958 and 中国斜学我术大学 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 2本书的使用说明 5 2.1本书的设计初衷 2.2本书的设计思想 2.3本书的使用方法 6 3课程综述 8 3.1课程主要内容 3.2课程学习日标 8 3.3课程学习方法… 9 3.4课程学习中蕴含的转化思想… 9 3.5定解问题求解方法的使用条件 10 3.6数理方程课程中的三步走战略 小4小小444小…小0 11 4第一章综合复习 12 41主要内容… 12 4.2学习目标 12 4.3学习方法… 多 4.4应用变量代换求解偏微分方程通解… 13 4.5定解问题的书写 14 4.6行波法求解一维无界区域弦振动问题… 15 4.7一维半无界区域的弦振动方程的处理之通解法和延拓法 16 4.8可以通过函数变换转化为一维无界区域波动方程问题 19 4.9通解法求解定解问题 21 5第二章综合复习… 22 5.1主要内容 22 5.2学习目标 22 5.3学习方法 23 5.4明确齐次方程的基本概念 23 1
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 ······················································ 4 2 本书的使用说明 ··················································································· 5 2.1 本书的设计初衷 ·········································································· 5 2.2 本书的设计思想 ·········································································· 5 2.3 本书的使用方法 ·········································································· 6 3 课程综述 ···························································································· 8 3.1 课程主要内容 ············································································· 8 3.2 课程学习目标 ············································································· 8 3.3 课程学习方法 ············································································· 9 3.4 课程学习中蕴含的转化思想 ··························································· 9 3.5 定解问题求解方法的使用条件 ························································ 10 3.6 数理方程课程中的三步走战略 ························································ 11 4 第一章综合复习 ··················································································· 12 4.1 主要内容 ··················································································· 12 4.2 学习目标 ··················································································· 12 4.3 学习方法 ··················································································· 13 4.4 应用变量代换求解偏微分方程通解 ·················································· 13 4.5 定解问题的书写 ·········································································· 14 4.6 行波法求解一维无界区域弦振动问题 ··············································· 15 4.7 一维半无界区域的弦振动方程的处理之通解法和延拓法 ······················· 16 4.8 可以通过函数变换转化为一维无界区域波动方程问题 ·························· 19 4.9 通解法求解定解问题 ···································································· 21 5 第二章综合复习 ··················································································· 22 5.1 主要内容 ··················································································· 22 5.2 学习目标 ··················································································· 22 5.3 学习方法 ··················································································· 23 5.4 明确齐次方程的基本概念 ······························································ 23 1
数理方程复习指导 2020春数理方程08班 5.5对于不满足施刘定理的问题的处理 24 5.6根据自然语言描述的物理问题书写定解问题并求解 26 5.7验证固有值问题是否满足施刘定理使用条件 28 5.8非齐次方程的求解 29 5.9非齐次边界的处理 31 6第三章综合复习33 6.1主要内容 33 6.2学习目标 33 6.3学习方法 34 6.4应用贝塞尔函数的母函数及其积分表示进行积分求解 34 6.5利用贝塞尔函数的递推关系进行积分求解 6.6给定函数的贝塞尔级数展开 36 6.7使用分离变量法结合贝塞尔函数求解定解问题 6 6.8应用勒让德多项式的性质和递推关系求解积分 37 6.9勒让德多项式的重要积分…38 6.10给定函数的勒让德级数展开… 39 6.11利用分离变量法结合勒让德函数求解定解问题 39 7第四章综合复习… 41 7.1主要内容 41 7.2学习目标 41 7.3学习方法 7.4利用傅里叶变换求解定解问题 7.5利用正余弦变换求解定解问题 43 7.6利用拉普拉斯变换求解定解问题 44 77利用傅里叶变换和拉普拉斯变换进行求解… 5 8第五章综合复习… 48 8.1主要内容 …………0 48 8.2学习目标 48 8.3学习方法 49 8.4关于6函数的等式的证明 49 8.56函数积分表示的应用 49 8.6利用镜像法求解格林函数 50 87利用分离变量法求解格林函数 52 8.8利用基本解方法求解定解问题 53
数理方程复习指导 2020 春数理方程 08 班 5.5 对于不满足施刘定理的问题的处理 ·················································· 24 5.6 根据自然语言描述的物理问题书写定解问题并求解 ····························· 26 5.7 验证固有值问题是否满足施刘定理使用条件 ······································ 28 5.8 非齐次方程的求解 ······································································· 29 5.9 非齐次边界的处理 ······································································· 31 6 第三章综合复习 ··················································································· 33 6.1 主要内容 ··················································································· 33 6.2 学习目标 ··················································································· 33 6.3 学习方法 ··················································································· 34 6.4 应用贝塞尔函数的母函数及其积分表示进行积分求解 ·························· 34 6.5 利用贝塞尔函数的递推关系进行积分求解 ········································· 35 6.6 给定函数的贝塞尔级数展开 ··························································· 36 6.7 使用分离变量法结合贝塞尔函数求解定解问题 ··································· 36 6.8 应用勒让德多项式的性质和递推关系求解积分 ··································· 37 6.9 勒让德多项式的重要积分 ······························································ 38 6.10 给定函数的勒让德级数展开 ·························································· 39 6.11 利用分离变量法结合勒让德函数求解定解问题 ·································· 39 7 第四章综合复习 ··················································································· 41 7.1 主要内容 ··················································································· 41 7.2 学习目标 ··················································································· 41 7.3 学习方法 ··················································································· 41 7.4 利用傅里叶变换求解定解问题 ························································ 42 7.5 利用正余弦变换求解定解问题 ························································ 43 7.6 利用拉普拉斯变换求解定解问题 ····················································· 44 7.7 利用傅里叶变换和拉普拉斯变换进行求解 ········································· 45 8 第五章综合复习 ··················································································· 48 8.1 主要内容 ··················································································· 48 8.2 学习目标 ··················································································· 48 8.3 学习方法 ··················································································· 49 8.4 关于 δ 函数的等式的证明 ······························································ 49 8.5 δ 函数积分表示的应用 ·································································· 49 8.6 利用镜像法求解格林函数 ······························································ 50 8.7 利用分离变量法求解格林函数 ························································ 52 8.8 利用基本解方法求解定解问题 ························································ 53 2
数理方程复习指导 2020春数理方程08班 9综合复习课讲义… 55 9.1定解问题的书写 55 9.2行波法求解定解问题（并和积分变换法作比较） 57 9.3齐次化原理的应用 59 9.4分离变量法求解定解问题 62 9.5积分变换法求解定解问题 67 9.66函数的性质 68 9.7基本解方法求解定解问题 69 10经典问题专题 . 71 10.1简介… 71 10.2函数变换法的应用 10.3微元法分析书写定解问题…73 10.4阻尼振动问题… 4………75 10.5勒让德多项式的递推公式推导 77 10.6一类重要的函数的傅里叶变换求解的特殊方法 78 10.7分离变量法求解基本解 80 11期末复习试卷 … 82 12期末模拟试卷 89 13期末模拟试卷参考答案 95 14总结… .103 15致谢…104
数理方程复习指导 2020 春数理方程 08 班 9 综合复习课讲义 ··················································································· 55 9.1 定解问题的书写 ·········································································· 55 9.2 行波法求解定解问题（并和积分变换法作比较） ································ 57 9.3 齐次化原理的应用 ······································································· 59 9.4 分离变量法求解定解问题 ······························································ 62 9.5 积分变换法求解定解问题 ······························································ 67 9.6 δ 函数的性质 ·············································································· 68 9.7 基本解方法求解定解问题 ······························································ 69 10 经典问题专题 ···················································································· 71 10.1 简介 ························································································ 71 10.2 函数变换法的应用 ······································································ 71 10.3 微元法分析书写定解问题 ····························································· 73 10.4 阻尼振动问题 ············································································ 75 10.5 勒让德多项式的递推公式推导 ······················································· 77 10.6 一类重要的函数的傅里叶变换求解的特殊方法 ·································· 78 10.7 分离变量法求解基本解 ································································ 80 11 期末复习试卷 ···················································································· 82 12 期末模拟试卷 ···················································································· 89 13 期末模拟试卷参考答案 ········································································ 95 14 总结 ································································································ 103 15 致谢 ································································································ 104 3
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。 2020
2020 春数理方程 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。 4