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所在平台: Coursera |
课程主页: https://www.coursera.org/learn/differential-equations-engineers
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课程名称：工程师的微分方程 课程概述：本课程专注于微分方程的基础理论和应用，旨在教授所有工程师应掌握的内容。前五周将学习常微分方程，最后一周将介绍偏微分方程。课程包含56段短讲座视频，每段讲座后有简单的习题，并且在每个重要主题后会有短小的实践测验。所有习题和测验的答案均可以在讲师提供的讲座笔记中找到。课程总共为六周，每周结束时都有一次评估测验。 课程大纲： 1. **一阶微分方程**：介绍微分方程的类型及分类，讨论利用欧拉方法求解一阶常微分方程的数值解，学习解析解法，并探讨实际应用实例。 2. **齐次线性微分方程**：对二阶常微分方程进行一般化，发展线性微分方程的理论概念，使用超位置原理和Wronskian，找到常系数齐次二阶微分方程的解析解。 3. **非齐次线性微分方程**：在常系数微分方程中添加非齐次项，研究共振现象并探讨RLC电路、弹簧上的质量和摆等重要应用。 4. **拉普拉斯变换和级数解法**：介绍两种新的解析解法，使用拉普拉斯变换解决具有不连续或冲击非齐次项的常系数微分方程，并简要介绍级数解法。 5. **微分方程组**：学习如何求解具有常系数的齐次一阶微分方程组，将这些方程转化为标准的矩阵代数特征值问题，并可视化二维解。 6. **偏微分方程**：定义傅里叶级数，推导一维扩散方程，使用分离变量法解决PDE，并利用解重新获得原始PDE的解。 更多信息和课程资料可通过以下链接获取： - 讲座笔记下载： [讲座笔记](http://www.math.ust.hk/~machas/differential-equations-for-engineers.pdf) - 观看宣传视频： [宣传视频](https://youtu.be/eSty7oo09ZI)
Name:First-Order Differential Equations
Description:A differential equation is an equation for a function with one or more of its derivatives. We introduce different types of differential equations and how to classify them. We then discuss the Euler method for numerically solving a first-order ordinary differential equation (ODE). We learn analytical methods for solving separable and linear first-order ODEs, with an explanation of the theory followed by illustrative solutions of some simple ODEs. Finally, we explore three real-world examples of first-order ODEs: compound interest, the terminal velocity of a falling mass, and the resistor-capacitor electrical circuit.
Name:Homogeneous Linear Differential Equations
Description:We generalize the Euler numerical method to a second-order ODE. We then develop two theoretical concepts used for linear equations: the principle of superposition and the Wronskian. Using these concepts, we can find analytical solutions to a homogeneous second-order ODE with constant coefficients. We make use of an exponential ansatz and transform the constant-coefficient ODE to a second-order polynomial equation called the characteristic equation of the ODE. The characteristic equation may have real or complex roots and we learn solution methods for the different cases.
Name:Inhomogeneous Linear Differential Equations
Description:We now add an inhomogeneous term to the constant-coefficient ODE. The inhomogeneous term may be an exponential, a sine or cosine, or a polynomial. We also study the phenomena of resonance, when the forcing frequency is equal to the natural frequency of the oscillator. Finally, we learn about three important applications: the RLC electrical circuit, a mass on a spring, and the pendulum.
Name:The Laplace Transform and Series Solution Methods
Description:We present two new analytical solution methods for solving linear ODEs. The first is the Laplace transform method, which is used to solve the constant-coefficient ODE with a discontinuous or impulsive inhomogeneous term. The Laplace transform is a good vehicle in general for introducing sophisticated integral transform techniques within an easily understandable context. We also introduce the solution of a linear ODE by a series solution. Although we do not go deeply into it here, an introduction to this technique may be useful to students who encounter it again in more advanced courses.
Name:Systems of Differential Equations
Description:We learn how to solve a coupled system of homogeneous first-order differential equations with constant coefficients. This system of ODEs can be written in matrix form, and we learn how to convert these equations into a standard matrix algebra eigenvalue problem. The two-dimensional solutions are then visualized using phase portraits. We next learn about the important application of coupled harmonic oscillators and the calculation of normal modes. The normal modes are those motions for which the individual masses that make up the system oscillate with the same frequency. We then apply the theory to solve a system of two coupled harmonic oscillators, and use the normal modes to analyze the motion of the system.
Name:Partial Differential Equations
Description:To learn how to solve a partial differential equation (PDE), we first define a Fourier series. We then derive the one-dimensional diffusion equation, which is a PDE describing the diffusion of a dye in a pipe. We then proceed to solve this PDE using the method of separation of variables. This involves dividing the PDE into two ordinary differential equations (ODEs), which can then be solved using the standard techniques of solving ODEs. We then use the solutions of these two ODEs, and our definition of a Fourier series, to recover the solution of the original PDE.
This course is about differential equations and covers material that all engineers should know. Both basic theory and applications are taught. In the first five weeks we will learn about ordinary differential equations, and in the final week, partial differential equations. The course is composed of 56 short lecture videos, with a few simple problems to solve following each lecture. And after each substantial topic, there is a short practice quiz. Solutions to the problems and practice quizzes can be found in instructor-provided lecture notes. There are a total of six weeks in the course, and at the end of each week there is an assessed quiz. Download the lecture notes: http://www.math.ust.hk/~machas/differential-equations-for-engineers.pdf Watch the promotional video: https://youtu.be/eSty7oo09ZI
