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

1. 引言

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

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

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

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

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

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26. NOTES

*通讯作者。