﻿ 偏微分方程的满足部分边界条件的解的多样性 The Diversity of Solutions Satisfied Partial Boundary Conditions for a Partial Differential Equation

Advances in Applied Mathematics
Vol.07 No.05(2018), Article ID:25079,5 pages
10.12677/AAM.2018.75073

The Diversity of Solutions Satisfied Partial Boundary Conditions for a Partial Differential Equation

Dandan Li, Shan Yin*

College of Sciences, Inner Mongolia University of Technology, Hohhot Inner Mongolia

Received: May 1st, 2018; accepted: May 18th, 2018; published: May 25th, 2018

ABSTRACT

For (initial) boundary value problems of partial differential equations, most of current methods are based on their partial (initial) boundary conditions. Then the solution obtained in this way could satisfy all the boundary conditions? For this reason, we based on Adomian decomposition method to solve the Dirichlet boundary value problem of groundwater recharge effect model on triangular area. We find that: 1) the solution satisfies all boundary conditions sometimes, sometimes not satisfies; 2) the solution satisfied partial boundary conditions is not unique; 3) the solution obtained by the Adomian decomposition method is a particular solution that satisfies partial boundary conditions.

Keywords:Partial Differential Equations, Dirichlet Boundary Value Problem, Adomian Decomposition Method

Copyright © 2018 by authors and Hans Publishers Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY).

http://creativecommons.org/licenses/by/4.0/

1. 引言

Adomian分解法是美国数学物理学家Georgie Adomian [18] [19] 提出的一种分解法。该方法克服了传统摄动方法对小参数的依赖性，并且具有很好的收敛性和易计算性。对偏微分方程的(初)边值问题，传统的Adomian分解法只基于部分边界条件，不是所有边界条件(见文 [20] [21] [22] [23] [24] )。Serrano [24] [25] 等人研究地下水流补给效应模型。地下水流模型控制微分方程为：

$\frac{{\partial }^{2}h}{\partial {x}^{2}}+\frac{{\partial }^{2}h}{\partial {y}^{2}}=-\frac{Rg}{T},\text{}0\le x\le 600,x\le y\le 600-x$ (1.1)

$h\left(x,x\right)={f}_{1}\left(x\right),$ (1.2)

$h\left(x,600-x\right)={f}_{2}\left(x\right),$ (1.3)

$h\left(x,0\right)={f}_{3}\left(x\right).$ (1.4)

${f}_{1}\left(x\right)=100+\frac{2x}{125}-\frac{{x}^{2}}{50000},$ (1.5)

${f}_{2}\left(x\right)=\frac{448}{5}+\frac{103x}{1500}-\frac{{x}^{2}}{12500},$ (1.6)

${f}_{3}\left(x\right)=100+\frac{17x}{3750}-\frac{{x}^{2}}{500000}$ (1.7)

${L}_{x}h\left(x,y\right)+{L}_{y}h\left(x,y\right)=-\frac{Rg}{T}$ (1.8)

2. 基于边界条件(1.2)和(1.3)的Adomian近似解

${L}_{1}^{-1}={\int }_{y}^{x}{\int }_{y}^{x}\cdot \text{d}x\text{d}x-\frac{x-y}{600-2y}{\int }_{y}^{600-y}{\int }_{y}^{x}\cdot \text{d}x\text{d}x$

$\sum _{n=0}^{\infty }{h}_{n}=\left(1-\frac{x-y}{600-2y}\right){f}_{1}\left(y\right)+\frac{x-y}{600-2y}{f}_{2}\left(600-y\right)-{L}_{1}^{-1}\frac{Rg}{T}-{L}_{1}^{-1}{L}_{y}\sum _{n=0}^{\infty }{h}_{n}$ (2.1)

${h}_{0}\left(x,y\right)=\left(1-\frac{x-y}{600-2y}\right){f}_{1}\left(y\right)+\frac{x-y}{600-2y}{f}_{2}\left(600-y\right)-{L}_{x}^{-1}\frac{rg}{t},$

${h}_{n}\left(x,y\right)=-{L}_{x}^{-1}{L}_{y}{h}_{n-1}$ (2.2)

${h}_{0}\left(x,y\right)=100-\frac{{x}^{2}}{20000}+x\left(\frac{1}{30}+\frac{3y}{100000}\right)-\frac{13y}{750},{h}_{n}=0,\left(n\ge 1\right)$

${H}_{1}\left(x,y\right)=100-\frac{{x}^{2}}{20000}+x\left(\frac{1}{30}+\frac{3y}{100000}\right)-\frac{13y}{750}$ (2.3)

3. 基于(1.3)和(1.4)的Adomian近似解

${L}_{2}^{-1}={\int }_{0}^{y}{\int }_{0}^{y}\cdot \text{d}y\text{d}y-\frac{y}{600-x}{\int }_{0}^{600-x}{\int }_{0}^{y}\cdot \text{d}y\text{d}y$

$\sum _{n=0}^{\infty }{h}_{n}=\left(1-\frac{y}{600-x}\right){f}_{3}\left(x\right)+\frac{y}{600-x}{f}_{2}\left(x\right)-{L}_{2}^{-1}\frac{Rg}{T}-{L}_{2}^{-1}{L}_{x}\sum _{n=0}^{\infty }{h}_{n},$ (3.1)

${h}_{0}\left(x,y\right)=\left(1-\frac{y}{600-x}\right){f}_{3}\left(x\right)+\frac{y}{600-x}{f}_{2}\left(x\right)-{L}_{y}^{-1}\frac{rg}{t},$

${h}_{n}\left(x,y\right)=-{L}_{y}^{-1}{L}_{x}{h}_{n-1}$ (3.2)

${h}_{0}\left(x,y\right)=\frac{-3{x}^{2}+x\left(6800+42y\right)+25\left(6000000+760y-3{y}^{2}\right)}{1500000}$ (3.3)

${h}_{1}\left(x,y\right)=\frac{y\left(-600+x+y\right)}{500000}$ (3.4)

${h}_{n}\left(x,y\right)=0,\left(n\ge 2\right)$ (3.5)

${H}_{2}\left(x,y\right)=\frac{-3{x}^{2}+5x\left(1360+9y\right)+8\left(18750000+2150y-9{y}^{2}\right)}{1500000}$ (3.6)

4. 总结

The Diversity of Solutions Satisfied Partial Boundary Conditions for a Partial Differential Equation[J]. 应用数学进展, 2018, 07(05): 617-621. https://doi.org/10.12677/AAM.2018.75073

1. 1. Hirota, R. and Satsuma, J. (1976) N-Soliton Solutions of Model Equations for Shallow Water Waves. Journal of the Physical Society of Japan, 40, 611-612.
https://doi.org/10.1143/JPSJ.40.611

2. 2. Hirota, R. (2004) The Direct Method in Soliton Theory. Cam-bridge University Press, Cambridge.
https://doi.org/10.1017/CBO9780511543043

3. 3. Wang, M.L. (1996) Exact Solutions for a Compound KdV-Burgers Equation. Physics Letters A, 213, 279-287.
https://doi.org/10.1016/0375-9601(96)00103-X

4. 4. Chen, Y., Li, B. and Zhang, H.Q. (2003) Generalized Riccati Equation Expansion Method and Its Application to the Bogoyavlenskiis Generalized Breaking Soliton Equation. Chinese Physics, 12, 940-945.
https://doi.org/10.1088/1009-1963/12/9/303

5. 5. 伊丽娜, 套格图桑. 非线性耦合KdV方程组的一种新求解法[J]. 数学杂志, 2017, 37(4): 823-832.

6. 6. 李宁, 套格图桑. 几种广义非线性发展方程的新解[J]. 数学杂志, 2016, 36(5): 1103-1110.

7. 7. Bluman, G.W. and Kumei, S. (1989) Symmetries and Di Erential Equations. Springer-Verlag, New York.
https://doi.org/10.1007/978-1-4757-4307-4

8. 8. Yun, Y.S. and Chaolu, T. (2015) Classical and Nonclassical Symmetry Clas-sifications of Nonlinear Wave Equation with Dissipation. Applied Mathematics and Mechanics (English Edition), 36, 365-378.
https://doi.org/10.1007/s10483-015-1910-6

9. 9. 白月星, 苏道毕力格. Poisson方程的一维最优系统和不变解[J]. 数学杂志, 2018, 38(4): 706-712.

10. 10. 谷超豪, 胡和生, 周子翔. 孤立子理论中的达布变换及其几何应用[M]. 上海: 科学技术出版社, 2005.

11. 11. Fan, E.G. (2000) Extended Tanh-Function Method and Its Applications to Nonlinear Equations. Physics Letters A, 277, 212-218.
https://doi.org/10.1016/S0375-9601(00)00725-8

12. 12. 王鑫, 邢文雅, 李胜军. 一类推广的KdV方程的新行波解[J]. 数学杂志, 2017, 37(4): 859-864.

13. 13. Nayfeh, A.H. (1973) Perturbation Methods. John Wiley and Sons, New York.

14. 14. Adomian, G. (1988) A Review of the Decomposition Method in Applied Mathematics. Journal of Mathematical Analysis and Applications, 135, 501-544.
https://doi.org/10.1016/0022-247X(88)90170-9

15. 15. He, J.H. (2000) A Coupling Method of a Homotopy Technique and a Perturbation Technique for Non-Linear Problems. International Journal of Non-Linear Mechanics, 35, 37-43.
https://doi.org/10.1016/S0020-7462(98)00085-7

16. 16. Yun, Y. and Temuer, C. (2015) Application of the Homotopy Perturbation Method for the Large Deflection Problem of a Circular Plate. Applied Mathematical Modelling, 39, 1308-1316.
https://doi.org/10.1016/j.apm.2014.09.001

17. 17. Liao, S. (1992) The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems. PhD Thesis, Shanghai Jiao Tong University, Shanghai.

18. 18. Adomian, G. (1998) Solutions of Nonlinear P.D.E. Applied Mathematics Letters, 11, 121-123.
https://doi.org/10.1016/S0893-9659(98)00043-3

19. 19. Adomian, G. (1983) Stochastic Systems. Academic Press, Pitts-burgh.

20. 20. Benneouala, T., Cherruault, Y. and Abbaoui, K. (2005) New Methods for Applying the Adomian Method to Partial Dif-ferential Equations with Boundary Conditions. Kybernetes, 34, 924-933.
https://doi.org/10.1108/03684920510605740

21. 21. Patel, A. and Serrano, S.E. (2011) Decomposition Solution of Multidimen-sional Groundwater Equations. Journal of Hydrology, 397, 202-209.
https://doi.org/10.1016/j.jhydrol.2010.11.032

22. 22. Shidfar, G. (2009) A Weighted Algorithm Based on Adomian Decomposition Method for Solving an Special Class of Evolution Equations. Communications in Nonlinear Science and Numerical Simulation, 14, 1146-1151.
https://doi.org/10.1016/j.cnsns.2008.04.004

23. 23. Yun, Y., Temuer, C. and Duan, J. (2014) A Segmented and Weighted Adomian Decomposition Algorithm for Boundary Value Problem of Nonlinear Groundwater Equation. Mathematical Methods in the Applied Sciences, 37, 2406-2418.
https://doi.org/10.1002/mma.2986

24. 24. Syafrin, T. and Serrano, S.E. (2015) Regional Groundwater Flow in the Louisville Aquifer. Ground Water, 53, 550-557.
https://doi.org/10.1111/gwat.12242

25. 25. Serrano, S. (2013) A Simple Approach to Groundwater Modelling with Decomposition. International Association of Scientific Hydrology Bulletin, 58, 177-187.
https://doi.org/10.1080/02626667.2012.745938

26. NOTES

*通讯作者。