ce and Te 中国辩学我术大学 数理方程复习指导 学院：信息科学技术学院 专业：信息安全 姓名：高源 指导教师：谢如龙老师
数理方程复习指导 学 院：信息科学技术学院 专 业：信息安全 姓 名：高源 指导教师：谢如龙老师
数理方程复习指导 2020春数理方程08班 目录 1写给数理方程08班的同学们的一封信。 3 2课程综述. 4 2.1课程主要内容 4 2.2课程学习目标 .4.4+44. 2.3课程学习方法 2.4课程学习中蕴含的转化思想. 5 2.5定解问题求解方法的使用条件 6 2.6数理方程课程中的三步走战略 3第一章综合复习. 8 3.1主要内容. 8 3.2学习目标 3.3学习方法. 9 3.4应用变量代换求解偏微分方程通解 3.5定解问题的书写 10 3.6行波法求解一维无界区域弦振动问题 11 3.7一维半无界区域的弦振动方程的处理之通解法和延拓法. 12 3.8可以通过函数变换转化为一维无界区域波动方程问题. 15 3.9通解法求解定解问题 17 4第二章综合复习 18 4.1主要内容 18 4.2学习目标 18 4.3学习方法 4.4明确齐次方程的基本概念. 19 4.5对于不满足施刘定理的问题的处理.。 20 4.6根据自然语言描述的物理问题书写定解问题并求解 22 4.7验证固有值问题是否满足施刘定理使用条件 24 4.8非齐次方程的求解 25
数理方程复习指导 2020 春数理方程 08 班 目录 1 写给数理方程 08 班的同学们的一封信 ······················································ 3 2 课程综述 ···························································································· 4 2.1 课程主要内容 ············································································· 4 2.2 课程学习目标 ············································································· 4 2.3 课程学习方法 ············································································· 5 2.4 课程学习中蕴含的转化思想 ··························································· 5 2.5 定解问题求解方法的使用条件 ························································ 6 2.6 数理方程课程中的三步走战略 ························································ 7 3 第一章综合复习 ··················································································· 8 3.1 主要内容 ··················································································· 8 3.2 学习目标 ··················································································· 8 3.3 学习方法 ··················································································· 9 3.4 应用变量代换求解偏微分方程通解 ·················································· 9 3.5 定解问题的书写 ·········································································· 10 3.6 行波法求解一维无界区域弦振动问题 ··············································· 11 3.7 一维半无界区域的弦振动方程的处理之通解法和延拓法 ······················· 12 3.8 可以通过函数变换转化为一维无界区域波动方程问题 ·························· 15 3.9 通解法求解定解问题 ···································································· 17 4 第二章综合复习 ··················································································· 18 4.1 主要内容 ··················································································· 18 4.2 学习目标 ··················································································· 18 4.3 学习方法 ··················································································· 19 4.4 明确齐次方程的基本概念 ······························································ 19 4.5 对于不满足施刘定理的问题的处理 ·················································· 20 4.6 根据自然语言描述的物理问题书写定解问题并求解 ····························· 22 4.7 验证固有值问题是否满足施刘定理使用条件 ······································ 24 4.8 非齐次方程的求解 ······································································· 25 1
数理方程复习指导 2020春数理方程08班 4.9非齐次边界的处理 27 5第三章综合复习 29 5.1主要内容 29 5.2学习目标 29 5.3学习方法 5.4应用贝塞尔函数的母函数及其积分表示进行积分求解 30 5.5利用贝塞尔函数的递推关系进行积分求解 5.6给定函数的贝塞尔级数展开. 32 5.7使用分离变量法结合贝塞尔函数求解定解问题 32 5.8应用勒让德多项式的性质和递推关系求解积分 33 5.9勒让德多项式的重要积分 34 5.10给定函数的勒让德级数展开. 35 5.11利用分离变量法结合勒让德函数求解定解问题. 35 6第四章综合复习 37 6.1主要内容 6.2学习目标 37 6.3学习方法 37 6.4利用傅里叶变换求解定解问题 38 6.5利用正余弦变换求解定解问题 39 6.6利用拉普拉斯变换求解定解问题. 40 6.7利用傅里叶变换和拉普拉斯变换进行求解. 4 7第五章综合复习 7.1主要内容 44 7.2学习目标 7.3学习方法 7.4关于6函数的等式的证明 5 7.56函数积分表示的应用. 45 7.6利用镜像法求解格林函数 LA0 46 7.7利用分离变量法求解格林函数. 48 78利用基本解方法求解定解问题 8期末摸拟试卷 51 9期末模拟试卷参考答案 10总结 65 11致谢. 66
数理方程复习指导 2020 春数理方程 08 班 4.9 非齐次边界的处理 ······································································· 27 5 第三章综合复习 ··················································································· 29 5.1 主要内容 ··················································································· 29 5.2 学习目标 ··················································································· 29 5.3 学习方法 ··················································································· 30 5.4 应用贝塞尔函数的母函数及其积分表示进行积分求解 ·························· 30 5.5 利用贝塞尔函数的递推关系进行积分求解 ········································· 31 5.6 给定函数的贝塞尔级数展开 ··························································· 32 5.7 使用分离变量法结合贝塞尔函数求解定解问题 ··································· 32 5.8 应用勒让德多项式的性质和递推关系求解积分 ··································· 33 5.9 勒让德多项式的重要积分 ······························································ 34 5.10 给定函数的勒让德级数展开 ·························································· 35 5.11 利用分离变量法结合勒让德函数求解定解问题 ·································· 35 6 第四章综合复习 ··················································································· 37 6.1 主要内容 ··················································································· 37 6.2 学习目标 ··················································································· 37 6.3 学习方法 ··················································································· 37 6.4 利用傅里叶变换求解定解问题 ························································ 38 6.5 利用正余弦变换求解定解问题 ························································ 39 6.6 利用拉普拉斯变换求解定解问题 ····················································· 40 6.7 利用傅里叶变换和拉普拉斯变换进行求解 ········································· 41 7 第五章综合复习 ··················································································· 44 7.1 主要内容 ··················································································· 44 7.2 学习目标 ··················································································· 44 7.3 学习方法 ··················································································· 45 7.4 关于 δ 函数的等式的证明 ······························································ 45 7.5 δ 函数积分表示的应用 ·································································· 45 7.6 利用镜像法求解格林函数 ······························································ 46 7.7 利用分离变量法求解格林函数 ························································ 48 7.8 利用基本解方法求解定解问题 ························································ 49 8 期末模拟试卷 ······················································································ 51 9 期末模拟试卷参考答案 ·········································································· 57 10 总结 ································································································ 65 11 致谢 ································································································ 66 2
数理方程复习指导 2020春数理方程08班 写给数理方程08班的同学们的一封信 亲爱的2020春数理方程08班的同学们，你们好 这本《数理方程复习指导》在几个月的努力下终于和大家见面了。这学期是 我第一次当助牧，而且由于特殊情况我们的课堂教学、习题课讨论等过程只能在 线上进行。考虑到各种原因，这一个学期在和大家的交流中我也在一直寻找合适 的方法能够尽自己所能为大家提供帮助，最终能够和大家一起顺利完成这门课程 的学习。 综合各种考虑，我决定制作这本《数理方程复习指导》，希望能够和大家分享 学习数理方程的方法、经验，以及遇到的困难。衷心希望这本复习指导能够对大 家有所帮助，也希望大家都能够圆满地完成这学期的学习。 遇到我们这个大家庭的每一个成员都让我感到幸运，衷心希望能够和大家一 起变得更好。 数理
数理方程 08 班 数理方程复习指导 2020 春数理方程 08 班 写给数理方程 08 班的同学们的一封信 亲爱的 2020 春数理方程 08 班的同学们，你们好: 这本《数理方程复习指导》在几个月的努力下终于和大家见面了。这学期是 我第一次当助教，而且由于特殊情况我们的课堂教学、习题课讨论等过程只能在 线上进行。考虑到各种原因，这一个学期在和大家的交流中我也在一直寻找合适 的方法能够尽自己所能为大家提供帮助，最终能够和大家一起顺利完成这门课程 的学习。 综合各种考虑，我决定制作这本《数理方程复习指导》，希望能够和大家分享 学习数理方程的方法、经验，以及遇到的困难。衷心希望这本复习指导能够对大 家有所帮助，也希望大家都能够圆满地完成这学期的学习。 遇到我们这个大家庭的每一个成员都让我感到幸运，衷心希望能够和大家一 起变得更好。 3
数理方程复习指导 2020春数理方程08班 课程综述 2.1课程主要内容 偏微分方程的基本概念 常见偏微分方程的通解的求解 数理方程的建立过程 三类常见方程及其对应的定解问题的书写及对应的物理意义 行波法求解一维无界区域波动方程问题的基本操作 延拓法的基本思想和应用 通解法求解一类定解问题的基本操作 分离变量法求解有界区域问题的基本操作 应用特殊函数实现分离变量法在柱坐标和球坐标系下分离变量得到的固有值问题 求解 傅里叶变换法求解无界区域问题的基本操作 正余弦变换的应用 拉普拉斯变换求解半无界区域问题的基本操作 基本解方法求解定解问题的基本操作 2.2课程学习目标 掌握数理方程的基本概念 理解数理方程的建立过程 理解三类重要方程及其对应的定解问题中的元素的物理意义
数理方程 08 班 数理方程复习指导 2020 春数理方程 08 班 课程综述 2.1 课程主要内容 偏微分方程的基本概念 常见偏微分方程的通解的求解 数理方程的建立过程 三类常见方程及其对应的定解问题的书写及对应的物理意义 行波法求解一维无界区域波动方程问题的基本操作 延拓法的基本思想和应用 通解法求解一类定解问题的基本操作 分离变量法求解有界区域问题的基本操作 应用特殊函数实现分离变量法在柱坐标和球坐标系下分离变量得到的固有值问题 求解 傅里叶变换法求解无界区域问题的基本操作 正余弦变换的应用 拉普拉斯变换求解半无界区域问题的基本操作 基本解方法求解定解问题的基本操作 2.2 课程学习目标 掌握数理方程的基本概念 理解数理方程的建立过程 理解三类重要方程及其对应的定解问题中的元素的物理意义 4