ODE/3.2.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from sympy import * # 导入 sympy 中的所有功能\n",
    "init_printing(use_latex=True) \n",
    "from IPython.display import display # 显示公式\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "* 解的局部存在问题\n",
    "* 解的最大存在区间问题"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 一、解的延拓的引入"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\frac{\\mathrm dy}{\\mathrm dx}=f(x,y)\n",
    "$$\n",
    "\n",
    "右端函数 $f(x,y)$ 在某一有界区域 $G$ 中有意义."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1、局部利普希茨条件"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "假设方程 \n",
    "\n",
    "$$\n",
    "\\frac{\\mathrm dy}{\\mathrm dx}=f(x,y)\n",
    "$$ \n",
    "\n",
    "右端函数 $f(x,y)$ 在某一区域 $G$ 连续，且关于 $y$ 满足局部的利普希茨条件，即对于区域 $G$ 内的每一点，有以其为中心的完全含于 $G$ 内的闭矩形 $R$ 存在，在 $R$ 上 $f(x,y)$ 关于** $y$ 满足利普希茨条件**（关于不同的点, 域 $R$的大小和常数 $L$ 可能不同).\n",
    " "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2、解的延拓"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "设 $y=\\varphi(x),\\, x\\in [a,b]$ 是 \n",
    "\n",
    "$$\n",
    "\\begin{cases}\\frac{\\mathrm dy}{\\mathrm dx}=f(x,y)\\\\\n",
    "\\varphi (x_0)=y_0\\\n",
    "\\end{cases}\n",
    "$$\n",
    "\n",
    "的解，若 $y=\\psi (x) , x\\in [a_1,b_1]$ 也是初值问题的解, 且 $[a,b]\\subset [a_1,b_1]$, \n",
    "\n",
    "当 $x\\in [a,b]$时， $\\varphi (x)=\\psi(x)$\n",
    "\n",
    "则称 $\\psi(x)$ 是解 $\\varphi (x)$ 在区间 $[a,b]$ 上的**延拓**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3、延拓方法"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "设方程 \n",
    "\n",
    "$$\n",
    "\\frac{\\mathrm dy}{\\mathrm dx}=f(x,y)\n",
    "$$ \n",
    "\n",
    "的解为\n",
    "\n",
    "$$\n",
    "y=\\varphi (x), x\\in[x_0 - h, x_0 + h]\n",
    "$$\n",
    "\n",
    "现取 \n",
    "\n",
    "$$\n",
    "x_1=x_0+h , y_1=\\varphi (x_1)=\\varphi (x_0+h)\n",
    "$$\n",
    "\n",
    "然后以 $Q_1(x_1,y_1)$ 中心，作一小矩形，使它连同边界都含在区域 $G$ 的内部，再用解的存在唯一性定理，存在 $h_1 >0$ 使得在区间 \n",
    "\n",
    "$$\n",
    "|x-x_1|\\leq h_1\n",
    "$$ \n",
    "\n",
    "方程 \n",
    "$$\n",
    "\\frac{\\mathrm dy}{\\mathrm dx}=f(x,y)\n",
    "$$ \n",
    "\n",
    "有过 $(x_1,y_1)$ 的解 $y=\\psi (x)$ 且在 $x=x_1$ 处有 \n",
    "\n",
    "$$\n",
    "\\psi (x)=\\varphi (x)\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "由于唯一性，显然解 $y=\\psi (x)$ 和解 $y=\\varphi (x)$ 都在定义的区间 \n",
    "\n",
    "$$\n",
    "x_1-h\\leq x\\leq x_1\n",
    "$$\n",
    "\n",
    "上，\n",
    "\n",
    "$$\n",
    "\\psi (x)=\\varphi (x)\n",
    "$$\n",
    "\n",
    "但是在区间 \n",
    "$$\n",
    "x_1-h\\leq x\\leq x_1\n",
    "$$ \n",
    "\n",
    "上 ， 解  \n",
    "\n",
    "$$\n",
    "y=\\varphi (x)\n",
    "$$\n",
    "\n",
    "向右方的延拓，即将延拓到较大的区间\n",
    "\n",
    "$$\n",
    "x_0-h\\leq x\\leq x_0+h+h_1\n",
    "$$ "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "再令 \n",
    "\n",
    "$$\n",
    "x_2=x_1+h_1 , y_2=\\psi (x_2)\n",
    "$$ \n",
    "\n",
    "如果 $(x_2,y_2)\\in G$ 我们又可以取 $(x_2,y_2)$ 为中心，作一小矩形，使它连同其边界都含在区域 $G$ 内。 仿前，又可以将解延拓到更大的区间\n",
    "\n",
    "$$\n",
    "x_0-h\\leq x\\leq x_0+h+h_1\\leq x+h+h_1+h_2 \n",
    "$$ \n",
    "\n",
    "上 ， 其中 $h_2$ 是某一个正常数. 对于 $x$ 值减小的一边可以进行同样讨论, 使解向左方延拓. 就是在原来的积分曲线 $y=\\varphi (x)$ 左右端个接上一个积分的曲线段. \n",
    "\n",
    "上述解的延拓的方法还可继续进行. 那么, $y=\\varphi (x)$ 向两边延拓的最终情况如何呢？"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 二、解的延拓定理及其推论"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1、解的延拓定理"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "如果\n",
    "* 方程右端的函数 $f(x,y)$ 在有界区域 $G$ 中连续, \n",
    "* 且在 $G$ 内关于 $y$ 满足局部的利普希茨条件, \n",
    "\n",
    "那么方程的解通过 $G$ 内任何一点 $(x_0,y_0)$ 的解 $y=\\varphi (x)$ 可以延拓, 直到点 $(x,\\varphi (x))$ 任意接近区域 $G$ 的边界. 以向 $x$ 增大的一方的延拓来说, 如果 $y=\\varphi (x)$ 只能延拓到区间 $x_0\\leq x<d$ 上, 则当 $x\\to d$ 时, $(x,\\varphi (x))$ 趋于区域 $G$ 的边界。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2、推论"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "  如果 $G$ 是无界区域, 在上面解的延拓定理的条件下, 方程的通过点 $(x_0,y_0)$ 的解 $y=\\varphi (x)$ 可\n",
    "  \n",
    "以延拓，以向 $x$ 增大的一方延拓来说，有下面两种情况：\n",
    "  \n",
    "1. 解 $y=\\varphi (x)$ 可以延拓到区间 $[x_0,+\\infty)$ ;\n",
    "2. 解 $y=\\varphi (x)$ 可以延拓到区间 $[x_0,d]$ , 其中 $d$ 为有限数，则当 $x\\to d$ 时，或者 $y=\\varphi (x)$ 无界，或者点 $(x,\\varphi (x))$ 趋于区域 $G$ 的边界;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**例1**  讨论方程 \n",
    "$$\n",
    "\\frac{\\mathrm dy}{\\mathrm dx}=\\frac{y^2-1}{2}\n",
    "$$\n",
    "\n",
    "的分别通过点 $(0,0),(\\ln 2,-3)$ 的解的存在区间。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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Ijg/j3/dczqMz09h0rIapv/2MN7ecwOly6x1NCF3IGrQwpJK6VpZ9fJTl24rJtIay9KZM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\nKWghhDAoKWghhDAoKWghhDCo/w9Y1ee0dEMPgAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f0e65012278>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from sympy.abc import x\n",
    "from sympy.plotting import plot\n",
    "p = plot(((1-exp(x))/(1+exp(x)), (x, -4, 4)), \n",
    "         ((1+exp(x))/(1-exp(x)), (x, 1e-1, 4)), ylim=(-2, 2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fe15bf76e80>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x7fe15bf76550>"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(x*ln(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Help on built-in function min in module builtins:\n",
      "\n",
      "min(...)\n",
      "    min(iterable, *[, default=obj, key=func]) -> value\n",
      "    min(arg1, arg2, *args, *[, key=func]) -> value\n",
      "    \n",
      "    With a single iterable argument, return its smallest item. The\n",
      "    default keyword-only argument specifies an object to return if\n",
      "    the provided iterable is empty.\n",
      "    With two or more arguments, return the smallest argument.\n",
      "\n"
     ]
    }
   ],
   "source": [
    "from sympy.plotting import plot3d\n",
    "from sympy.abc import x, y\n",
    "f = y*x**y*(1-x)"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.5.2"
  },
  "latex_envs": {
   "bibliofile": "biblio.bib",
   "cite_by": "apalike",
   "current_citInitial": 1,
   "eqLabelWithNumbers": true,
   "eqNumInitial": 0
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}