﻿ 格林公式在微分方程中的应用

# 格林公式在微分方程中的应用Application of Green’s Formula in Differential Equations

Abstract: This paper investigates the Green’s function solution of common non-homogeneous ordinary dif-ferential equations and partial differential equations. We first obtain Green’s functions according to the physical meanings of special partial differential equations. Then we yield their solutions. Considering non-homogeneous ordinary differential equations with insignificant physical meanings, this paper begins with the simplest first-order linear equations. Furthermore, the Green’s function of higher-order linear equations is derived.

1. 引言

1828年，乔治·格林(1793~1841)发表了《论数学分析在电磁理论上的应用》 [1] ，在这篇著作中，格林试图确定由特定电势的导体在真空中的电势。用已知的符号，我们可以说格林想要得到在区域V中且满足特定边界条件S， ${\nabla }^{2}u=-f$ 的解。

${\nabla }^{2}G\left(r,{r}_{0}\right)=-4\text{π}\delta \left(r-{r}_{0}\right)$ (1)

${\iiint }_{V}\left(\phi {\nabla }^{2}\chi -\chi {\nabla }^{2}\phi \right)\text{d}V={∯}_{S}\left(\phi \nabla \chi -\chi \nabla \phi \right)\cdot n\text{d}S$ (2)

${\iiint }_{V}G{\nabla }^{2}u\text{d}V+{∯}_{S}G\nabla u\cdot n\text{d}S={\iiint }_{V}u{\nabla }^{2}G\text{d}V+{∯}_{S}u\nabla G\cdot n\text{d}S-4\text{π}u\left(r0\right)$

$u\left(r\right)=\frac{1}{\text{4π}}{∯}_{S}\stackrel{¯}{u}\nabla G\cdot n\text{d}S$

2. 格林公式

Gaussy公式：设空间闭域是由光滑或分段光滑的双侧封闭曲面 $\Sigma$ 所围成，函数 $P\left(x,y,z\right)$$Q\left(x,y,z\right)$$R\left(x,y,z\right)$ 上具有一阶连续偏导数，则成立如下Gauss公式：

${\iiint }_{\Omega }\left(\frac{\partial P}{\partial x}\text{+}\frac{\partial Q}{\partial y}\text{+}\frac{\partial R}{\partial z}\right)\text{d}\Omega ={∯}_{\Sigma }\left[P\mathrm{cos}\left(n,x\right)+Q\mathrm{cos}\left(n,y\right)+R\mathrm{cos}\left(n,z\right)\right]\text{d}S$ (3)

2.1. 第一格林公式

$P=u\frac{\partial v}{\partial x},Q=u\frac{\partial v}{\partial y},R=u\frac{\partial v}{\partial z}$

$\begin{array}{l}{\iiint }_{\Omega }\left(\frac{\partial P}{\partial x}\text{+}\frac{\partial Q}{\partial y}\text{+}\frac{\partial R}{\partial z}\right)\text{d}\Omega \text{\hspace{0.17em}}={\iiint }_{\Omega }\left(\frac{\partial }{\partial x}\left(u\frac{\partial v}{\partial x}\right)\text{+}\frac{\partial }{\partial y}\left(u\frac{\partial v}{\partial y}\right)\text{+}\frac{\partial }{\partial z}\left(u\frac{\partial v}{\partial z}\right)\right)\text{d}x\text{d}y\text{d}z\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}={∯}_{\Sigma }\left[u\frac{\partial v}{\partial x}\mathrm{cos}\left(n,x\right)+u\frac{\partial v}{\partial y}\mathrm{cos}\left(n,y\right)+u\frac{\partial v}{\partial z}\mathrm{cos}\left(n,z\right)\right]\text{d}S\end{array}$

$\frac{\partial v}{\partial n}=\frac{\partial v}{\partial x}\mathrm{cos}\left(n,x\right)+\frac{\partial v}{\partial y}\mathrm{cos}\left(n,y\right)+\frac{\partial v}{\partial z}\mathrm{cos}\left(n,z\right)$

${\iiint }_{\Omega }u\Delta v\text{d}x\text{d}y\text{d}z={∯}_{\Sigma }u\frac{\partial v}{\partial n}\text{d}S-{\iiint }_{\Omega }\nabla u\cdot \nabla v\text{d}x\text{d}y\text{d}z$ (4)

2.2. 第二格林公式

${\iiint }_{\Omega }v\Delta u\text{d}x\text{d}y\text{d}z={∯}_{\Sigma }v\frac{\partial u}{\partial n}\text{d}S-{\iiint }_{\Omega }\nabla v\cdot \nabla u\text{d}x\text{d}y\text{d}z$ (5)

 (6)

3. 格林函数

3.1. 基本积分公式

$u\left({M}_{0}\right)=\frac{1}{\text{4π}}{\iint }_{\Sigma }\left[\frac{1}{{r}_{M{M}_{0}}}\frac{\partial u}{\partial n}-u\frac{\partial }{\partial n}\left(\frac{1}{{r}_{M{M}_{0}}}\right)\right]\text{d}S-\frac{1}{\text{4π}}{\iiint }_{\Omega }\frac{\Delta u}{{r}_{M{M}_{0}}}\text{d}x\text{d}y\text{d}z$

$u\left({M}_{0}\right)=\frac{1}{4\text{π}}{\iint }_{\partial \Omega }\left[\frac{1}{{r}_{M{M}_{0}}}\frac{\partial u}{\partial n}-u\frac{\partial }{\partial n}\left(\frac{1}{{r}_{M{M}_{0}}}\right)\right]\text{d}S.$ (7)

3.2. 格林函数的定义

$\left\{\begin{array}{l}\Delta u=0,\left(x,y,z\right)\in \Omega \\ u|{}_{\partial \Omega }=f\left(x,y,z\right),\left(x,y,z\right)\in \partial \Omega \end{array}$ (8)

${\iint }_{\partial \Omega }\left(u\frac{\partial v}{\partial n}-v\frac{\partial u}{\partial n}\right)\text{d}S=0$ (9)

(7) + (9)，得

$u\left({M}_{0}\right)={\iint }_{\partial \Omega }\left[\left(\frac{1}{4\text{π}{r}_{M{M}_{0}}}-v\right)\frac{\partial u}{\partial n}-u\frac{\partial }{\partial n}\left(\frac{1}{4\text{π}{r}_{M{M}_{0}}}-v\right)\right]\text{d}S$

$v|{}_{\partial \Omega }=\frac{1}{4\text{π}{r}_{M{M}_{0}}},M\in \partial \Omega$

$u\left({M}_{0}\right)=-{\iint }_{\partial \Omega }u\frac{\partial }{\partial n}\left(\frac{1}{4\text{π}{r}_{M{M}_{0}}}-v\right)\text{d}S$

$G\left(M,{M}_{0}\right)=\frac{1}{4\text{π}{r}_{M{M}_{0}}}-v$

$u\left({M}_{0}\right)=-{\iint }_{\partial \Omega }f\frac{\partial G}{\partial n}\text{d}S$ (10)

$G\left(M,{M}_{0}\right)$ 为Laplace方程Dirichlet问题在区域 $\Omega$ 上的格林函数。

4. 格林函数的性质

5. 格林函数在偏微分方程中的应用

$G\left(M,{M}_{0}\right)=\frac{1}{4\text{π}{r}_{M{M}_{0}}}-v\left(M,{M}_{0}\right)$

5.1. 半空间的Green函数及Dirichlet边值问题

$z>0$ 半空间的Green函数，并求解下列Dirichlet边值问题

$\left\{\begin{array}{l}\Delta u\left(x,y,z\right)=0,-\infty 0\\ u\left(x,y,0\right)=f\left(x,y\right),-\infty (11)

$G\left(M,{M}_{0}\right)=\frac{1}{4\text{π}}\left(\frac{1}{{r}_{M{M}_{0}}}-\frac{1}{{r}_{M{M}_{1}}}\right)$

$\begin{array}{c}{\frac{\partial G}{\partial n}|}_{z=0}={-\frac{\partial G}{\partial z}|}_{z=0}\\ ={\frac{1}{4\text{π}}\left[\frac{z-{z}_{0}}{{\left[{\left(x-{x}_{0}\right)}^{2}+{\left(y-{y}_{0}\right)}^{2}+{\left(z-{z}_{0}\right)}^{2}\right]}^{\frac{3}{2}}}-\frac{z+{z}_{0}}{{\left[{\left(x-{x}_{0}\right)}^{2}+{\left(y-{y}_{0}\right)}^{2}+{\left(z+{z}_{0}\right)}^{2}\right]}^{\frac{3}{2}}}\right]|}_{z=0}\\ =\frac{-1}{2\text{π}}\frac{{z}_{0}}{{\left[{\left(x-{x}_{0}\right)}^{2}+{\left(y-{y}_{0}\right)}^{2}+{z}_{0}^{2}\right]}^{\frac{3}{2}}}\end{array}$ (12)

Figure 1. The mirror image method

$u\left({M}_{0}\right)=\frac{{z}_{0}}{2\text{π}}{\int }_{-\infty }^{+\infty }{\int }_{-\infty }^{+\infty }\frac{f\left(x,y\right)}{{\left[{\left(x-{x}_{0}\right)}^{2}+{\left(y-{y}_{0}\right)}^{2}+{z}_{0}^{2}\right]}^{\frac{3}{2}}}\text{d}x\text{d}y$

5.2. 球体区域的Green函数及Dirichlet边值问题

$\left\{\begin{array}{l}\Delta u\left(x,y,z\right)=0,\left(x,y,z\right)\in \Omega \\ {u|}_{\Gamma }=f\left(x,y,z\right),\left(x,y,z\right)\in \Gamma \end{array}$ (13)

${r}_{0}{r}_{1}={R}^{2}$ (14)

$\frac{{r}_{M{}_{1}P}}{{r}_{{M}_{0}P}}=\frac{R}{{r}_{0}}$

$v\left(M,{M}_{0}\right)=\frac{R}{4\text{π}{r}_{0}{r}_{M{M}_{1}}}$

$G\left(M,{M}_{0}\right)=\frac{1}{4\text{π}{r}_{M{M}_{0}}}-\frac{1}{4\text{π}{r}_{0}{r}_{M{M}_{1}}}$ (15)

Figure 2. Sphere area

${r}_{{M}_{0}M}^{2}={r}_{0}^{2}+{r}^{2}-2{r}_{0}r\mathrm{cos}a,\text{\hspace{0.17em}}{r}_{M{M}_{1}}^{2}={r}_{1}^{2}+{r}^{2}-2{r}_{1}r\mathrm{cos}a$

$G\left(M,{M}_{0}\right)=\frac{1}{\text{4π}}\left(\frac{1}{\sqrt{{r}_{0}^{2}+{r}^{2}-2{r}_{0}r\mathrm{cos}a}}-\frac{R}{\sqrt{{R}^{4}+{r}_{0}^{2}{r}^{2}-2{R}^{2}{r}_{0}r\mathrm{cos}a}}\right)$

$\begin{array}{c}{\frac{\partial G}{\partial n}|}_{\Gamma }={\frac{\partial G}{\partial n}|}_{r=R}={-\frac{1}{\text{4π}}\left(\frac{r-{r}_{0}\mathrm{cos}a}{{\left({r}_{0}^{2}+{r}^{2}-2{r}_{0}r\mathrm{cos}a\right)}^{3/2}}-\frac{R\left({r}_{0}^{2}r-{R}^{2}{r}_{0}\mathrm{cos}a\right)}{{\left({R}^{4}+{r}_{0}^{2}{r}^{2}-2{R}^{2}{r}_{0}r\mathrm{cos}a\right)}^{3/2}}\right)|}_{r=R}\\ =-\frac{1}{4\text{π}R}\frac{{R}^{2}-{r}_{0}^{2}}{{\left({R}^{2}+{r}_{0}^{2}-2Rr\mathrm{cos}a\right)}^{3/2}}\end{array}$

$u\left({M}_{0}\right)=\frac{1}{4\text{π}R}{\iint }_{\Gamma }f\left(x,y,z\right)\frac{{R}^{2}-{r}_{0}^{2}}{{\left({R}^{2}+{r}_{0}^{2}-2R{r}_{0}\mathrm{cos}a\right)}^{3/2}}\text{d}S$

6. 格林函数在常微分方程中的应用

6.1. 预备知识

$\delta$ 函数是由物理学家狄拉克首先引进的。在数学上，可以把 $\delta$ 函数看成广义函数。传统意义上， $\delta$ 函数的定义 [17] 是

$\delta \left(x\right)=\left\{\begin{array}{l}0,x\ne 0\\ \infty ,x=0\end{array}$

${\int }_{-\infty }^{+\infty }\delta \left(x\right)\text{d}x=1$ .

$\delta$ 函数一般具有下列性质 [18] ：

1) ${\int }_{{\text{x}}_{1}}^{{x}_{2}}f\left(x\right)\delta \left(x\right)\text{d}x=\left\{\begin{array}{l}f\left(0\right),{x}_{1}<0<{x}_{2}\\ 0,0\notin \left({x}_{1},{x}_{2}\right)\end{array}$

2) ${\int }_{-\infty }^{+\infty }f\left(x\right)\delta \left(x-a\right)=f\left(a\right)$

3) $\delta \left(-x\right)=\delta \left(x\right)$

$\frac{{\text{d}}^{2}y}{\text{d}{x}^{2}}=f\left(x\right),a

$y\left(x\right)=-{\int }_{a}^{x}\frac{\left(b-x\right)\left(\xi -a\right)}{b-a}f\left(\xi \right)\text{d}\xi -{\int }_{x}^{b}\frac{\left(b-\xi \right)\left(x-a\right)}{b-a}f\left(\xi \right)\text{d}\xi$

$f\left(x\right)=\int f\left(\xi \right)\delta \left(x-\xi \right)\text{d}\xi$

6.2. 一阶非齐次常微分方程

$L\left[y\right]\equiv {y}^{\prime }+p\left(x\right)y=f\left(x\right),x>a$ (16)

$B\left[y\right]\equiv y\left(a\right)=0$

$\left\{\begin{array}{l}L\left[G\left(x,\xi \right)\right]=\delta \left(x-\xi \right)\\ G\left(a,\xi \right)=0\end{array}$

$\begin{array}{c}L\left[{\int }_{a}^{\infty }G\left(x,\xi \right)f\left(\xi \right)\text{d}\xi \right]={\int }_{a}^{\infty }L\left[G\left(x,\xi \right)\right]f\left(\xi \right)\text{d}\xi \\ ={\int }_{a}^{\infty }\delta \left(x-\xi \right)f\left(\xi \right)\text{d}\xi \\ =f\left(x\right)\end{array}$

$\begin{array}{c}B\left[{\int }_{a}^{\infty }G\left(x,\xi \right)f\left(\xi \right)\text{d}\xi \right]={\int }_{a}^{\infty }B\left[G\left(x,\xi \right)\right]f\left(\xi \right)\text{d}\xi \\ ={\int }_{a}^{\infty }0\cdot f\left(\xi \right)\text{d}\xi \\ =0\end{array}$

$\begin{array}{l}{G}^{\prime }+p\left(x\right)G=\delta \left(x-\xi \right),\\ G\left({\xi }^{+},\xi \right)-G\left({\xi }^{-},\xi \right)+{\int }_{{\xi }^{-}}^{{\xi }^{+}}p\left(x\right)G\left(x,\xi \right)\text{d}x=1,\end{array}$

$G\left({\xi }^{+},\xi \right)-G\left({\xi }^{-},\xi \right)=1$ (17)

$G\left(x,\xi \right)=\left\{\begin{array}{l}{c}_{1}{\text{e}}^{-\int p\left(x\right)\text{d}x},a\xi \end{array}$

$G\left(x,\xi \right)=\left\{\begin{array}{l}0,a

$G\left(x,\xi \right)=\left\{\begin{array}{l}0,a\xi \end{array}$

$G\left(x,\xi \right)={\text{e}}^{-{\int }_{\xi }^{x}p\left(t\right)\text{d}t}H\left(x-\xi \right)$

6.3. 二阶非齐次常微分方程

$\left\{\begin{array}{l}L\left[y\right]={y}^{″}+p\left(x\right){y}^{\prime }+q\left(x\right)y=f\left(x\right),x\in \left(a,b\right)\\ y\left(a\right)=y\left(b\right)=0\end{array}$

$\left\{\begin{array}{l}L\left[G\left(x,\xi \right)\right]=\delta \left(x-\xi \right)\\ G\left(a,\xi \right)=G\left(b,\xi \right)=0\end{array}$ (18)

$\begin{array}{c}L\left[{\int }_{a}^{b}G\left(x,\xi \right)f\left(\xi \right)\text{d}\xi \right]={\int }_{a}^{b}L\left[G\left(x,\xi \right)\right]f\left(\xi \right)\text{d}\xi \\ ={\int }_{a}^{b}\delta \left(x-\xi \right)f\left(\xi \right)\text{d}\xi \\ =f\left(x\right)\end{array}$

$\begin{array}{c}{B}_{i}\left[{\int }_{a}^{\infty }G\left(x,\xi \right)f\left(\xi \right)\text{d}\xi \right]={\int }_{a}^{\infty }{B}_{i}\left[G\left(x,\xi \right)\right]f\left(\xi \right)\text{d}\xi \\ ={\int }_{a}^{b}0\cdot f\left(\xi \right)\text{d}\xi \\ =0\end{array}$

$L\left[G\right]=\delta \left(x-\xi \right),{B}_{1}\left[G\right]={B}_{2}\left[G\right]=0$

$L\left[f\right]=f\left(x\right),{B}_{1}\left[y\right]={B}_{2}\left[y\right]=0$

$\delta \left(x\right)$ 进行积分得到赫维赛德函数

$H\left(x\right)={\int }_{-\infty }^{x}\delta \left(t\right)\text{d}t=\left\{\begin{array}{l}0,x<0\\ 1,x>0\end{array}$

$r\left(x\right)={\int }_{-\infty }^{x}H\left(t\right)\text{d}t=\left\{\begin{array}{l}0,x<0\\ x,x>0\end{array}$

$x\ne 0$ 时， $\delta \left(x\right)$ 的导数为0；在 $x=0$ 处，它的导数从0上升到 $+\infty$ ，下降到 $-\infty$ 然后回到0。

${G}^{″}\left(x,\xi \right)+p\left(x\right){G}^{\prime }\left(x,\xi \right)+q\left(x\right)G\left(x,\xi \right)=\delta \left(x-\xi \right)$

${y}_{1},{y}_{2}$ 是齐次微分方程 $L\left[y\right]=0$ 的两个线性无关解。当 $x\ne \xi$ 时，格林函数满足齐次微分方程，因此它是齐次解的线性组合。

$G\left(x,\xi \right)=\left\{\begin{array}{l}{c}_{1}{y}_{1}+{c}_{2}{y}_{2},x<\xi \\ {d}_{1}{y}_{1}+{d}_{2}{y}_{2},x>\xi \end{array}$

$L\left[G\left(x,\xi \right)\right]=\delta \left(x-\xi \right)$$\left({\xi }^{-},{\xi }^{+}\right)$ 上积分，得

${\int }_{{\xi }^{-}}^{{\xi }^{+}}\left[{G}^{″}\left(x,\xi \right)+p\left(x\right){G}^{\prime }\left(x,\xi \right)+q\left(x\right)G\left(x,\xi \right)\right]\text{d}x={\int }_{{\xi }^{-}}^{{\xi }^{+}}\delta \left(x-\xi \right)\text{d}x$

$\begin{array}{l}{\int }_{{\xi }^{-}}^{{\xi }^{+}}p\left(x\right){G}^{\prime }\left(x,\xi \right)\text{d}x=0,{\int }_{{\xi }^{-}}^{{\xi }^{+}}q\left(x\right)G\left(x,\xi \right)\text{d}x=0,\\ {\int }_{{\xi }^{-}}^{{\xi }^{+}}{G}^{″}\left(x,\xi \right)\text{d}x={\int }_{{\xi }^{-}}^{{\xi }^{+}}\delta \left(x-\xi \right)\text{d}x,\\ ⇒{{G}^{\prime }\left(x,\xi \right)|}_{{\xi }^{-}}^{{\xi }^{+}}={\left[H\left(x-\xi \right)\right]|}_{{\xi }^{-}}^{{\xi }^{+}},\\ ⇒{G}^{\prime }\left({\xi }^{+},\xi \right)-{G}^{\prime }\left({\xi }^{-},\xi \right)=1.\end{array}$

${d}_{1}{{y}^{\prime }}_{1}\left(\xi \right)+{d}_{2}{{y}^{\prime }}_{2}\left(\xi \right)-{c}_{1}{{y}^{\prime }}_{1}\left(\xi \right)-{c}_{2}{{y}^{\prime }}_{2}\left(\xi \right)=1$

6.4. 斯特姆-刘维尔问题

$L\left[y\right]=\frac{\text{d}}{\text{d}x}\left[p\left(x\right)\frac{\text{d}y}{\text{d}x}\right]+q\left(x\right)y=f\left(x\right)$ (19)

$L\left[G\left(x,\xi \right)\right]=\delta \left(x-\xi \right),{B}_{1}\left[G\left(x,\xi \right)\right]={B}_{2}\left[G\left(x,\xi \right)\right]=0$

${y}_{1},{y}_{2}$ 是方程(19)对应的齐次方程的两个非零解(分别满足左、右边值条件)，即

$\left\{\begin{array}{l}L\left[{y}_{1}\right]=0\\ {B}_{1}\left[{y}_{1}\right]=0\end{array},\left\{\begin{array}{l}L\left[{y}_{2}\right]=0\\ {B}_{2}\left[{y}_{2}\right]=0\end{array}$

$x\ne \xi$ 时，格林函数满足的微分方程退化为齐次方程，因此可以改写如下形式

$G\left(x,\xi \right)=\left\{\begin{array}{l}{c}_{1}\left(\xi \right){y}_{1}\left(x\right),a\le x\le \xi \\ {c}_{2}\left(\xi \right){y}_{2}\left(x\right),\xi \le x\le b\end{array}$

$\begin{array}{l}G\left({\xi }^{-},\xi \right)=G\left({\xi }^{+},\xi \right)\\ {c}_{1}\left(\xi \right){y}_{1}\left(\xi \right)={c}_{2}\left(\xi \right){y}_{2}\left(\xi \right)\end{array}$

${G}^{″}\left(x,\xi \right)+\frac{{p}^{\prime }}{p}{G}^{\prime }\left(x,\xi \right)+\frac{q}{p}G\left(x,\xi \right)=\frac{\delta \left(x-\xi \right)}{p}$

$\begin{array}{l}{G}^{\prime }\left({\xi }^{+},\xi \right)-{G}^{\prime }\left({\xi }^{-},\xi \right)=\frac{1}{p\left(\xi \right)},\\ {c}_{2}\left(\xi \right){{y}^{\prime }}_{2}\left(\xi \right)-{c}_{1}\left(\xi \right){{y}^{\prime }}_{1}\left(\xi \right)=\frac{1}{p\left(\xi \right)},\end{array}$

$\left\{\begin{array}{l}{c}_{1}\left(\xi \right){y}_{1}\left(\xi \right)-{c}_{2}\left(\xi \right){y}_{2}\left(\xi \right)=0\\ {c}_{1}\left(\xi \right){{y}^{\prime }}_{1}\left(\xi \right)-{c}_{2}\left(\xi \right){{y}^{\prime }}_{2}\left(\xi \right)=-\frac{1}{p\left(\xi \right)}\end{array}$

${c}_{1}\left(\xi \right)=-\frac{{y}_{2}\left(\xi \right)}{p\left(\xi \right)\left(-W\left(\xi \right)\right)},{c}_{2}\left(\xi \right)=-\frac{{y}_{1}\left(\xi \right)}{p\left(\xi \right)\left(-W\left(\xi \right)\right)}$

$G\left(x,\xi \right)=\left\{\begin{array}{l}\frac{{y}_{1}\left(x\right){y}_{2}\left(\xi \right)}{p\left(\xi \right)W\left(\xi \right)},a\le x\le \xi \\ \frac{{y}_{2}\left(x\right){y}_{1}\left(\xi \right)}{p\left(\xi \right)W\left(\xi \right)},\xi \le x\le b\end{array}$

6.5. 初值问题

$\left\{\begin{array}{l}L\left[y\right]={y}^{″}+p\left(x\right){y}^{\prime }+q\left(x\right)y=f\left(x\right),x\in \left(a,b\right)\\ y\left(a\right)={y}_{1},{y}^{\prime }\left(a\right)={y}_{2}\end{array}$

${u}^{″}+p\left(x\right){u}^{\prime }+q\left(x\right)u=f\left(x\right),u\left(a\right)=0,{u}^{\prime }\left(a\right)=0$

${u}^{″}+p\left(x\right){u}^{\prime }+q\left(x\right)u=0,u\left(a\right)={\gamma }_{1},{u}^{\prime }\left(a\right)={\gamma }_{2}$

$W\left(x\right)=c\mathrm{exp}\left(-\int p\left(x\right)\text{d}x\right)$

$\left\{\begin{array}{l}{G}^{″}\left(x,\xi \right)+p\left(x\right){G}^{\prime }\left(x,\xi \right)+q\left(x\right)G\left(x,\xi \right)=\delta \left(x-\zeta \right)\\ G\left(a,\xi \right)=0,{G}^{\prime }\left(a,\xi \right)=0\end{array}$

$G\left({\xi }^{-},\xi \right)=G\left({\xi }^{+},\xi \right),{G}^{\prime }\left({\xi }^{-},\xi \right)+1={G}^{\prime }\left({\xi }^{+},\xi \right)$

${u}_{1},{u}_{2}$ 为微分方程的两个线性无关解，当 $x<\xi$ 时， $G\left(x,\xi \right)$ 是这些解的线性组合，因为伏朗斯基行列式非零，故只有平凡解满足齐次初值条件，格林函数为

$G\left(x,\xi \right)=\left\{\begin{array}{l}0,x<\xi \\ {u}_{\xi }\left(x\right),x>\xi \end{array}$

${u}_{\xi }\left(\xi \right)=0,{{u}^{\prime }}_{\xi }\left(\xi \right)=1$

$G\left(x,\xi \right)=H\left(x-\xi \right){u}_{\xi }\left(x\right)$

$\begin{array}{c}u={\int }_{a}^{b}G\left(x,\text{}\xi \right)f\left(\xi \right)\text{d}\xi \\ ={\int }_{a}^{b}H\left(x-\xi \right){u}_{\xi }\left(x\right)f\left(\xi \right)\text{d}\xi \\ ={\int }_{a}^{x}{u}_{\xi }\left(x\right)f\left(\xi \right)\text{d}\xi \end{array}$

$y=v+{\int }_{a}^{x}{u}_{\xi }\left(x\right)f\left(\xi \right)\text{d}\xi$

6.6. 混合边值问题

$\left\{\begin{array}{l}{B}_{1}\left[y\right]={\alpha }_{11}y\left(a\right)+{\alpha }_{12}{y}^{\prime }\left(a\right)+{\beta }_{11}y\left(b\right)+{\beta }_{12}{y}^{\prime }\left(b\right)={\gamma }_{1}\\ {B}_{2}\left[y\right]={\alpha }_{21}y\left(a\right)+{\alpha }_{22}{y}^{\prime }\left(a\right)+{\beta }_{21}y\left(b\right)+{\beta }_{22}{y}^{\prime }\left(b\right)={\gamma }_{2}\end{array}$

$\begin{array}{l}{u}^{″}+p\left(x\right){u}^{\prime }+q\left(x\right)u=f\left(x\right),{B}_{1}\left[u\right]={B}_{2}\left[u\right]=0,\\ {v}^{″}+p\left(x\right){v}^{\prime }+q\left(x\right)v=0,{B}_{1}\left[v\right]={y}_{1},{B}_{2}\left[v\right]={y}_{2},\end{array}$

${y}_{1}$${y}_{2}$ 是满足齐次边界条件 ${B}_{1}\left[{y}_{1}\right]=0$${B}_{2}\left[{y}_{2}\right]=0$ 的齐次常微分方程的解，又因为完全齐次问题无解，则 ${B}_{1}\left[{y}_{2}\right]$${B}_{2}\left[{y}_{1}\right]$ 非零。解v有如下形式

$v={c}_{1}{y}_{1}+{c}_{2}{y}_{2}$

$v=\frac{{\gamma }_{2}}{{B}_{2}\left[{y}_{1}\right]}{y}_{1}+\frac{{\gamma }_{1}}{{B}_{1}\left[{y}_{2}\right]}{y}_{2}$

${G}^{″}\left(x,\xi \right)+p\left(x\right){G}^{\prime }\left(x,\xi \right)+q\left(x\right)G\left(x,\xi \right)=\delta \left(x-\xi \right),{B}_{1}\left[G\right]={B}_{2}\left[G\right]=0$

$G\left({\xi }^{-},\xi \right)=G\left({\xi }^{+},\xi \right),{G}^{\prime }\left({\xi }^{-},\xi \right)+1={G}^{\prime }\left({\xi }^{+},\xi \right)$

$G\left(x,\xi \right)=H\left(x-\xi \right){y}_{\xi }\left(x\right)+{c}_{1}{y}_{1}\left(x\right)+{c}_{2}{y}_{2}\left(x\right)$

$\begin{array}{l}{B}_{1}\left[G\right]={B}_{1}\left[H\left(x-\xi \right){y}_{\xi }\right]+{c}_{2}{B}_{1}\left[{y}_{2}\right]=0,\\ {B}_{2}\left[G\right]={B}_{2}\left[H\left(x-\xi \right){y}_{\xi }\right]+{c}_{1}{B}_{2}\left[{y}_{1}\right]=0,\end{array}$

$G\left(x,\xi \right)=H\left(x-\xi \right){y}_{\xi }\left(x\right)-\frac{{\beta }_{21}{y}_{\xi }\left(b\right)+{\beta }_{22}{{y}^{\prime }}_{\xi }\left(b\right)}{{B}_{2}\left[{y}_{1}\right]}{y}_{1}\left(x\right)-\frac{{\beta }_{11}{y}_{\xi }\left(b\right)+{\beta }_{12}{{y}^{\prime }}_{\xi }\left(b\right)}{{B}_{1}\left[{y}_{2}\right]}{y}_{2}\left(x\right)$

$u={\int }_{a}^{b}G\left(x,\xi \right)f\left(\xi \right)\text{d}\xi$

$y={\int }_{a}^{b}G\left(x,\xi \right)f\left(\xi \right)\text{d}\xi +\frac{{\gamma }_{2}}{{B}_{2}\left[{y}_{1}\right]}{y}_{1}+\frac{{\gamma }_{1}}{{B}_{1}\left[{y}_{2}\right]}{y}_{2}$

6.7. 高阶非齐次常微分方程

$L\left[y\right]={y}^{\left(n\right)}+{p}_{n-1}\left(x\right){y}^{\left(n-1\right)}+\cdots +{p}_{1}\left(x\right){y}^{\prime }+{p}_{0}y=f\left(x\right)$

${B}_{j}\left[y\right]={y}_{j}$

$B\left[y\right]=\underset{k=0}{\overset{n-1}{\sum }}{\alpha }_{k}{y}^{\left(k\right)}\left(a\right)+\underset{k=0}{\overset{n-1}{\sum }}{\beta }_{k}{y}^{\left(k\right)}\left(b\right)$

$\begin{array}{l}L\left[u\right]=f\left(x\right),{B}_{j}\left[u\right]=0,\\ L\left[v\right]=0,{B}_{j}\left[v\right]={y}_{j}.\end{array}$

$\left(\begin{array}{cccc}{B}_{1}\left[{y}_{1}\right]& {B}_{1}\left[{y}_{2}\right]& \cdots & {B}_{1}\left[{y}_{n}\right]\\ {B}_{2}\left[{y}_{1}\right]& {B}_{2}\left[{y}_{2}\right]& \cdots & {B}_{2}\left[{y}_{n}\right]\\ ⋮& ⋮& \ddots & ⋮\\ {B}_{n}\left[{y}_{1}\right]& {B}_{n}\left[{y}_{2}\right]& \cdots & {B}_{n}\left[{y}_{n}\right]\end{array}\right)\left(\begin{array}{c}{c}_{1}\\ {c}_{2}\\ ⋮\\ {c}_{n}\end{array}\right)=\left(\begin{array}{c}{y}_{1}\\ {y}_{2}\\ ⋮\\ {y}_{n}\end{array}\right)$

$\left\{\begin{array}{l}L\left[G\left(x,\xi \right)\right]=\delta \left(x-\xi \right)\\ {B}_{j}\left[G\right]=0\end{array}$

${y}_{\xi }\left(x\right)$ 是满足下列条件的齐次解的组合：

$\begin{array}{l}{y}_{\xi }\left(\xi \right)=0,\\ {{y}^{\prime }}_{\xi }\left(\xi \right)=0,\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}⋮\\ {y}_{\xi }^{\left(n-2\right)}\left(\xi \right)=0,\\ {y}_{\xi }^{\left(n-1\right)}\left(\xi \right)=1,\end{array}$

${y}_{c}\left(x\right)=H\left(x-\xi \right){y}_{\xi }\left(x\right)$

$G\left(x,\xi \right)=H\left(x-\xi \right){y}_{\xi }\left(x\right)+{d}_{1}{y}_{1}\left(x\right)+\cdots +{d}_{n}{y}_{n}\left(x\right)$

$\left(\begin{array}{cccc}{B}_{1}\left[{y}_{1}\right]& {B}_{1}\left[{y}_{2}\right]& \cdots & {B}_{1}\left[{y}_{n}\right]\\ {B}_{2}\left[{y}_{1}\right]& {B}_{2}\left[{y}_{2}\right]& \cdots & {B}_{2}\left[{y}_{n}\right]\\ ⋮& ⋮& \ddots & ⋮\\ {B}_{n}\left[{y}_{1}\right]& {B}_{n}\left[{y}_{2}\right]& \cdots & {B}_{n}\left[{y}_{n}\right]\end{array}\right)\left(\begin{array}{c}{d}_{1}\\ {d}_{2}\\ ⋮\\ {d}_{n}\end{array}\right)=\left(\begin{array}{c}-{B}_{1}\left[H\left(x-\xi \right){y}_{\xi }\left(x\right)\right]\\ -{B}_{2}\left[H\left(x-\xi \right){y}_{\xi }\left(x\right)\right]\\ ⋮\\ -{B}_{n}\left[H\left(x-\xi \right){y}_{\xi }\left(x\right)\right]\end{array}\right)$

$u={\int }_{a}^{b}G\left(x,\text{}\xi \right)f\left(\xi \right)\text{d}\xi$

7. 结论

$y={\int }_{a}^{b}G\left(x,\xi \right)f\left(\xi \right)\text{d}\xi$

NOTES

*通讯作者。

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