基于Bernstein多项式和样条函数的高阶常微分方程数值解法研究
|
摘要
科学和工程技术中的许多实际问题都可以转化为微分方程的求解问题,而大部分的微分方程很难求出其解析解,因此,微分方程的数值解法的研究就具有重要的意义。由于样条函数具有许多好的性质而被广泛用于数值逼近、计算几何和微分方程数值解等领域。本文主要任务是利用几类样条函数来构造常微分方程的数值解。
本文章节内容安排如下:
第一、二两章介绍了本文的研究背景、几类样条函数和几种经典的微分方程数值解方法。
第三章在总结了近年来Bernstein多项式在微分方程数值解中的应用的基础上,用Bernstein多项式构造了一类两点边值问题方程组和奇异边值问题的数值解法,并用数值实例说明方法的有效性。
第四章总结了已有的两点边值问题的数值解法,在此基础上,给出了一类带有小参数ξ的两点边值问题的三次多项式样条解法,并分析了截断误差,用所给的方法计算了相关参考文献中的数值例子,与其他方法的结果进行了比较。
第五章讨论了高阶微分方程的数值解法,主要是对已有的方法特别近几年的新方法作了总结,在此基础上,给出了任意高阶微分方程的一种三次B样条解法,用所给的方法计算了参考文献中的数值例子,与已有的结果进行了比较,说明本文方法的有效性。
第六章总结了全文并进行了展望。
There are practical problems occurring in many areas of science and engineering, which can be transformed into solve differential equations. Most of differential equations are difficult to obtain their analytical solutions. Therefore, the research of numerical solution of differential equations is very important and necessary. The spline has many useful properties, such as the continuity, unity partition. So they are applied widely in function approximation, computational geometry, and numerical solution of differential equations and so on. In this paper, we study the numerical solution methods based on various kinds of spline function for initial and boundary value problems of ordinary differential equations.
This paper is organized as follows:
Chapter1and 2: Investigative background of numerical solution of differential equations, and introduce various types of spline functions and several classical numerical methods for solving differential equations.
Chapter 3: Summarize the application of Bernstein polynomial to solve differential equations, especially the recent existent numerical methods. Based on the result, we constructed the numerical method of a linear system of second-order boundary value problems and a kind of singular boundary value problem. Several numerical examples are presented to illustrate the efficiency of the methods.
Chapter 4: Summarize the existing numerical solution of second-order boundary value problems, based on the result, we developed a numerical technique of a class of two-point boundary value problem with parametricξusing cubic polynomial spline function, analysis the error and compute the numerical examples of the relevant references using the given method, and compare with other articles.
Chapter 5: Discuss the solution of high-order boundary value problems, sum up the recent existent numerical methods, based on the result, we constructed the numerical method of any high-order boundary value problems using third B-spline, we applied the new method to compute several numerical examples of the reference literature, and compared the result with other articles, it is shown that the new method is efficient.
Chapter 6: Summarizes the whole dissertation and give some expect ion.
本文章节内容安排如下:
第一、二两章介绍了本文的研究背景、几类样条函数和几种经典的微分方程数值解方法。
第三章在总结了近年来Bernstein多项式在微分方程数值解中的应用的基础上,用Bernstein多项式构造了一类两点边值问题方程组和奇异边值问题的数值解法,并用数值实例说明方法的有效性。
第四章总结了已有的两点边值问题的数值解法,在此基础上,给出了一类带有小参数ξ的两点边值问题的三次多项式样条解法,并分析了截断误差,用所给的方法计算了相关参考文献中的数值例子,与其他方法的结果进行了比较。
第五章讨论了高阶微分方程的数值解法,主要是对已有的方法特别近几年的新方法作了总结,在此基础上,给出了任意高阶微分方程的一种三次B样条解法,用所给的方法计算了参考文献中的数值例子,与已有的结果进行了比较,说明本文方法的有效性。
第六章总结了全文并进行了展望。
There are practical problems occurring in many areas of science and engineering, which can be transformed into solve differential equations. Most of differential equations are difficult to obtain their analytical solutions. Therefore, the research of numerical solution of differential equations is very important and necessary. The spline has many useful properties, such as the continuity, unity partition. So they are applied widely in function approximation, computational geometry, and numerical solution of differential equations and so on. In this paper, we study the numerical solution methods based on various kinds of spline function for initial and boundary value problems of ordinary differential equations.
This paper is organized as follows:
Chapter1and 2: Investigative background of numerical solution of differential equations, and introduce various types of spline functions and several classical numerical methods for solving differential equations.
Chapter 3: Summarize the application of Bernstein polynomial to solve differential equations, especially the recent existent numerical methods. Based on the result, we constructed the numerical method of a linear system of second-order boundary value problems and a kind of singular boundary value problem. Several numerical examples are presented to illustrate the efficiency of the methods.
Chapter 4: Summarize the existing numerical solution of second-order boundary value problems, based on the result, we developed a numerical technique of a class of two-point boundary value problem with parametricξusing cubic polynomial spline function, analysis the error and compute the numerical examples of the relevant references using the given method, and compare with other articles.
Chapter 5: Discuss the solution of high-order boundary value problems, sum up the recent existent numerical methods, based on the result, we constructed the numerical method of any high-order boundary value problems using third B-spline, we applied the new method to compute several numerical examples of the reference literature, and compared the result with other articles, it is shown that the new method is efficient.
Chapter 6: Summarizes the whole dissertation and give some expect ion.
引文
[1] Shahid S.Siddiqi, Ghazala Akram. Septic spline solutions of sixth-order boundary value problems [J]. Journal of Computational and Applied Mathematics, 2008, 215(1): 288– 301.
[2] Eisa A.Al-Said, Muhammad Aslam Noor. Quartic spline method for solving fo urth order obstacle boundary value problems [J]. Journal of Computational and Applied Mathematics, 2002, 143(1): 107–116.
[3] S. I. Zaki. A quintic B-spline finite elements scheme for the KdVB equation [J]. Computer Methods in Applied Mechanics and Engineering, 2000, 188(1-3):121-134.
[4] Siraj-ul-Islam, Ikram A. Tirmizi. Non-polynomial splines approach to the solution of sixth order boundary-value problems [J]. Applied Mathematics and Computation, 2008, 195(1): 270–284.
[5] Shahid S. Siddiqi Solution of 10th-order boundary value problems using nonpolynomial spline technique [J]. Applied Mathematics and Computation, 2007, 190(1): 641–651.
[6] Junfeng Lu.Variational iteration method for solving a nonlinear system of second-order boundary value problems [J]. Computers and Mathematics with Applications, 2007, 54(7-8):1133-1138.
[7] Muhammad Aslam Noor, Syed Tauseef Mohyud-Din. Variational iteration technique for solving higher order boundary value problems [J]. Applied Mathematics and Computation.2007, 189(2): 1929–1942.
[8] Bülent Saka, ?dris Da?. Quartic B-spline Galerkin approach to the numerical solution of the KdVB equation [J]. Applied Mathematics andComputation, 2009, 215(2): 746-758.
[9] A.S.V.Ravi Kanth, K.Aruna. He’s homotopy-perturbation method for solving higher-order boundary value problems [J]. Chaos, Solitons and Fractals.doi:10 16/j.chaos.2008.07.044. 73 (247).
[10] Muhammad Aslam Noor, Syed Tauseef Mohyud-Din. Homotopy perturbation method for solving sixth-order boundary value problems [J]. Computers and Mathematics with Applications, 2008, 55(12): 2953–2972.
[11] Golbabai, M.Javidi. Application of homotopy perturbation method for solving eighth-order boundary value problems [J]. Applied Mathematics and Comput-ation, 2007, 191(2): 334–346.
[12] M.K. Kadalbajoo, Puneet Arora. B-spline collocation method for the singular perturbation problem using artificial viscosity [J]. Computers and Mathemati- cs with Applications, 2009, 57(4):650-663.
[13] Samuel Jator. A high order B-spline collocation method for linear boundary value problems [J]. Applied Mathematics and Computation, 2007, 191(1):100-11 6
[14] Mehdi Dehghan, Abbas Saadatmandi. The numerical solution of a nonlinear system of second-order boundary value problems using the Sinc-collocation method [J]. Mathematical and Computer Modeling, 2007, 46(11-12):143-41.
[15] Lei Wu, Li-dan Xie, Jie-fang Zhang. Adnominal decomposition method for nonlinear differential-difference equations [J]. Communications in Nonlinear Science and Numerical Simulation, 2009, 14(1):12-18.
[16] Dambaru D. Bhatta, Muhammad I. Bhatti. Numerical solution of KdV equation using modified Bernstein polynomials [J]. Applied Mathematics and Computation, 2006, 174(2): 1255–1268.
[17] B.N. Mandal a*, Subhra Bhattacharya. Numerical solution of some classes of integral equations using Bernstein polynomials [J]. Applied Mathematics and Computation, 2007, 190(2):1707–1716.
[18] M.A.Ramadan, I.F.Lashien, W.K.Zahra. Quintic nopolynomial spline solutions for fourth order two-point boundary value problem [J]. Communications in Nonlinear Science and Numerical Simulation, 2009, 14(4):1105–1114.
[19] Hengfei Ding, Yuxin Zhang. Parameters spline methods for the solution of hyperbolic equations [J]. Applied Mathematics and Computation, 2008, 204(2):938-941.
[20] Arshad Khan. Parametric cubic spline solution of two point boundary value problems [J]. Applied Mathematics and Computation, 2004, 54(1): 175- 182.
[21] S. I. Zaki. A quintic B-spline finite elements scheme for the KdVB equation [J]. Computer Methods in Applied Mechanics and Engineering, 2000, 188(1-3): 121-134.
[22] P.M.普伦特著,样条函数与变分方法[M].柴家根,江伯南译.上海科技大学出版社,1980:90-95.
[23]李庆杨,王超能,易大义编.数值分析(第四版)[M].华中科技大学出版社,2006:35-38.
[24]孙家旭.样条函数和计算几何[M].北京:科学出版社,1982:231-238
[25]莫容.计算几何辅助设计[M].科学出版社,2004:83.
[26] Nazan Caglar, Hikmet Caglar. B-spline solution of singular boundary val- ue problems [J]. Applied Mathematics and Computation, 2006, 182(2):15 09–1513.
[27] M. Idrees Bhatti, P. Bracken. Solutions of differential equations in a Bernstein polynomial basis [J]. Journal of Computational and Applied Mathematics, 2007, 205 (1):272– 280.
[28]田振夫.两点边值问题的一种高精度方法[J].贵州大学学报, 1997, 14(2): 19-23.
[29]徐长发.偏微分方程数值解法[M].华中理工大学出版社,2006:137- 141
[30]胡建伟,汤怀民.微分方程数值解法[M].科学出版社,2007:65-70.
[31] Kumar M, Aziz T. A non uniform mesh finite difference method and its convergence for a class of singular two-point boundary-value problems [J]. Int JComput Math, 2004, 81(2):1507–12.
[32] Abbas Saadatmandi, Jalal Askari Farsangi. Chebyshev finite difference method for a nonlinear system of second-order boundary value problems [J]. Applied Mathematics and Computation, 2007,192(2):586-591.
[33]李瑞霞何志庆.微分方程数值解法[M].华东理工大学出版社,2005:172-216.
[34] A.S.V. Ravi Kanth, Y.N. Reddy, Cubic spline for a class of singular two-point boundary value problems [J]. Appl. Math. Comput, 2005, 170(2): 733–740.
[35] Manoj Kumar, Neelima Singh. A collection of computational techniques for solving singular boundary-value problems [J]. Advances in Engineering Softw- are, 2009, 40(4):288–297
[36] Manoj Kumar, Hradyesh Kumar Mishra, Peetam Singh. A boundary value approach for a class of linear singularly perturbed boundary value problems [J]. Advances in Engineering Software, 2009, 40 (4):298–304.
[37] A.S.V. Ravi Kanth, Y.N. Reddy, A numerical method for singular boundary value problems via Chebyshev economization [J]. Appl.Math.Comput, 2003, 146: 691–700.
[38] M.K. Kadalbajoo, K.C. Patidar. Variable mesh spline in compression for the numerical solution of singular perturbation problems [J], Int. J. Comput. Math, 2003, 80 (l):83-93.
[39] Nazan Caglar, Hikmet Caglar. B-spline method for solving linear system of second-order boundary value problems [J] .Computers and Mathematics with Applications, 2009, 57(7):757-762.
[40] Siraj-ul-IslamMuhammad Aslam NoorIkram A. Tirmizi. Quadratic non-polynomial spline approach to the solutionof a system of second-order boundary-value problems [J]. Applied Mathematics and Computation, 2006, 179(1): 153–160.
[41] Manoj Kumar. Higher order method for singular boundary-value problems by using spline function [J]. Applied Mathematics and Computa, 2007, 192(1): 175–179.
[42] Abbas Saadatmandia, Mehdi Dehghanb, Ali Eftekhari. Application of He’s homotopy perturbation method for non-linear system of second-order bound ary value problems [J]. Nonlinear Analysis: Real World Applications, 2009, 10(3): 1912-1922.
[43] Junfeng Lu.Variational iteration method for solving a nonlinear system of second-order boundary value problems [J]. Computers and Mathematics with Applications, 2007, 54(7-8):1133-1138.
[44] Abbas Saadatmandi, Jalal Askari Farsangi. Chebyshev finite difference method for a nonlinear system of second-order boundary value problems [J]. Applied Mathematics and Computation, 2007,192(2):586-591.
[45] Fazhan Geng, Minggen Cui. Solving a nonlinear system of second order boundary value problems [J]. J. Math. Anal. Appl, 2007,327 (2): 1167–1181.
[46] X. Cheng, C. Zhong, Existence of positive solutions for a second order ordinary differential system [J]. Math.Anal.Appl. 2005, 312(1):14-23.
[47] M.A. Ramadan, I.F. Lashien.High order accuracy nonpolynomial spline solutions for 2lth ordertwo point boundary value problems [J] Applied Mathematics and Computation[J] .2008,204(2): 920–927.
[48] Shahid S. Siddiqi Solution of 10th-order boundary value problems using non-polynomial spline technique [J]. Applied Mathematics and Computation, 2007, 190(1): 641–651.
[49] Shahid S. Siddiqi. Solutions of 12th order boundary value problems using nonpolynomial spline technique [J]. Applied Mathematics and Computation, 2008, 199(2): 559–571.
[50] Shahid S. Siddiqi, Ghazala Akram.Numerical solution of a system of fourth order boundaryvalue problems using cubic non-polynomial spline method [J].Applied Mathematics and Computation 2007, 190(1): 652–661.
[51] G.B. Loghmani, M. Ahmadinia. Numerical solution of sixth order boundary value problems with sixth degree B-spline functions [J]. Applied Mathematics and Computation, 2007, 186(2): 992–999.
[52] Al-Said EA, Noor MA, Rassias TM. Cubic splines method forsolvingfourth-order obstacle boundary value problems. Appl Math Comput[J] 2006;174(3):180–7.
[53] M.E. Gamel, J.R. Cannon, J.LatourA, I Zayed. Sinc-Galerkin method for solv- ing linear sixth-order boundary-value problems [J], Math.Comput, 2003, 73 (247):1325–1343
[54] M. Hesaaraki, Y. Jillian. A numerical method for solving nth-order boundary-value problems [J]. Applied Mathematics and Computation, 2008, 196(2): 889–897.
[55] Muhammad Aslam Noor. Variational iteration method for solving sixth-order boundary value problems [J]. Commun Nonlinear Sci Numer Simulat, 2009, 14(6): 2571-2580
[56] Siraj-Ul Islam, Sirajul Haq, Javid Ali. Numerical solution of special 12th-order boundary value problems using differential transform method [J]. Applied Mathematics and Computation, 2009, 14(4): 1132–1138.
[2] Eisa A.Al-Said, Muhammad Aslam Noor. Quartic spline method for solving fo urth order obstacle boundary value problems [J]. Journal of Computational and Applied Mathematics, 2002, 143(1): 107–116.
[3] S. I. Zaki. A quintic B-spline finite elements scheme for the KdVB equation [J]. Computer Methods in Applied Mechanics and Engineering, 2000, 188(1-3):121-134.
[4] Siraj-ul-Islam, Ikram A. Tirmizi. Non-polynomial splines approach to the solution of sixth order boundary-value problems [J]. Applied Mathematics and Computation, 2008, 195(1): 270–284.
[5] Shahid S. Siddiqi Solution of 10th-order boundary value problems using nonpolynomial spline technique [J]. Applied Mathematics and Computation, 2007, 190(1): 641–651.
[6] Junfeng Lu.Variational iteration method for solving a nonlinear system of second-order boundary value problems [J]. Computers and Mathematics with Applications, 2007, 54(7-8):1133-1138.
[7] Muhammad Aslam Noor, Syed Tauseef Mohyud-Din. Variational iteration technique for solving higher order boundary value problems [J]. Applied Mathematics and Computation.2007, 189(2): 1929–1942.
[8] Bülent Saka, ?dris Da?. Quartic B-spline Galerkin approach to the numerical solution of the KdVB equation [J]. Applied Mathematics andComputation, 2009, 215(2): 746-758.
[9] A.S.V.Ravi Kanth, K.Aruna. He’s homotopy-perturbation method for solving higher-order boundary value problems [J]. Chaos, Solitons and Fractals.doi:10 16/j.chaos.2008.07.044. 73 (247).
[10] Muhammad Aslam Noor, Syed Tauseef Mohyud-Din. Homotopy perturbation method for solving sixth-order boundary value problems [J]. Computers and Mathematics with Applications, 2008, 55(12): 2953–2972.
[11] Golbabai, M.Javidi. Application of homotopy perturbation method for solving eighth-order boundary value problems [J]. Applied Mathematics and Comput-ation, 2007, 191(2): 334–346.
[12] M.K. Kadalbajoo, Puneet Arora. B-spline collocation method for the singular perturbation problem using artificial viscosity [J]. Computers and Mathemati- cs with Applications, 2009, 57(4):650-663.
[13] Samuel Jator. A high order B-spline collocation method for linear boundary value problems [J]. Applied Mathematics and Computation, 2007, 191(1):100-11 6
[14] Mehdi Dehghan, Abbas Saadatmandi. The numerical solution of a nonlinear system of second-order boundary value problems using the Sinc-collocation method [J]. Mathematical and Computer Modeling, 2007, 46(11-12):143-41.
[15] Lei Wu, Li-dan Xie, Jie-fang Zhang. Adnominal decomposition method for nonlinear differential-difference equations [J]. Communications in Nonlinear Science and Numerical Simulation, 2009, 14(1):12-18.
[16] Dambaru D. Bhatta, Muhammad I. Bhatti. Numerical solution of KdV equation using modified Bernstein polynomials [J]. Applied Mathematics and Computation, 2006, 174(2): 1255–1268.
[17] B.N. Mandal a*, Subhra Bhattacharya. Numerical solution of some classes of integral equations using Bernstein polynomials [J]. Applied Mathematics and Computation, 2007, 190(2):1707–1716.
[18] M.A.Ramadan, I.F.Lashien, W.K.Zahra. Quintic nopolynomial spline solutions for fourth order two-point boundary value problem [J]. Communications in Nonlinear Science and Numerical Simulation, 2009, 14(4):1105–1114.
[19] Hengfei Ding, Yuxin Zhang. Parameters spline methods for the solution of hyperbolic equations [J]. Applied Mathematics and Computation, 2008, 204(2):938-941.
[20] Arshad Khan. Parametric cubic spline solution of two point boundary value problems [J]. Applied Mathematics and Computation, 2004, 54(1): 175- 182.
[21] S. I. Zaki. A quintic B-spline finite elements scheme for the KdVB equation [J]. Computer Methods in Applied Mechanics and Engineering, 2000, 188(1-3): 121-134.
[22] P.M.普伦特著,样条函数与变分方法[M].柴家根,江伯南译.上海科技大学出版社,1980:90-95.
[23]李庆杨,王超能,易大义编.数值分析(第四版)[M].华中科技大学出版社,2006:35-38.
[24]孙家旭.样条函数和计算几何[M].北京:科学出版社,1982:231-238
[25]莫容.计算几何辅助设计[M].科学出版社,2004:83.
[26] Nazan Caglar, Hikmet Caglar. B-spline solution of singular boundary val- ue problems [J]. Applied Mathematics and Computation, 2006, 182(2):15 09–1513.
[27] M. Idrees Bhatti, P. Bracken. Solutions of differential equations in a Bernstein polynomial basis [J]. Journal of Computational and Applied Mathematics, 2007, 205 (1):272– 280.
[28]田振夫.两点边值问题的一种高精度方法[J].贵州大学学报, 1997, 14(2): 19-23.
[29]徐长发.偏微分方程数值解法[M].华中理工大学出版社,2006:137- 141
[30]胡建伟,汤怀民.微分方程数值解法[M].科学出版社,2007:65-70.
[31] Kumar M, Aziz T. A non uniform mesh finite difference method and its convergence for a class of singular two-point boundary-value problems [J]. Int JComput Math, 2004, 81(2):1507–12.
[32] Abbas Saadatmandi, Jalal Askari Farsangi. Chebyshev finite difference method for a nonlinear system of second-order boundary value problems [J]. Applied Mathematics and Computation, 2007,192(2):586-591.
[33]李瑞霞何志庆.微分方程数值解法[M].华东理工大学出版社,2005:172-216.
[34] A.S.V. Ravi Kanth, Y.N. Reddy, Cubic spline for a class of singular two-point boundary value problems [J]. Appl. Math. Comput, 2005, 170(2): 733–740.
[35] Manoj Kumar, Neelima Singh. A collection of computational techniques for solving singular boundary-value problems [J]. Advances in Engineering Softw- are, 2009, 40(4):288–297
[36] Manoj Kumar, Hradyesh Kumar Mishra, Peetam Singh. A boundary value approach for a class of linear singularly perturbed boundary value problems [J]. Advances in Engineering Software, 2009, 40 (4):298–304.
[37] A.S.V. Ravi Kanth, Y.N. Reddy, A numerical method for singular boundary value problems via Chebyshev economization [J]. Appl.Math.Comput, 2003, 146: 691–700.
[38] M.K. Kadalbajoo, K.C. Patidar. Variable mesh spline in compression for the numerical solution of singular perturbation problems [J], Int. J. Comput. Math, 2003, 80 (l):83-93.
[39] Nazan Caglar, Hikmet Caglar. B-spline method for solving linear system of second-order boundary value problems [J] .Computers and Mathematics with Applications, 2009, 57(7):757-762.
[40] Siraj-ul-IslamMuhammad Aslam NoorIkram A. Tirmizi. Quadratic non-polynomial spline approach to the solutionof a system of second-order boundary-value problems [J]. Applied Mathematics and Computation, 2006, 179(1): 153–160.
[41] Manoj Kumar. Higher order method for singular boundary-value problems by using spline function [J]. Applied Mathematics and Computa, 2007, 192(1): 175–179.
[42] Abbas Saadatmandia, Mehdi Dehghanb, Ali Eftekhari. Application of He’s homotopy perturbation method for non-linear system of second-order bound ary value problems [J]. Nonlinear Analysis: Real World Applications, 2009, 10(3): 1912-1922.
[43] Junfeng Lu.Variational iteration method for solving a nonlinear system of second-order boundary value problems [J]. Computers and Mathematics with Applications, 2007, 54(7-8):1133-1138.
[44] Abbas Saadatmandi, Jalal Askari Farsangi. Chebyshev finite difference method for a nonlinear system of second-order boundary value problems [J]. Applied Mathematics and Computation, 2007,192(2):586-591.
[45] Fazhan Geng, Minggen Cui. Solving a nonlinear system of second order boundary value problems [J]. J. Math. Anal. Appl, 2007,327 (2): 1167–1181.
[46] X. Cheng, C. Zhong, Existence of positive solutions for a second order ordinary differential system [J]. Math.Anal.Appl. 2005, 312(1):14-23.
[47] M.A. Ramadan, I.F. Lashien.High order accuracy nonpolynomial spline solutions for 2lth ordertwo point boundary value problems [J] Applied Mathematics and Computation[J] .2008,204(2): 920–927.
[48] Shahid S. Siddiqi Solution of 10th-order boundary value problems using non-polynomial spline technique [J]. Applied Mathematics and Computation, 2007, 190(1): 641–651.
[49] Shahid S. Siddiqi. Solutions of 12th order boundary value problems using nonpolynomial spline technique [J]. Applied Mathematics and Computation, 2008, 199(2): 559–571.
[50] Shahid S. Siddiqi, Ghazala Akram.Numerical solution of a system of fourth order boundaryvalue problems using cubic non-polynomial spline method [J].Applied Mathematics and Computation 2007, 190(1): 652–661.
[51] G.B. Loghmani, M. Ahmadinia. Numerical solution of sixth order boundary value problems with sixth degree B-spline functions [J]. Applied Mathematics and Computation, 2007, 186(2): 992–999.
[52] Al-Said EA, Noor MA, Rassias TM. Cubic splines method forsolvingfourth-order obstacle boundary value problems. Appl Math Comput[J] 2006;174(3):180–7.
[53] M.E. Gamel, J.R. Cannon, J.LatourA, I Zayed. Sinc-Galerkin method for solv- ing linear sixth-order boundary-value problems [J], Math.Comput, 2003, 73 (247):1325–1343
[54] M. Hesaaraki, Y. Jillian. A numerical method for solving nth-order boundary-value problems [J]. Applied Mathematics and Computation, 2008, 196(2): 889–897.
[55] Muhammad Aslam Noor. Variational iteration method for solving sixth-order boundary value problems [J]. Commun Nonlinear Sci Numer Simulat, 2009, 14(6): 2571-2580
[56] Siraj-Ul Islam, Sirajul Haq, Javid Ali. Numerical solution of special 12th-order boundary value problems using differential transform method [J]. Applied Mathematics and Computation, 2009, 14(4): 1132–1138.