非线性偏微分方程及其数值计算
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摘要
本文的研究对象是非线性偏微分方程,由于这些偏微分方程来源于物理和其它应用学科,具有鲜明的物理意义,因此又称为非线性数学物理方程。本文讨论几个经典的非线性偏微分方程及他们的孤立波解,特别是较为详细地介绍了反演散射方法,以及利用这一方法来求KdV方程的单孤立波解和多孤立波解。反演散射法是解非线性偏微分方程的最常用,也是最普遍的方法,许多方程都可以利用这种方法来求解,目前也取得了一些结果。
本文概述了非线性偏微分方程的一种数值解法——Adomian分解方法(ADM法),包括基本原理,Adomian多项式,噪声现象和收敛性分析。这种方法是比较简单实用的,它对方程和解法的要求都不高,但是它的缺点也是明显的,就是收敛区间比较小,我们通过对ADM法解出的级数解使用Padé逼近,有效地改进ADM法的这一缺陷,取得了良好的效果。通过对形变Boussinesq方程的实验,我们验证了ADM方法的应用,同时,通过这一例子,也说明了Padé逼近对ADM法的改进效果是非常明显的。
In this thesis,the main research is about nonlinear partial differential equations.As these equations originate from physics and other applied subjects,with obvious physical meaning,they are also called nonlinear mathematical physics equations.We mainly summarizes some classical nonlinear partial differential equations and their solitary wave solutions,especially emphasizes the inverse scattering transform method(IST).Using this method,we get the single soliton solution and multiple soliton solutions of KdV equation.IST is the most common method in solving nonlinear partial differential equations.
We also consider a numerical method of nonlinear partial differential equations-------- Adomian decomposition method(ADM).The main points of the decomposition method is:the principle,the Adomian polynomials,the noise terms and the convergence results.This method is very easy and practical,but the disadvantage of this method is very obvious-----the convergent domain is very narrow.Using ADM,we get a series solution.In this paper,we used the Padéapproximant with ADM in overcoming this drawback.The ADM method together with Padéapproximant extends the domain of solution and gives better accuracy and better convergence than using ADM alone.
本文概述了非线性偏微分方程的一种数值解法——Adomian分解方法(ADM法),包括基本原理,Adomian多项式,噪声现象和收敛性分析。这种方法是比较简单实用的,它对方程和解法的要求都不高,但是它的缺点也是明显的,就是收敛区间比较小,我们通过对ADM法解出的级数解使用Padé逼近,有效地改进ADM法的这一缺陷,取得了良好的效果。通过对形变Boussinesq方程的实验,我们验证了ADM方法的应用,同时,通过这一例子,也说明了Padé逼近对ADM法的改进效果是非常明显的。
In this thesis,the main research is about nonlinear partial differential equations.As these equations originate from physics and other applied subjects,with obvious physical meaning,they are also called nonlinear mathematical physics equations.We mainly summarizes some classical nonlinear partial differential equations and their solitary wave solutions,especially emphasizes the inverse scattering transform method(IST).Using this method,we get the single soliton solution and multiple soliton solutions of KdV equation.IST is the most common method in solving nonlinear partial differential equations.
We also consider a numerical method of nonlinear partial differential equations-------- Adomian decomposition method(ADM).The main points of the decomposition method is:the principle,the Adomian polynomials,the noise terms and the convergence results.This method is very easy and practical,but the disadvantage of this method is very obvious-----the convergent domain is very narrow.Using ADM,we get a series solution.In this paper,we used the Padéapproximant with ADM in overcoming this drawback.The ADM method together with Padéapproximant extends the domain of solution and gives better accuracy and better convergence than using ADM alone.
引文
[1] TamerA.Abassy, Magdy,A.El-Tawil, Hassan K.Saleh. TheSolution of KdV and mKdV Equations Using Adomian-PadéApproximation [J].International Journal of Nonlinear Science and Numerical Simulation,5(4)(2004),327-339.
[2] Y.C.Jiao, Y.Yamamoto, C.Dang, Y.Hao. An Aftertreatment Technique for Improving the Accuracy of Adomian’s Decomposition Method[J].Computers and Mathematics with Applications,43(2003),783-798.
[3] Tamer A.Abassy, Magdy,A.El-Tawil, Hassan K.Saleh. The Solution of Burgers and good Boussinesq equations using ADM-Padétechnique[J].Chaos,Solitons and Fractals,32(2007),1008-1026.
[4] Abdul-Majid Wazwaz. Abundant solutions for several forms of the fifth-order KdV equation by using the tanh method[J].Applied mathematics and Computation,2006.
[5] V.V.Sokolov, Thomas Wolf. A symmetry test for quasilinear coupled systems[J].Inverse Problems,15(1999),L5-L11.
[6] Mehdi Dehghan, Asgar Hamidi, Mohammad Shakourifar. The solution of coupled Burgers’equations using Adomian-Padétechnique[J].Applied Mathematics and Computation,189(2007),1034-1047.
[7] Abir Alharbi, E.S.Fahmy, ADM-Padésolutions for generalized Burgers and Burgers-Huxley systems with two coupled equations[J].Journal of Computational and Applied Mathematics,2009.
[8] S.A.El-Wakil, M.A.Adou, New applications of Adomian decomposition method.[J].Chaos,Soliton and Fractals,33(2007),513-522.
[9] Mehdi Dehghan, Mohammad Shakourifar, Asgar Hamidi. The solution linear and nonlinear systems of Volterra functional equations using Adomian-Padétechnique[J]. Chaos,Soliton and Fractals,39(2009),2509-2521.
[10] Peiling Li, Zhenli Xu. Numerical Study of Two Classes of Nonlinearly Dispersive Equationswith“Compacton”Solution[J].2006.
[11] Mario Basto, Viriato Semiao, Francisco L.Calheiros. Numerical study of modified Adomian’s method applied to Burgers equation[J]Journal of Computational and Applied Mathematics 206(2007),927-949.
[12] M.A.Helal, M.S.Mahanna. The tanh method and Adomian decomposition method for solving the foam drainage eqution[J] Applied Mathematics and Computation.190(2007),599-609.
[13] Y.Cherruault, G.Adomian, K.Abbaoui, R.Rach. Further remarks on convergence of decompositionmethod[J]InternatinalJournalofBio-MedicalComputing.38(1995),89-93
[14] M.M.Hosseini, H.Nasabzadeh. On the convergence of Adomian decomposition method[J] Applied Mathematics and Computation.182(2006),536-543.
[15] D.Lesnic. Convergence of Adomian Decomposition Method:Periodic Tempratuers[J]Computers and Mathematics with Applications.44(2002),13-24.
[16] I.L.El-Kalla. Convergence of Adomian decomposition method applied to a class of integral equations[J] Applied Mathematics Letters.21(2008),372-376.
[17] D.K.R.Babajee, M.Z.Dauhoo, M.T.Darvishi, A.Barati. A note on the local convergence of interative methods based on Adomian decomposition method and 3-node quadrature rule[J] Applied Mathematics and Computation.200(2008),452-458.
[18] I Hashim, M.S.M.Noorani, R.Ahmad, S.A.Bakar. E.S.Ismail, A.M.Zakaria. Accuracy of Adomian decomposition method applied to the Lorenz system[J] Chaos,Soliton and Fractals.28(2006),1149-1158.
[19] E.Babolian, J.Biazar. On the order of convergence of Adomian method[J] Applied Mathematics and Computation.130(2002),383-387.
[20] Dogan Kaya ,Ibrahim E Inan .A convergence analysis of the ADM and an application[J].Applied Mathematics and Computation.161(2005),1015-1025.
[21] Peter Vadasz, Shmual Olek. Convergence and accuracy of Adomian decomposition method for the solution of Lorenz equations[J]International Journal of Heat and Mass Transfer.43(2000),1715-1734.
[22] Abdulaziz O, N.F.M.Noor, I.Hashim, M.S.M.Noorani. Further Accuracy tests on Adomian decomposition method for chaotic systems[J]. Chaos,Soliton and Fractals.36(2008),1405-1411.
[23] K.Abbaoui, Y.Cherruault. New Ideas for Proving Convergence of Decomposition Methods[J]. Computers and Mathematics with Applications.Vol 29,No 7(1995),103-108.
[24] Jose Marcio Machado, Sergio Luis Lopes Verardi, Yang Shiyou. An application of the Adomian Decomposition Method to the analysis of MHD Duct Flows[J]IEEE TRANSCATION AND MAGNETICS,VOL 41,NO 5,MAY 2005.
[25] T Badredine, K.Abbaoui, Y.Cherruault. Convergence of Adomian’s method applied to integral equations[J]Kybernetes.Vol 28,No 5(1999),557-564.
[26] Y.Cherruault. Convergence of Adomian’s method[J] Kybernetes.18,2(1988),31-38
[27] Z M Gharsseldien, K Hemida. New technique to avoid“noise terms”on the solutions of inhomogeneous differential equations by using Adomian decomposition method[J]Conmunications in Nonlinear Sciense and Numerical Simulation.14(2009),685-696.
[28] Abdul-Majid Wazwaz. The existence of noise terms for systems of inhomogeneous differential and integral equations[J] Applied Mathematics and Computation.146(2003),81-92.
[29] Abdul-Majid Wazwaz, Salah M .El-Sayed. A new modification of the Adomian decomposition method for linear and nonlinear operators[J] Applied Mathematics and Computation.122(2001),393-405.
[30] Abdul-Majid Wazwaz. A reliable modification of the Adomian decomposition method[J] Applied Mathematics and Computation.102(1999),77-86.
[31] Abdul-Majid Wazwaz. Necessary Conditions for the Appearance of Noise Terms in Decomposition Solution Series[J] Applied Mathematics and Computation.81(1997),265-274.
[32] Yonggui Zhu, Qianshun Chang, Shengchang Wu. A new algorithm for calculating Adomian polynomials[J] Applied Mathematics and Computation.169(2005),402-416.
[33] E Babolian, Sh.Javadi. New method for calculating Adomian polynomials[J] Applied Mathematics and Computation.153(2004),253-259.
[34]潘祖梁,陈仲慈.工程技术中的偏微分方程[M].杭州:浙江大学出版社,1995.269-280.
[35]陈祖墀.偏微分方程[M].合肥:中国科技大学出版社,2004.1-7.
[36]徐献瑜,李家楷,徐国良. Padé逼近逼近概论[M].上海:上海科学技术出版社,1988.1-48.
[37]范恩贵.可积系统与计算机代数[M].北京:科学出版社,2004.14-15.
[38]阳明盛,林建华.Mathematica基础及数学软件[M].大连:大连理工大学出版社,2003.
[39]李兰勤,Padé逼近及其应用[D].合肥工业大学硕士学位论文,2008.
[40]刘春,现代势论及其在数值计算中的应用[D].合肥工业大学硕士学位论文,2008.
[41]李兰勤,林京。ADM-Padé法解fKdV方程[J].合肥工业大学学报(自然科学版),31(10)(2008),1703-1705.
[42]骆兴国,Adomian分解方法及其最新进展[D].浙江大学硕士学位论文,2006.
[43]郑成德,Padé逼近若干问题研究[D].大连理工大学博士学位论文,2004.
[44]高峰,代数函数逼近及其在微分数值解中的应用[D].大连理工大学博士学位论文2004.
[45]沈进东,函数值部分Padé-型逼近及在积分方程中的应用[D].上海大学硕士学位论文,2006.
[46]严芹,有理样条逼近与插值[D].合肥工业大学硕士学位论文,2006.
[47]顾惠峰,快速Adomian方法及其在非线性系统中的应用[D].华东师范大学硕士学位论文,2007
[48]陈登远,孤子引论[M].北京:科学出版社,2006.46-48.
[49]张鸿林,葛显良。英汉数学词汇[M].北京:清华大学出版社,2005.
[50]杨伯君,赵玉芳。高等数学物理方法[M].北京:北京邮电大学出版社,2003.51-56,60-91.
[51]郭玉翠。非线性偏微分方程引论[M].北京:清华大学出版社,2008.1-5,40-44.
[52]郭柏灵,庞小峰。孤立子[M].北京:科学出版社,1987,1-5.
[53]朱永贵,高小山。计算Adomian多项式的新算法[J].系统科学与数学,2005(2),18-28.
[54]朱永贵,朝鲁。Adomian分解法在求解微分方程定解中的应用[J].内蒙古大学学报(自然科学版),2004,35(4),381-385.
[2] Y.C.Jiao, Y.Yamamoto, C.Dang, Y.Hao. An Aftertreatment Technique for Improving the Accuracy of Adomian’s Decomposition Method[J].Computers and Mathematics with Applications,43(2003),783-798.
[3] Tamer A.Abassy, Magdy,A.El-Tawil, Hassan K.Saleh. The Solution of Burgers and good Boussinesq equations using ADM-Padétechnique[J].Chaos,Solitons and Fractals,32(2007),1008-1026.
[4] Abdul-Majid Wazwaz. Abundant solutions for several forms of the fifth-order KdV equation by using the tanh method[J].Applied mathematics and Computation,2006.
[5] V.V.Sokolov, Thomas Wolf. A symmetry test for quasilinear coupled systems[J].Inverse Problems,15(1999),L5-L11.
[6] Mehdi Dehghan, Asgar Hamidi, Mohammad Shakourifar. The solution of coupled Burgers’equations using Adomian-Padétechnique[J].Applied Mathematics and Computation,189(2007),1034-1047.
[7] Abir Alharbi, E.S.Fahmy, ADM-Padésolutions for generalized Burgers and Burgers-Huxley systems with two coupled equations[J].Journal of Computational and Applied Mathematics,2009.
[8] S.A.El-Wakil, M.A.Adou, New applications of Adomian decomposition method.[J].Chaos,Soliton and Fractals,33(2007),513-522.
[9] Mehdi Dehghan, Mohammad Shakourifar, Asgar Hamidi. The solution linear and nonlinear systems of Volterra functional equations using Adomian-Padétechnique[J]. Chaos,Soliton and Fractals,39(2009),2509-2521.
[10] Peiling Li, Zhenli Xu. Numerical Study of Two Classes of Nonlinearly Dispersive Equationswith“Compacton”Solution[J].2006.
[11] Mario Basto, Viriato Semiao, Francisco L.Calheiros. Numerical study of modified Adomian’s method applied to Burgers equation[J]Journal of Computational and Applied Mathematics 206(2007),927-949.
[12] M.A.Helal, M.S.Mahanna. The tanh method and Adomian decomposition method for solving the foam drainage eqution[J] Applied Mathematics and Computation.190(2007),599-609.
[13] Y.Cherruault, G.Adomian, K.Abbaoui, R.Rach. Further remarks on convergence of decompositionmethod[J]InternatinalJournalofBio-MedicalComputing.38(1995),89-93
[14] M.M.Hosseini, H.Nasabzadeh. On the convergence of Adomian decomposition method[J] Applied Mathematics and Computation.182(2006),536-543.
[15] D.Lesnic. Convergence of Adomian Decomposition Method:Periodic Tempratuers[J]Computers and Mathematics with Applications.44(2002),13-24.
[16] I.L.El-Kalla. Convergence of Adomian decomposition method applied to a class of integral equations[J] Applied Mathematics Letters.21(2008),372-376.
[17] D.K.R.Babajee, M.Z.Dauhoo, M.T.Darvishi, A.Barati. A note on the local convergence of interative methods based on Adomian decomposition method and 3-node quadrature rule[J] Applied Mathematics and Computation.200(2008),452-458.
[18] I Hashim, M.S.M.Noorani, R.Ahmad, S.A.Bakar. E.S.Ismail, A.M.Zakaria. Accuracy of Adomian decomposition method applied to the Lorenz system[J] Chaos,Soliton and Fractals.28(2006),1149-1158.
[19] E.Babolian, J.Biazar. On the order of convergence of Adomian method[J] Applied Mathematics and Computation.130(2002),383-387.
[20] Dogan Kaya ,Ibrahim E Inan .A convergence analysis of the ADM and an application[J].Applied Mathematics and Computation.161(2005),1015-1025.
[21] Peter Vadasz, Shmual Olek. Convergence and accuracy of Adomian decomposition method for the solution of Lorenz equations[J]International Journal of Heat and Mass Transfer.43(2000),1715-1734.
[22] Abdulaziz O, N.F.M.Noor, I.Hashim, M.S.M.Noorani. Further Accuracy tests on Adomian decomposition method for chaotic systems[J]. Chaos,Soliton and Fractals.36(2008),1405-1411.
[23] K.Abbaoui, Y.Cherruault. New Ideas for Proving Convergence of Decomposition Methods[J]. Computers and Mathematics with Applications.Vol 29,No 7(1995),103-108.
[24] Jose Marcio Machado, Sergio Luis Lopes Verardi, Yang Shiyou. An application of the Adomian Decomposition Method to the analysis of MHD Duct Flows[J]IEEE TRANSCATION AND MAGNETICS,VOL 41,NO 5,MAY 2005.
[25] T Badredine, K.Abbaoui, Y.Cherruault. Convergence of Adomian’s method applied to integral equations[J]Kybernetes.Vol 28,No 5(1999),557-564.
[26] Y.Cherruault. Convergence of Adomian’s method[J] Kybernetes.18,2(1988),31-38
[27] Z M Gharsseldien, K Hemida. New technique to avoid“noise terms”on the solutions of inhomogeneous differential equations by using Adomian decomposition method[J]Conmunications in Nonlinear Sciense and Numerical Simulation.14(2009),685-696.
[28] Abdul-Majid Wazwaz. The existence of noise terms for systems of inhomogeneous differential and integral equations[J] Applied Mathematics and Computation.146(2003),81-92.
[29] Abdul-Majid Wazwaz, Salah M .El-Sayed. A new modification of the Adomian decomposition method for linear and nonlinear operators[J] Applied Mathematics and Computation.122(2001),393-405.
[30] Abdul-Majid Wazwaz. A reliable modification of the Adomian decomposition method[J] Applied Mathematics and Computation.102(1999),77-86.
[31] Abdul-Majid Wazwaz. Necessary Conditions for the Appearance of Noise Terms in Decomposition Solution Series[J] Applied Mathematics and Computation.81(1997),265-274.
[32] Yonggui Zhu, Qianshun Chang, Shengchang Wu. A new algorithm for calculating Adomian polynomials[J] Applied Mathematics and Computation.169(2005),402-416.
[33] E Babolian, Sh.Javadi. New method for calculating Adomian polynomials[J] Applied Mathematics and Computation.153(2004),253-259.
[34]潘祖梁,陈仲慈.工程技术中的偏微分方程[M].杭州:浙江大学出版社,1995.269-280.
[35]陈祖墀.偏微分方程[M].合肥:中国科技大学出版社,2004.1-7.
[36]徐献瑜,李家楷,徐国良. Padé逼近逼近概论[M].上海:上海科学技术出版社,1988.1-48.
[37]范恩贵.可积系统与计算机代数[M].北京:科学出版社,2004.14-15.
[38]阳明盛,林建华.Mathematica基础及数学软件[M].大连:大连理工大学出版社,2003.
[39]李兰勤,Padé逼近及其应用[D].合肥工业大学硕士学位论文,2008.
[40]刘春,现代势论及其在数值计算中的应用[D].合肥工业大学硕士学位论文,2008.
[41]李兰勤,林京。ADM-Padé法解fKdV方程[J].合肥工业大学学报(自然科学版),31(10)(2008),1703-1705.
[42]骆兴国,Adomian分解方法及其最新进展[D].浙江大学硕士学位论文,2006.
[43]郑成德,Padé逼近若干问题研究[D].大连理工大学博士学位论文,2004.
[44]高峰,代数函数逼近及其在微分数值解中的应用[D].大连理工大学博士学位论文2004.
[45]沈进东,函数值部分Padé-型逼近及在积分方程中的应用[D].上海大学硕士学位论文,2006.
[46]严芹,有理样条逼近与插值[D].合肥工业大学硕士学位论文,2006.
[47]顾惠峰,快速Adomian方法及其在非线性系统中的应用[D].华东师范大学硕士学位论文,2007
[48]陈登远,孤子引论[M].北京:科学出版社,2006.46-48.
[49]张鸿林,葛显良。英汉数学词汇[M].北京:清华大学出版社,2005.
[50]杨伯君,赵玉芳。高等数学物理方法[M].北京:北京邮电大学出版社,2003.51-56,60-91.
[51]郭玉翠。非线性偏微分方程引论[M].北京:清华大学出版社,2008.1-5,40-44.
[52]郭柏灵,庞小峰。孤立子[M].北京:科学出版社,1987,1-5.
[53]朱永贵,高小山。计算Adomian多项式的新算法[J].系统科学与数学,2005(2),18-28.
[54]朱永贵,朝鲁。Adomian分解法在求解微分方程定解中的应用[J].内蒙古大学学报(自然科学版),2004,35(4),381-385.