad range of analytical solution methods have been propsed,such as Bucklund transformation method[1],Hirota’s bilinear scheme[2],inverse scattering method[3],the homotopy analytic method[4-5], the homotopy perturbation method[6],Adomian Pade approximation[7],the extan
ded tanh method[8],variational method[9]and others.
In this paper, the homotopy analytic method(HAM)is applied to give analytic solutions of a porous medium equation[10]with a source term, which is proved to be very powerful,effective, and convenient to solve nonlinear equations with boundary value.
2.The solution of the porous medium equation
In this section,we consider the Cauchy problem for a porous medium equation with a source term.This equation has been widely used as a simple model for a non-linear heat propagation in a reactive medium. The porous medium equation with a source term［10］is
=aw+bw (2.1)
For simplicity we take ｆ（ω）＝ω,ｇ（ω）＝bω.Here ｍ，ｋ are rational numbers,a,b are parameters.The case when a=1,b=0 is studied in[11]. Also a=1,b≠0,m=0 is studied in[12].Here we consider the porous equation with m=-2,k=1.Exact solution of this equation is given w(x,t)=exp(bt) as with the initial conditions w(x,0)=w=[12].
Then equation can be rewritten as
=aw+bw (2.2)
Firstly we make the transformation z=x-ct,c≠0.Substituting z=x-ct,c≠0 into (2.2), we have
bw+cw'-a(-+)=0 (2.3)
Now we employ the HAM to solve it.We at first construct a following continue mapping namely the zero-order deformation equation:
(1-p)(bW(z,p,h)+c-bW-c)-ph[bW(z,p,h)+c-a(+)]=0 (2.4)
Then,the approximate solution of Eq.(2.4) can be written as:
w(z)=W+ (2.5) where h is a non-zero real number.Assuming that W(z,p,h)=,then we take first order differential with respect to p in Eq.(2.4),we finally have the following first-order deformation equation:
-+-b(1+h)W(z,p,h)-b(hp+p-1)-c(1+h)-+c(1-p+hp)+[2ah(()+2p+2p+p)]-=0 (2.6)
Let p=0,Eq.(2.6) can be rewritten as
+b+c=0 （2.7)
Solving it, we have
=h(-e) (2.8)
Differentiating about the first-order deformation equation (2.6) with respect to p,we obtain the second-order deformation equation
-{-24ahp+W(z,p,h)[2b(1+h)+b(ph+p-1)+c(2(1+h)+(ph+p-1)]+6ahW(z,p,h)[p()+2(+2p)+p()]-2ahW(z,p,h)[2p()+（2+p+2+p+2p]+ahW(z,p,h)3[2+p]}=0 (2.9)
Let p=0,Eq.(2.9) can be rewritten as
4aeh++---++b+c=0 (2.10)
Solving it, we have
=[2eh(-6aceh(x-1)x+3c(h+1)(e-xe) +6abcehx(x-1)-abehx(x-1))] (2.11)
So,from Eq.(2.8),Eq.(2.11),we have
w(x,t)≈W++= (2.12)
By using mathematic software mathematica,we obtain other terms of series solution.Assuming that a=0,b=1,c=1,h=1.5,we compare exact solution with the solution by HAM at x=2.5,t∈[4,5] by in Fig.1.When x=2,t=4.5, the series solution is convergent quickly to the exact one, as shown in Table. 1.
Fig. 1. Comparison of the exact solution with analytic approximation solution of Eq.(2.2)
Table.1. Comparison between the exact solution and approximation solutions of Eq.(2.2)
3.Conclusions
In order to verify numerically whether the proposed methodology leads to higher accuracy, we study the porous medium equation by the method.We demonstrate the higher efficiency of the corresponding analytical approximate solutions of Eq.(2.2) in the Fig.1 and Table.1.It is to be noted that 13 terms approximate solutions can achieved a very good approximation with the actual solution of the equations by the HAM. It is evident that the solutions are very rapidly convergent to the exact one by the HAM.
【References】
［１］Wadati M.Introduction to solitons [J].Pramana:J Phys.,2001,57: 841－7.
［２］Hirota R. Exact solution of the Korteweg-de Vries equation for multiple collisions of soliton[J].Phys Rev Lett．1971,7：192-201.
［３］Blowitz M J,Clarkson PA.Nonlinear evolution and inverse scattering[M].Cambridge: Cambridge University Press．1991.
［４］S.J. Liao,The proposed homotopy analysis technique for the solution of nonlinear problems[D].PhD thesis, Shanghai Jiao Tong University,1992.
［５］S.J. Liao,An explicit, totally analytic approximate solution for Blasius’ viscous flow problems[J].Int. J. Non-Linear Mech.,1999,34：759-778.
［６］He JH. Homotopy perturbation technique [J]. Comput. Meth .Appl .Mech .Eng,1999,178:257–62.
［７］G. Adomian,A review of the decomposition method in applied mathematics [J].J Math.Anal. Appl.,1998,135:501-504.
［８］De-Sheng L,Feng G,Hong-Qing Z. Solving the(2+1)-dimensional higher order Broer–Kaup system via a transformation and tanh-function method [J].Chaos,Solitons & Fractals,２004,20(5):1021－5.
［９］He JH.Variational principles for some nonlinear partial differential equations with variable coefficients [J].Chaos,Solitons &Fractals,2004,19(4): 847–51.
［１０］Nevin Pamuk.Series solution for porous medium equation with a source term by Adomian”s decomposition method,Appl.Math.Comput[J].2006, 178:480-485.
［１１］S.Pamuk.Solution of the porous media equation by Adomian’s decomposition method[J].Phys.Lett.A,2005,344:184–188.
［１２］A.D. Polyanin,V.F.Zaitsev,Handbook of Nonlinear Partial Differential Equations[M].Chapman and Hall/CRC Press,Boca Raton,2004.