AAM  >> Vol. 7 No. 5 (May 2018)

    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

  • 全文下载: PDF(431KB) HTML   XML   PP.617-621   DOI: 10.12677/AAM.2018.75073  
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作者:  

李丹丹,银 山:内蒙古工业大学理学院,内蒙古 呼和浩特

关键词:
偏微分方程Dirichlet边值问题Adomian分解法Partial Differential Equations Dirichlet Boundary Value Problem Adomian Decomposition Method

摘要:

对偏微分方程的(初)边值问题,目前的大多数求解方法基于其部分(初)边界条件。那么这样得到的解能否满足所有边界条件呢?为此,我们基于Adomian分解法,求解三角形地下水域上补给效应模型的Dirichlet边值问题。我们发现:1) 求出的解有时满足所有边界条件,有时不满足;2) 满足部分边界条件的解不是唯一;3) Adomian分解法得到的解是满足部分边界条件的某一个特解。

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.

1. 引言

求解偏微分方程精确解的方法有很多,如双线性导数法 [1] [2] ,齐次平衡法 [3] ,辅助方程法 [4] [5] [6] ,Lie对称 [7] [8] [9] ,达布变换 [10] ,tanh函数法 [11] ,G展开法 [12] 等等。但这些方法很难应用于偏微分方程的(初)边值问题。对偏微分方程(初)边值问题,近期人们提出并发展了很多方法,如摄动法 [13] ,Adomian分解法 [14] 、同伦摄动法 [15] [16] 、同伦分析法 [17] 等等。而这些求解偏微分方程的边值问题的方法,求解过程中主要依赖部分边界条件,不是所有边界条件。那么,基于部分边界条件能否得到满足所有边界条件的解呢?即满足部分边界条件的解是否唯一呢?针对该问题,我们基于Adomian分解考虑三角形地下水域上的补给效应模型。

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:

2 h x 2 + 2 h y 2 = R g T , 0 x 600 , x y 600 x (1.1)

其中 T = 100 m 2 /month , R g = 10 mm/month ,假设附加的边界条件为

h ( x , x ) = f 1 ( x ) , (1.2)

h ( x , 600 x ) = f 2 ( x ) , (1.3)

h ( x , 0 ) = f 3 ( x ) . (1.4)

其中

f 1 ( x ) = 100 + 2 x 125 x 2 50000 , (1.5)

f 2 ( x ) = 448 5 + 103 x 1500 x 2 12500 , (1.6)

f 3 ( x ) = 100 + 17 x 3750 x 2 500000 (1.7)

为了方便令其中 L x = 2 x 2 , L y = 2 y 2 则方程(1.1)可改写为

L x h ( x , y ) + L y h ( x , y ) = R g T (1.8)

2. 基于边界条件(1.2)和(1.3)的Adomian近似解

根据Adomian算法 [23] ,把逆算子

L 1 1 = y x y x d x d x x y 600 2 y y 600 y y x d x d x

作用于方程(1.8)两边,并考虑边界条件(1.2),(1.3)可得

n = 0 h n = ( 1 x y 600 2 y ) f 1 ( y ) + x y 600 2 y f 2 ( 600 y ) L 1 1 R g T L 1 1 L y n = 0 h n (2.1)

由上式构造迭代公式

h 0 ( x , y ) = ( 1 x y 600 2 y ) f 1 ( y ) + x y 600 2 y f 2 ( 600 y ) L x 1 r g t ,

h n ( x , y ) = L x 1 L y h n 1 (2.2)

从迭代公式(2.2),我们得到

h 0 ( x , y ) = 100 x 2 20000 + x ( 1 30 + 3 y 100000 ) 13 y 750 , h n = 0 , ( n 1 )

由此得到方程(1.1)的满足边界条件(1.2)和(1.3)的精确解:

H 1 ( x , y ) = 100 x 2 20000 + x ( 1 30 + 3 y 100000 ) 13 y 750 (2.3)

经验证 H 1 ( x , y ) 不满足边界条件(1.4)。

3. 基于(1.3)和(1.4)的Adomian近似解

将逆算子

L 2 1 = 0 y 0 y d y d y y 600 x 0 600 x 0 y d y d y

作用于方程(1.8)两侧,考虑边界条件(1.3)和(1.4),可得

n = 0 h n = ( 1 y 600 x ) f 3 ( x ) + y 600 x f 2 ( x ) L 2 1 R g T L 2 1 L x n = 0 h n , (3.1)

从上式,我们构造迭代公式:

h 0 ( x , y ) = ( 1 y 600 x ) f 3 ( x ) + y 600 x f 2 ( x ) L y 1 r g t ,

h n ( x , y ) = L y 1 L x h n 1 (3.2)

从(3.2),我们得到

h 0 ( x , y ) = 3 x 2 + x ( 6800 + 42 y ) + 25 ( 6000000 + 760 y 3 y 2 ) 1500000 (3.3)

h 1 ( x , y ) = y ( 600 + x + y ) 500000 (3.4)

h n ( x , y ) = 0 , ( n 2 ) (3.5)

由此得到方程满足边界条件(1.3)和(1.4)的精确解:

H 2 ( x , y ) = 3 x 2 + 5 x ( 1360 + 9 y ) + 8 ( 18750000 + 2150 y 9 y 2 ) 1500000 (3.6)

通过验证发现 H 2 ( x , y ) 满足所有边界条件。

4. 总结

从上述求解结果可以发现: 基于(1.2),(1.3)用Adomian算法求出的解(2.3)只满足边界条件(1.2),(1.3),不满足(1.4);而基于(1.3),(1.4)的Adomian解满足所有边界条件(1.2)~(1.3)。这说明,利用部分边界条件求出的解有时满足所有的边界条件,有时只满足部分边界条件,不是所有边界条件。Adomian解(2.3)和(3.6)都满足边界条件(1.2)与(1.3),从而得知满足偏微分方程部分边界条件的解不是唯一,是多样的。并且说明了有Adomian分解法得到的解是满足部分边界条件的某一个特解。

致谢

感谢银山老师的支持与帮助。

基金项目

内蒙古自治区项目(2016MS0109),内蒙古自治区高等学校科研研究项目(NJZY094),内蒙古工业大学重点项目(ZD201515)。

NOTES

*通讯作者。

文章引用:
李丹丹, 银山. 偏微分方程的满足部分边界条件的解的多样性[J]. 应用数学进展, 2018, 7(5): 617-621. https://doi.org/10.12677/AAM.2018.75073

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