广义Burgers-Fisher方程的Haar小波有限差分法
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摘要
本论文研究了广义Burgers-Fisher方程初边值问题的Haar小波有限差分法。广义Burgers-Fisher方程作为描述反应机理、对流效应和扩散传播之间相互作用的典型模型,在现代物理学中具有重要的意义。本文利用周期小波定义了闭区间上的Haar小波系,对Haar小波系进行积分,从而建立了新的Haar小波n重积分矩阵,并将该矩阵应用到Haar小波有限差分法中。采用一阶精度的向后差分逼近时间微分,而空间导数则利用Haar小波系来展开,提出了一个求解广义Burgers-Fisher方程初边值问题的Haar小波有限差分法,进而给出了相应的算法和程序框图,编制了MATLAB程序代码。
本文还分析了Haar小波有限差分法的稳定性,数值验证了文中所给出的算法是条件稳定的,并对算法进行了正定性和有界性测试,结果表明文中算法能够保持数值解的正定性和有界性。
本文算法充分结合了Haar小波多分辨率分析计算灵活、计算效率高的特性和有限差分法易于实现的优点,同时,由于利用了Haar小波n重积分矩阵的稀疏性,因而有效地提高了计算速度和精度。通过计算机模拟所获得的数值结果,与有限差分法和Adomian分解法等方法进行比较,显示本文所给出的算法精度高于有限差分法,对于求解较小的时间问题时,精度高于Adomian分解法,这说明文中所给出的算法是可行的、有效的,并且精度较高,为研究广义Burgers-Fisher方程初边值问题的数值解法提供了新的途径和新的思路。
In this paper, the numerical solution of the generalized Burgers-Fisher equation with the initial boundary value conditions based on the Haar wavelet-finite difference method is studied. The generalized Burgers-Fisher equation shows a prototypical model for describing the interaction between the reaction mechanism, convection effect, and diffusion transport. It has the extremely vital significance in the modern physics. The Haar wavelet family on the closed interval is given. The Haar wavelet matrix of n-tuple integration is established. Haar wavelet-finite difference method adopted the matrix of n-tuple integration. The matrix is to convert the integral operations into the matrix operations. According to this method the spatial operators are approximated by the Haar wavelet family and the time derivation operators by the first-order precision backward difference quotient. The Haar wavelet-finite difference method is proposed to solve the generalized Burgers-Fisher equation with the general initial boundary value conditions. The paper gives the algorithm, program diagram and MATLAB code.
This paper analyzes the stability of Haar wavelet-finite difference method, and numerical proves that the algorithm is conditional stability. The algorithm property of the positivity of the numerical solutions and their boundedness is testing. The results show that the algorithm can maintain property of the positivity of the numerical solutions and their boundedness.
This paper fully combines the features of Haar wavelet multi-resolution analysis flexible, efficiency and the advantage of finite difference easy to realize. This method improves the computation speed and accuracy. It is due to the sparsity of the Haar wavelet matrix of n-tuple integration. Numerical results, obtained by computer simulation, are compared with the finite difference method and the Adomian decomposition method. The result showed that the method for solving problems of small time is more than Adomian decomposition method. The algorithm precision is higher than the finite difference method. The algorithm is feasible, effective and high accuracy. In order to study the numerical solution for the generalized Burgers-Fisher equation with initial boundary value problem providing new ideas and new methods.
本文还分析了Haar小波有限差分法的稳定性,数值验证了文中所给出的算法是条件稳定的,并对算法进行了正定性和有界性测试,结果表明文中算法能够保持数值解的正定性和有界性。
本文算法充分结合了Haar小波多分辨率分析计算灵活、计算效率高的特性和有限差分法易于实现的优点,同时,由于利用了Haar小波n重积分矩阵的稀疏性,因而有效地提高了计算速度和精度。通过计算机模拟所获得的数值结果,与有限差分法和Adomian分解法等方法进行比较,显示本文所给出的算法精度高于有限差分法,对于求解较小的时间问题时,精度高于Adomian分解法,这说明文中所给出的算法是可行的、有效的,并且精度较高,为研究广义Burgers-Fisher方程初边值问题的数值解法提供了新的途径和新的思路。
In this paper, the numerical solution of the generalized Burgers-Fisher equation with the initial boundary value conditions based on the Haar wavelet-finite difference method is studied. The generalized Burgers-Fisher equation shows a prototypical model for describing the interaction between the reaction mechanism, convection effect, and diffusion transport. It has the extremely vital significance in the modern physics. The Haar wavelet family on the closed interval is given. The Haar wavelet matrix of n-tuple integration is established. Haar wavelet-finite difference method adopted the matrix of n-tuple integration. The matrix is to convert the integral operations into the matrix operations. According to this method the spatial operators are approximated by the Haar wavelet family and the time derivation operators by the first-order precision backward difference quotient. The Haar wavelet-finite difference method is proposed to solve the generalized Burgers-Fisher equation with the general initial boundary value conditions. The paper gives the algorithm, program diagram and MATLAB code.
This paper analyzes the stability of Haar wavelet-finite difference method, and numerical proves that the algorithm is conditional stability. The algorithm property of the positivity of the numerical solutions and their boundedness is testing. The results show that the algorithm can maintain property of the positivity of the numerical solutions and their boundedness.
This paper fully combines the features of Haar wavelet multi-resolution analysis flexible, efficiency and the advantage of finite difference easy to realize. This method improves the computation speed and accuracy. It is due to the sparsity of the Haar wavelet matrix of n-tuple integration. Numerical results, obtained by computer simulation, are compared with the finite difference method and the Adomian decomposition method. The result showed that the method for solving problems of small time is more than Adomian decomposition method. The algorithm precision is higher than the finite difference method. The algorithm is feasible, effective and high accuracy. In order to study the numerical solution for the generalized Burgers-Fisher equation with initial boundary value problem providing new ideas and new methods.
引文
[1] A. M. Wazwaz. The tanh method for generalized forms of nonlinear heat conduction and Burgers-Fisher equations[J].Applied Mathematics and Computation, 2005,169(1):321-338.
[2] L. Wazzan. A modified tanh-coth method for solving the general Burgers-Fisher and the Kuramoto-Sivashinsky equations[J].Communications in Nonlinear Science and Numerical Simulation, 2009,14(6):2642-2652.
[3]段玲玲.广义Burgers-Fisher方程的数值逼近[D].厦门:厦门大学,2008.
[4]李立康,於崇华,朱政华.微分方程数值解法[M].上海:复旦大学出版社,2003.
[5]樊启斌.小波分析[M].武汉:武汉大学出版社,2008.
[6]何正嘉,陈雪峰.小波有限元理论研究与工程应用的进展[J].机械工程学报,2005,41(3):1-11.
[7] J. Satsuma. Exact solutions of Burgers equation with reaction terms[C]//M. Ablowitz, B. Fuchssteiner, M. Kruskal. Topics in soliton theory and exactly solvable nonlinear equations. Singapore: World Scientific, 1987: 255-262.
[8] Cai Guoliang, Wang Yan, Zhang Fengyun. Nonclassical symmetries and group invariant solutions of Burgers-Fisher equations[J].World Journal of Modelling and Simulation, 2007,3(4):305-309.
[9]王明亮,白雪.广义Burgers-Fisher方程的精确解[J].兰州大学学报(自然科学版), 1999,35(2):1-6.
[10]何宝钢.Burgers-Fisher方程的精确解[J].怀化师专学报,2002,21(2):17-19.
[11]姜璐.非线性偏微分方程精确解与孤波相互作用问题的研究[D].北京:北京邮电大学,2007.
[12] R. E. Mickens, A. Gumel. Construction and analysis of non-standard finite difference scheme for the Burgers-Fisher equation[J].Journal of Sound and Vibration, 2002,257(4),791-797.
[13]陈晓永.Burgers-Fisher方程的数值解法[D].南京:南京航空航天大学,2007.
[14]高钦姣.MQ拟插值在Burgers-Fisher方程数值解中的应用[D].大连:大连理工大学,2009.
[15] Sun Xinxiu, Lu Yiping, Liu Haiyang. Exact solutions to wick type stochastic generalized Burgers-Fisher equation[J].Journal of Xuzhou Normal University (Natural Science Edition), 2009,27(2):50-54.
[16] Wang Xinyi, Lu Yuekai. Exact solutions of the extended Burgers-Fisher equation[J].Chinese Physics Letters, 1990,7(4):145-147.
[17] Chen Huaitang, Zhang Hongqing. New multiple soliton solutions to the generalBurgers-Fisher equation and the Kuramoto-Sivashinsky equation[J].Chaos, Solitons and Fractals, 2004,19(1):71-76.
[18] S. A. Wakil, M. A. Abdou. Modified extended tanh-function method for solving nonlinear partial differential equation[J].Chaos, Solitons and Fractals, 2007,31(5):1256-1264.
[19] D. Kaya, S. M. Sayed. A numerical simulation and explicit solutions of the generalized Burgers-Fisher equation[J].Applied Mathematics and Computation, 2004,152(2):403-413.
[20] H. N. A. Ismail, et al. Adomian decomposition method for Burgers-Huxley and Burgers-Fisher equations[J].Applied Mathematics and Computation, 2004,159(1): 291-301.
[21] H. N. A. Ismail, A. A. A. Rabboh. A restrictive Padéapproximation for the solution of the generalized Fisher and Burger-Fisher equations[J].Applied Mathematics and Computation, 2004,154(1): 203-210.
[22] M. Javidi. Spectral collocation method for the solution of the generalized Burger-Fisher equation[J].Applied Mathematics and Computation, 2006,174(1): 345-352.
[23] A. Golbabai, M. Javidi. A spectral domain decomposition approach for the generalized Burgers-Fisher equation[J].Chaos, Solitons and Fractals, 2009,39(1): 385-392.
[24] M. Moghimi, F. S. A. Hejazi. Variational iteration method for solving generalized Burger-Fisher and Burger equations[J].Chaos, Solitons and Fractals, 2007,33(5): 1756-1761.
[25] M. Sari, G. Gürarslan, A. Zeytino?lu. High-order finite difference schemes for the solution of the generalized Burgers-Fisher equation[J].International Journal for Numerical Methods in Biomedical Engineering, 2009,27(5):125-132.
[26] M. Sari, G. Gürarslan, I. Da?. A compact finite difference method for the solution of the generalized Burgers-Fisher equation[J].Numerical Methods for Partial Differential Equations, 2010,26(1):125-134.
[27]高博,曲小钢.广义Burgers-Fisher方程的Haar小波有限差分法[J].纺织高校基础科学学报, 2010,23(2):170-173.
[28] Shi Zhi, Liu Tao, Gao Bo. Haar wavelet method for solving wave equation[C]//Yi Xie. The 2010 International Conference on Computer Application and System Modeling. Volume 12. Chengdu: Institute of Electrical and Electronics Engineers, Inc., 2010:561-564.
[29] Jüri, Majak et al. Weak formulation based Haar wavelet method for solving differential equations[J].Applied Mathematics and Computation, 2009,211(2): 488-494.
[30]ü. Lepik. Numerical solution of differential equations using Haar wavelets[J].Mathematicsand Computers in Simulation, 2005,68(2): 127-143.
[31]ü. Lepik. Numerical solution of evolution equations by the Haar wavelet method[J].Applied Mathematics and Computation, 2007,185(1): 695-704.
[32]ü. Lepik. Haar wavelet method for solving higher order differential equations[J].International Journal of Mathematics and Computation, 2008,1(N08): 84-94.
[33]ü. Lepik. Solving integral and differential equations by the aid of non-uniform Haar wavelets[J].Applied Mathematics and Computation, 2008,198(1): 326-332.
[34]ü. Lepik. Solving fractional integral equations by the Haar wavelet method[J].Applied Mathematics and Computation, 2009,214(2): 468-478.
[35] C. F. Chen, C. H. Hsiao. Haar wavelet method for solving lumped and distributed-parameter systems[J].IEE Proc. Control Theory and Appl., 1997,144(1): 87-94.
[36] C. H. Hsiao, et al. Optimal control of linear time-varying systems via Haar wavelets[J].Journal of Optimization Theory and Applications, 1999,103(3): 641-655.
[37] C. H. Hsiao, et al. State analysis and optimal control of time-varying discrete systems via Haar wavelets[J].Journal of Optimization Theory and Applications, 1999,103(3): 623-640.
[38] C. H. Hsiao, et al. Numerical solution of time-varying functional differential equations via Haar wavelets[J].Applied Mathematics and Computation, 2007,188(1): 1049-1058.
[39] M. Razzaghi, Y. Ordokhani. Solution of differential equations via rationalized Haar functions[J].Mathematics and Computers in Simulation, 2001,56(3): 235-246.
[40] N. M. Bujurke, et al. An application of single-term Haar wavelet series in the solution of nonlinear oscillator equations[J].Journal of Computational and Applied Mathematics, 2009,227(2): 234-244.
[41]王一博,杨慧珠.方位各向异性介质的多尺度有限差分方法波场模拟[J].清华大学学报(自然科学版), 2006,46(2):293-296.
[42]马坚伟,杨慧珠,朱亚平.多尺度有限差分法模拟复杂介质波传问题[J].物理学报, 2001,50(08):1415-1420.
[43] He Ying. A wavelet finite-difference method for numerical simulation of wave propagation in fluid-saturated porous media[J].Appl. Math. Mech. -Engl. Ed., 2008,29(11):1495-1504.
[44] A. Haar. Zur theorie der orthogonalen funktionensysteme[J].Mathematische Annalen, 1910,69(3): 331-371.
[45]王国秋,全宏跃.最优多进Haar小波及在图像压缩中的应用[J].计算数学,2009,31(1):99-110.
[46]刘明才.小波分析及其应用[M].北京:清华大学出版社,2005.
[47]冉启文,谭立英.小波分析与分数傅里叶变换及应用[M].北京:国防工业出版社,2002.
[48] S. Mallat著,杨力华等译.信号处理的小波导引[M].北京:机械工业出版社,2002.
[49] A. Boggess等著,芮国胜等译.小波与傅里叶分析基础[M].北京:电子工业出版社,2010.
[50] I. Daubechies. Ten lectures on wavelets[M].Philadelphia:SIAM,1992.
[51] M. A. Pinsky.傅里叶分析与小波分析导论(英文版)[M].北京:机械工业出版社,2003.
[52]吕同富,康兆敏,方秀男.数值计算方法[M].北京:清华大学出版社,2008.
[2] L. Wazzan. A modified tanh-coth method for solving the general Burgers-Fisher and the Kuramoto-Sivashinsky equations[J].Communications in Nonlinear Science and Numerical Simulation, 2009,14(6):2642-2652.
[3]段玲玲.广义Burgers-Fisher方程的数值逼近[D].厦门:厦门大学,2008.
[4]李立康,於崇华,朱政华.微分方程数值解法[M].上海:复旦大学出版社,2003.
[5]樊启斌.小波分析[M].武汉:武汉大学出版社,2008.
[6]何正嘉,陈雪峰.小波有限元理论研究与工程应用的进展[J].机械工程学报,2005,41(3):1-11.
[7] J. Satsuma. Exact solutions of Burgers equation with reaction terms[C]//M. Ablowitz, B. Fuchssteiner, M. Kruskal. Topics in soliton theory and exactly solvable nonlinear equations. Singapore: World Scientific, 1987: 255-262.
[8] Cai Guoliang, Wang Yan, Zhang Fengyun. Nonclassical symmetries and group invariant solutions of Burgers-Fisher equations[J].World Journal of Modelling and Simulation, 2007,3(4):305-309.
[9]王明亮,白雪.广义Burgers-Fisher方程的精确解[J].兰州大学学报(自然科学版), 1999,35(2):1-6.
[10]何宝钢.Burgers-Fisher方程的精确解[J].怀化师专学报,2002,21(2):17-19.
[11]姜璐.非线性偏微分方程精确解与孤波相互作用问题的研究[D].北京:北京邮电大学,2007.
[12] R. E. Mickens, A. Gumel. Construction and analysis of non-standard finite difference scheme for the Burgers-Fisher equation[J].Journal of Sound and Vibration, 2002,257(4),791-797.
[13]陈晓永.Burgers-Fisher方程的数值解法[D].南京:南京航空航天大学,2007.
[14]高钦姣.MQ拟插值在Burgers-Fisher方程数值解中的应用[D].大连:大连理工大学,2009.
[15] Sun Xinxiu, Lu Yiping, Liu Haiyang. Exact solutions to wick type stochastic generalized Burgers-Fisher equation[J].Journal of Xuzhou Normal University (Natural Science Edition), 2009,27(2):50-54.
[16] Wang Xinyi, Lu Yuekai. Exact solutions of the extended Burgers-Fisher equation[J].Chinese Physics Letters, 1990,7(4):145-147.
[17] Chen Huaitang, Zhang Hongqing. New multiple soliton solutions to the generalBurgers-Fisher equation and the Kuramoto-Sivashinsky equation[J].Chaos, Solitons and Fractals, 2004,19(1):71-76.
[18] S. A. Wakil, M. A. Abdou. Modified extended tanh-function method for solving nonlinear partial differential equation[J].Chaos, Solitons and Fractals, 2007,31(5):1256-1264.
[19] D. Kaya, S. M. Sayed. A numerical simulation and explicit solutions of the generalized Burgers-Fisher equation[J].Applied Mathematics and Computation, 2004,152(2):403-413.
[20] H. N. A. Ismail, et al. Adomian decomposition method for Burgers-Huxley and Burgers-Fisher equations[J].Applied Mathematics and Computation, 2004,159(1): 291-301.
[21] H. N. A. Ismail, A. A. A. Rabboh. A restrictive Padéapproximation for the solution of the generalized Fisher and Burger-Fisher equations[J].Applied Mathematics and Computation, 2004,154(1): 203-210.
[22] M. Javidi. Spectral collocation method for the solution of the generalized Burger-Fisher equation[J].Applied Mathematics and Computation, 2006,174(1): 345-352.
[23] A. Golbabai, M. Javidi. A spectral domain decomposition approach for the generalized Burgers-Fisher equation[J].Chaos, Solitons and Fractals, 2009,39(1): 385-392.
[24] M. Moghimi, F. S. A. Hejazi. Variational iteration method for solving generalized Burger-Fisher and Burger equations[J].Chaos, Solitons and Fractals, 2007,33(5): 1756-1761.
[25] M. Sari, G. Gürarslan, A. Zeytino?lu. High-order finite difference schemes for the solution of the generalized Burgers-Fisher equation[J].International Journal for Numerical Methods in Biomedical Engineering, 2009,27(5):125-132.
[26] M. Sari, G. Gürarslan, I. Da?. A compact finite difference method for the solution of the generalized Burgers-Fisher equation[J].Numerical Methods for Partial Differential Equations, 2010,26(1):125-134.
[27]高博,曲小钢.广义Burgers-Fisher方程的Haar小波有限差分法[J].纺织高校基础科学学报, 2010,23(2):170-173.
[28] Shi Zhi, Liu Tao, Gao Bo. Haar wavelet method for solving wave equation[C]//Yi Xie. The 2010 International Conference on Computer Application and System Modeling. Volume 12. Chengdu: Institute of Electrical and Electronics Engineers, Inc., 2010:561-564.
[29] Jüri, Majak et al. Weak formulation based Haar wavelet method for solving differential equations[J].Applied Mathematics and Computation, 2009,211(2): 488-494.
[30]ü. Lepik. Numerical solution of differential equations using Haar wavelets[J].Mathematicsand Computers in Simulation, 2005,68(2): 127-143.
[31]ü. Lepik. Numerical solution of evolution equations by the Haar wavelet method[J].Applied Mathematics and Computation, 2007,185(1): 695-704.
[32]ü. Lepik. Haar wavelet method for solving higher order differential equations[J].International Journal of Mathematics and Computation, 2008,1(N08): 84-94.
[33]ü. Lepik. Solving integral and differential equations by the aid of non-uniform Haar wavelets[J].Applied Mathematics and Computation, 2008,198(1): 326-332.
[34]ü. Lepik. Solving fractional integral equations by the Haar wavelet method[J].Applied Mathematics and Computation, 2009,214(2): 468-478.
[35] C. F. Chen, C. H. Hsiao. Haar wavelet method for solving lumped and distributed-parameter systems[J].IEE Proc. Control Theory and Appl., 1997,144(1): 87-94.
[36] C. H. Hsiao, et al. Optimal control of linear time-varying systems via Haar wavelets[J].Journal of Optimization Theory and Applications, 1999,103(3): 641-655.
[37] C. H. Hsiao, et al. State analysis and optimal control of time-varying discrete systems via Haar wavelets[J].Journal of Optimization Theory and Applications, 1999,103(3): 623-640.
[38] C. H. Hsiao, et al. Numerical solution of time-varying functional differential equations via Haar wavelets[J].Applied Mathematics and Computation, 2007,188(1): 1049-1058.
[39] M. Razzaghi, Y. Ordokhani. Solution of differential equations via rationalized Haar functions[J].Mathematics and Computers in Simulation, 2001,56(3): 235-246.
[40] N. M. Bujurke, et al. An application of single-term Haar wavelet series in the solution of nonlinear oscillator equations[J].Journal of Computational and Applied Mathematics, 2009,227(2): 234-244.
[41]王一博,杨慧珠.方位各向异性介质的多尺度有限差分方法波场模拟[J].清华大学学报(自然科学版), 2006,46(2):293-296.
[42]马坚伟,杨慧珠,朱亚平.多尺度有限差分法模拟复杂介质波传问题[J].物理学报, 2001,50(08):1415-1420.
[43] He Ying. A wavelet finite-difference method for numerical simulation of wave propagation in fluid-saturated porous media[J].Appl. Math. Mech. -Engl. Ed., 2008,29(11):1495-1504.
[44] A. Haar. Zur theorie der orthogonalen funktionensysteme[J].Mathematische Annalen, 1910,69(3): 331-371.
[45]王国秋,全宏跃.最优多进Haar小波及在图像压缩中的应用[J].计算数学,2009,31(1):99-110.
[46]刘明才.小波分析及其应用[M].北京:清华大学出版社,2005.
[47]冉启文,谭立英.小波分析与分数傅里叶变换及应用[M].北京:国防工业出版社,2002.
[48] S. Mallat著,杨力华等译.信号处理的小波导引[M].北京:机械工业出版社,2002.
[49] A. Boggess等著,芮国胜等译.小波与傅里叶分析基础[M].北京:电子工业出版社,2010.
[50] I. Daubechies. Ten lectures on wavelets[M].Philadelphia:SIAM,1992.
[51] M. A. Pinsky.傅里叶分析与小波分析导论(英文版)[M].北京:机械工业出版社,2003.
[52]吕同富,康兆敏,方秀男.数值计算方法[M].北京:清华大学出版社,2008.