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所在平台: Udemy |
课程主页: https://www.udemy.com/course/introduction-to-partial-differential-equations/
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Coursera 课程《偏微分方程导论》课程总结 本课程是关于偏微分方程（PDEs）的入门课程，PDEs 在纯数学和应用数学中都占据核心地位，是描述自然界中连续变化的量（如温度、磁场、流体速度、静电势、肿瘤密度、大脑活动、股票价格、物种数量等）的数学模型。通过现代软件、人工智能和深度学习的力量，PDEs 的应用已广泛渗透到现代科学和技术的各个领域。 本课程旨在为学习 PDEs 打下坚实基础，从其物理起源和动机出发，重点介绍数学物理中的三大经典方程：热方程、波动方程和拉普拉斯方程。 课程将涵盖学习 PDEs 的关键思想，包括： * **分离变量法（Separation of Variables）** * **积分变换（Integral Transforms）** * **数学物理中的特殊函数（Special Functions of Mathematical Physics）** * **傅里叶级数（Fourier Series）** * 以及相关的其他主题。 虽然课程内容侧重于基础数学，但会不断地将数学概念与它们所源于的物理背景联系起来。课程也包含大量的实践内容，将利用所学方法显式地解决各种物理问题。
Partial Differential Equations (PDEs) are pivotal in both pure and applied mathematics. They emerge as mathematical models of processes in nature in which some quantities change continuously in space and time. The temperature or the magnetic field on earth, the velocity of a fluid or gas, electrostatic potential of the conductor, the density of a cancerous tumor in the body, neuronal activity of the brain, stock price fluctuations, and the population of biological species are just a few of the complex systems modeled by this ubiquitous equations. Supported with the power of modern software and the emerging fields of Artificial Intelligence and Deep Learning, PDEs are expanding to all areas of modern science and technology. This course presents a foundation for PDEs, starting from their physical origin and motivation. In particular, it introduces the classical equations of mathematical physics, namely the heat, wave, and Laplace equations. In this course, you will learn key ideas critical to the study of PDEs - separation of variables, integral transforms, special functions of mathematical physics, Fourier series, and related topics.Though the topics focus on the foundational mathematics, connections are constantly made to the underlying physics from which they emerge. Substantial practical part of the course is dedicated to solving explicitly various physical problems by using these methods
