变分迭代法在抛物型方程反问题中的应用
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摘要
变分迭代算法是由何吉欢提出并广泛应用于一些微分方程的求解和一些特殊的非线性方程.变分迭代算法在非线性问题研究中有非常重要的作用,是解决一些非线性问题强有力的工具.
在求解抛物型方程反问题的同时,也必须决定一些未知的控制系数,这类问题在工程和科学的许多分支都起着重要的作用.变分迭代法应用于求解反问题中具有收敛序列的精确解.
本文共分为五章.
第一章叙述了变分迭代算法的研究现状和本文的研究内容.
第二章给出有关变分理论的预备知识及结论.
第三章首先介绍了变分迭代算法的基本思想,然后叙述变分迭代算法及其简单的应用,最后给出改进的变分迭代算法.
第四章介绍变分迭代算法应用于抛物型方程反问题.
第五章的主要内容是改进的变分迭代算法在多维抛物型方程反问题中的应用.
Variational iteration method which has been proposed by Ji-Huan He is widely applied to solve some differential equations and some special non-linear equations. Variational iteration method is very important in studying of the nonlinear problems and has been a powerful tool.
While determining the solutions of the problems, we shall determinate some unknown control parameters. These problems play a very important role in many branches of engineering and sciences. Application of the variational iteration tech-nique to these inverse problems shows the rapid convergence of the sequence con-structed to the exact solution.
This paper is divided into five sections.
In section 1, we briefly introduce the research status of variational iteration me-thod and the content of this paper.
In section 2, the preparation and the important conclusions of the variational theory are given.
In section 3, we first introduce the basic ideas of the variational iteration method, then its simple applications and the modified variational iteration method are de-scribed.
In section 4, the variational iteration method is applied to inverse parabolic problem.
In section 5, the modified variational iteration method is applied to the multi -dimensional inverse parabolic problem.
在求解抛物型方程反问题的同时,也必须决定一些未知的控制系数,这类问题在工程和科学的许多分支都起着重要的作用.变分迭代法应用于求解反问题中具有收敛序列的精确解.
本文共分为五章.
第一章叙述了变分迭代算法的研究现状和本文的研究内容.
第二章给出有关变分理论的预备知识及结论.
第三章首先介绍了变分迭代算法的基本思想,然后叙述变分迭代算法及其简单的应用,最后给出改进的变分迭代算法.
第四章介绍变分迭代算法应用于抛物型方程反问题.
第五章的主要内容是改进的变分迭代算法在多维抛物型方程反问题中的应用.
Variational iteration method which has been proposed by Ji-Huan He is widely applied to solve some differential equations and some special non-linear equations. Variational iteration method is very important in studying of the nonlinear problems and has been a powerful tool.
While determining the solutions of the problems, we shall determinate some unknown control parameters. These problems play a very important role in many branches of engineering and sciences. Application of the variational iteration tech-nique to these inverse problems shows the rapid convergence of the sequence con-structed to the exact solution.
This paper is divided into five sections.
In section 1, we briefly introduce the research status of variational iteration me-thod and the content of this paper.
In section 2, the preparation and the important conclusions of the variational theory are given.
In section 3, we first introduce the basic ideas of the variational iteration method, then its simple applications and the modified variational iteration method are de-scribed.
In section 4, the variational iteration method is applied to inverse parabolic problem.
In section 5, the modified variational iteration method is applied to the multi -dimensional inverse parabolic problem.
引文
[1]Cannon JR.Lin Y. Determination of parameter p(t) in somequasi-linear parabolic differential equation, Inverse Problems, 1988;4:35-45.
[2]Emine Can Baran. Numerical procedures for determining of an unknown para-meter in parabolic equation. Applied Mathematics and Computation 2005;162:1219-1226.
[3]Hasanov A., Liu ZH., An inverse coefficient problem for a nonlinear parabolic variational inequality, Applied Mathematics Letters, In Press.
[4]Liu ZH., Browder-Tikhonov Regularization of non-coercive evolution hemivatia-tional inequalities, Inverse problems 2005;21:13-20.
[5]Liu ZH., Identification of parameters in semilinear parabolic equations, Acta Mathematica Scientia, 19(2), 1999,175-180.
[6]Liu ZH., On the identification of coefficients of semilinear parabolic equations, Acta Mathematica Applicatae Sinica, English Series, 10(4),1994,356-367.
[7]Tatari M, Dehghan M, He's variational iteration method for computing a control parameter in a semi-linear inverse parabolic equation, Chaos Solitons Fractals 2007;33 (2): 671-677.
[8]M.Inokuti,H.Sekine,T.Mura.General use of the lagrange multiplier in nonlinear mathematical physics,.in:S.Nemat-Nasser(Ed.),Variational Method in the me-chanics of Solids[C], New York: Pergamon Press,1978,156-163.
[9]He,J.H.A new approach to nonlinear partial differential equations [J].Commun. Nonlin.Sci.& Numer.Simul.,1997,2:230-235.
[10]A.H.Nayfeh.Perturbation Methods[M],John Wiley & Sons,New York,1973.
[11]G.Adomian,Stochastic Systems[M],Academic Press,Inc.,New York,1983.
[12]He,J.H.Variationaliterationmethod:a kind of nonlinear analytical technique:some example[J].Int.J.Nonlinear Mech.1999,34(4):699-708.
[13]He.J.H.Homotopy perturbation technique[J],Comput.Meth.Appl.Mech.Eng. 1999,(178)257-262.
[14]S.J.Liao.Beyond Perturbation[M],CRC Press,Boca Raton,Florida,2003.
[15]He.J.H.Modified Linstedt-Poincare methods for some non-linear oscillations[J] Part I:expansion of constant,J.Non-linear Mech,2002,37:309-314.
[16]He.J.H.Some asymptotic methods for strongly nonlinear equations[J], Int.J.Modern.Phys,2006,20:1141-1199.
[17]Abdou,M.A.,Soliman,A.A..New applications of Vatiational iteration method[J] . Phys.2005,D211:1-8.
[18]Abdou,M.A.,Soliman,A.A..Variational iteration method for solving Burger’s and Coupled Burgder’s equations[J].J.Comput.Appl.Math,2005,181:245-251.
[19]Abulwafa,E.M,Abdou,M.A,Mahmoud,A.A.The solution of nonlinear Coagula-tion problem with mass loss[J].Chaos Solitons Fractals ,2006,29:301-330.
[20]Abulwafa,E.M.,Abdou,M.A.,Mahmoud,A.A..Nonlinear fluid flows in pipe-like domain problem using variational iteration method[J]. Chaos Solitons Fractals, 2007, 32:1384-1397.
[21]Soliman,A.A.,Abdou,M.A..Numerical solutions of nonlinear evelution iquations using variational iteration method[J].J. Comput.Appl.Math.2007,207(1):111-120.
[22]Bildik,N.,Konural.P.A..The use of variational iteration method,differential transform method and Adomian decomposition method for solving different types of nonlinear partial differential equations[J].Int.J.Nonlinear Sci.Numer- Simul.2006;7(1):65-70
[23]C.M.Bender,K.A.Milton,S.S.Pinsky,L.M.Simmons.A new perturbative approach to nonlinear problems[J],J.Math.Phys,1989,30:1447-1455.
[24]Campos.L.M.B.C.On the solution of some simple fractional differential equa-tions [J],International J.Math & Math.Sci.,1990,13(3):481-496.
[25]Delbosco.D.Existence and uniqueness for a nonlinear fractional differential eq-uation[J],J.Math.Anal & Appl.,1996,204(2):609-612.
[26]M.Tatari,M.Dehghan..On the convergence of He’s variational iteration method [J].Journal of Comput.and Appl.Math,2007,207:121-128.
[27]Finlayson,B.A.The method of weighted residuals and variational principles. Acad Press,1972.
[28]He,J.H,Wu,X.H:.Variational iteration method:New development and applica-tions [J].Computers and Mathematics with Applications .2007, 54: 881-894.
[29]T.A.Abassy,M.A.El-Tawil,H.El Zoheiry,Solving nonlinear partial differential equations using the modified vatiational iteration-Pade technique,Journal of Computational and Applied Mathematics,2007,207(1):73-91.
[30]S.Abbasbandy,A new application of He’s variational iteration method for quadratic Riccati differential equation by using Adomian’s polynomials,Journal of Computational and Applied Mathematics,2007,207(1):59-63.
[31]G.Adomian.A review of the decomposition method in applied mathematics[J].J.Math.Anal.Applic,1988,135:510-513.
[32]Zhongwei Cha.Partial differential equations of mathematics and physics[M] ,Traffi-c university of southwest press,2005.
[33]T.A.Abassy,M.A.EI-Tawil,H,EI Zoheiry,Solving nonlinear partial differential equations using the modified variational iteralion-Pade technique,J.C.and Appl.Math.2007;207(1) :73-91.
[34]N.Bildik,A.Konuralp,The use of variational iteration method, differential trans-form method and adomian decomposition method for solving different types of nonlinear partial differential equations,I.J.Nonlinear Sciences and Numerical Simulation 2006;7(1):65-70.
[35]S.Abbasbandy,A new application of He’s variational iteration method for quadratic Riccati differential equation by using Adomian’s polynomials, J.Compu.and Appl.Math. 2007;207(1):59-63.
[36]J.I.Ramos,On the variational iteration method and other iterative techniques for nonlinear differential equations.Appl.Math.Comput.(2007),doi:10.1016/j.amc. 2007.09.024
[37]J.Biazar,H.Ghazvini,Homotopy perturbation method for solving hyperbolic par-tial Differential equations,Comp. & Math. with Appl.2008,56(2):453-458.
[38]J.H.He,Approximate analytical solution for seepage flow with fractional deriva-tive in porous media,Computer Methods in Applied Mechanics and Engineering 1998;167(1-2):57-68.
[39]Odibat.ZM., Momani.S.,Application of variational iteration method to Nonlinear differential equations of fractional order, Int. J. Nonlinear Sci., 2006;7 (1): 27-34.
[40]Biazar J.,Ghazvini H., He's variational iteration method for solving hyperbolic differential equations, Int. J. Nonlinear Sci., 2007;8 (3):311-314.
[41]Odibat ZM., Momani S.,Application of variational iteration method to Nonlinear differential equations of fractional order, Int. J. Nonlinear Sci., 2006;7 (1): 27-34.
[42]Sadighi A., Ganji DD., Solution of the generalized nonlinear Boussinesq equa-tion using homotopy perturbation and variational iteration methods, Int. J. Non-linear Sci., 2007;8 (3): 435-443.
[43]Tari H., Ganji DD.,Rostamian M.,Approximate solutions of K (2,2),KdV and modified KdV equations by variational iteration method,homotopy perturbation method and homotopy analysis method,Int. J. Nonlinear Sci., 2007;8(2):203-210.
[44]胡海昌,变分学,中国建筑工业出版社,1987,7.
[2]Emine Can Baran. Numerical procedures for determining of an unknown para-meter in parabolic equation. Applied Mathematics and Computation 2005;162:1219-1226.
[3]Hasanov A., Liu ZH., An inverse coefficient problem for a nonlinear parabolic variational inequality, Applied Mathematics Letters, In Press.
[4]Liu ZH., Browder-Tikhonov Regularization of non-coercive evolution hemivatia-tional inequalities, Inverse problems 2005;21:13-20.
[5]Liu ZH., Identification of parameters in semilinear parabolic equations, Acta Mathematica Scientia, 19(2), 1999,175-180.
[6]Liu ZH., On the identification of coefficients of semilinear parabolic equations, Acta Mathematica Applicatae Sinica, English Series, 10(4),1994,356-367.
[7]Tatari M, Dehghan M, He's variational iteration method for computing a control parameter in a semi-linear inverse parabolic equation, Chaos Solitons Fractals 2007;33 (2): 671-677.
[8]M.Inokuti,H.Sekine,T.Mura.General use of the lagrange multiplier in nonlinear mathematical physics,.in:S.Nemat-Nasser(Ed.),Variational Method in the me-chanics of Solids[C], New York: Pergamon Press,1978,156-163.
[9]He,J.H.A new approach to nonlinear partial differential equations [J].Commun. Nonlin.Sci.& Numer.Simul.,1997,2:230-235.
[10]A.H.Nayfeh.Perturbation Methods[M],John Wiley & Sons,New York,1973.
[11]G.Adomian,Stochastic Systems[M],Academic Press,Inc.,New York,1983.
[12]He,J.H.Variationaliterationmethod:a kind of nonlinear analytical technique:some example[J].Int.J.Nonlinear Mech.1999,34(4):699-708.
[13]He.J.H.Homotopy perturbation technique[J],Comput.Meth.Appl.Mech.Eng. 1999,(178)257-262.
[14]S.J.Liao.Beyond Perturbation[M],CRC Press,Boca Raton,Florida,2003.
[15]He.J.H.Modified Linstedt-Poincare methods for some non-linear oscillations[J] Part I:expansion of constant,J.Non-linear Mech,2002,37:309-314.
[16]He.J.H.Some asymptotic methods for strongly nonlinear equations[J], Int.J.Modern.Phys,2006,20:1141-1199.
[17]Abdou,M.A.,Soliman,A.A..New applications of Vatiational iteration method[J] . Phys.2005,D211:1-8.
[18]Abdou,M.A.,Soliman,A.A..Variational iteration method for solving Burger’s and Coupled Burgder’s equations[J].J.Comput.Appl.Math,2005,181:245-251.
[19]Abulwafa,E.M,Abdou,M.A,Mahmoud,A.A.The solution of nonlinear Coagula-tion problem with mass loss[J].Chaos Solitons Fractals ,2006,29:301-330.
[20]Abulwafa,E.M.,Abdou,M.A.,Mahmoud,A.A..Nonlinear fluid flows in pipe-like domain problem using variational iteration method[J]. Chaos Solitons Fractals, 2007, 32:1384-1397.
[21]Soliman,A.A.,Abdou,M.A..Numerical solutions of nonlinear evelution iquations using variational iteration method[J].J. Comput.Appl.Math.2007,207(1):111-120.
[22]Bildik,N.,Konural.P.A..The use of variational iteration method,differential transform method and Adomian decomposition method for solving different types of nonlinear partial differential equations[J].Int.J.Nonlinear Sci.Numer- Simul.2006;7(1):65-70
[23]C.M.Bender,K.A.Milton,S.S.Pinsky,L.M.Simmons.A new perturbative approach to nonlinear problems[J],J.Math.Phys,1989,30:1447-1455.
[24]Campos.L.M.B.C.On the solution of some simple fractional differential equa-tions [J],International J.Math & Math.Sci.,1990,13(3):481-496.
[25]Delbosco.D.Existence and uniqueness for a nonlinear fractional differential eq-uation[J],J.Math.Anal & Appl.,1996,204(2):609-612.
[26]M.Tatari,M.Dehghan..On the convergence of He’s variational iteration method [J].Journal of Comput.and Appl.Math,2007,207:121-128.
[27]Finlayson,B.A.The method of weighted residuals and variational principles. Acad Press,1972.
[28]He,J.H,Wu,X.H:.Variational iteration method:New development and applica-tions [J].Computers and Mathematics with Applications .2007, 54: 881-894.
[29]T.A.Abassy,M.A.El-Tawil,H.El Zoheiry,Solving nonlinear partial differential equations using the modified vatiational iteration-Pade technique,Journal of Computational and Applied Mathematics,2007,207(1):73-91.
[30]S.Abbasbandy,A new application of He’s variational iteration method for quadratic Riccati differential equation by using Adomian’s polynomials,Journal of Computational and Applied Mathematics,2007,207(1):59-63.
[31]G.Adomian.A review of the decomposition method in applied mathematics[J].J.Math.Anal.Applic,1988,135:510-513.
[32]Zhongwei Cha.Partial differential equations of mathematics and physics[M] ,Traffi-c university of southwest press,2005.
[33]T.A.Abassy,M.A.EI-Tawil,H,EI Zoheiry,Solving nonlinear partial differential equations using the modified variational iteralion-Pade technique,J.C.and Appl.Math.2007;207(1) :73-91.
[34]N.Bildik,A.Konuralp,The use of variational iteration method, differential trans-form method and adomian decomposition method for solving different types of nonlinear partial differential equations,I.J.Nonlinear Sciences and Numerical Simulation 2006;7(1):65-70.
[35]S.Abbasbandy,A new application of He’s variational iteration method for quadratic Riccati differential equation by using Adomian’s polynomials, J.Compu.and Appl.Math. 2007;207(1):59-63.
[36]J.I.Ramos,On the variational iteration method and other iterative techniques for nonlinear differential equations.Appl.Math.Comput.(2007),doi:10.1016/j.amc. 2007.09.024
[37]J.Biazar,H.Ghazvini,Homotopy perturbation method for solving hyperbolic par-tial Differential equations,Comp. & Math. with Appl.2008,56(2):453-458.
[38]J.H.He,Approximate analytical solution for seepage flow with fractional deriva-tive in porous media,Computer Methods in Applied Mechanics and Engineering 1998;167(1-2):57-68.
[39]Odibat.ZM., Momani.S.,Application of variational iteration method to Nonlinear differential equations of fractional order, Int. J. Nonlinear Sci., 2006;7 (1): 27-34.
[40]Biazar J.,Ghazvini H., He's variational iteration method for solving hyperbolic differential equations, Int. J. Nonlinear Sci., 2007;8 (3):311-314.
[41]Odibat ZM., Momani S.,Application of variational iteration method to Nonlinear differential equations of fractional order, Int. J. Nonlinear Sci., 2006;7 (1): 27-34.
[42]Sadighi A., Ganji DD., Solution of the generalized nonlinear Boussinesq equa-tion using homotopy perturbation and variational iteration methods, Int. J. Non-linear Sci., 2007;8 (3): 435-443.
[43]Tari H., Ganji DD.,Rostamian M.,Approximate solutions of K (2,2),KdV and modified KdV equations by variational iteration method,homotopy perturbation method and homotopy analysis method,Int. J. Nonlinear Sci., 2007;8(2):203-210.
[44]胡海昌,变分学,中国建筑工业出版社,1987,7.