微分方程:傅立叶级数和偏微分方程

Differential Equations: Fourier Series and Partial Differential Equations

Learn to use Fourier series to solve differential equations with periodic input signals and to solve boundary value problems involving the heat equation and wave equation.

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麻省理工学院
edX
  • 完成时间大约为 8
  • 中级
  • 英语
注:因开课平台的各种因素变化,以上开课日期仅供参考

你将学到什么

How to describe periodic functions using Fourier series

How to solve ODEs with Fourier series input

How to solve separable PDEs using Fourier series inputs and boundary conditions

课程概况

Differential equations are the mathematical language we use to describe the world around us. Many phenomena are not modeled by differential equations, but by partial differential equations depending on more than one independent variable. In this course, we will use Fourier series methods to solve ODEs and separable partial differential equations (PDEs). You will learn how to describe any periodic function using Fourier series, and will be able to use resonance and to determine the behavior of systems with periodic input signals that can be described in terms of Fourier series. This course will use MATLAB to assist computations.

In this course we will explore:

How to process noisy sound files
The way a beam bends in response to external forces
How to design of ovens to create strong but lightweight composites
The motion of a violin string
The five modules in this series are being offered as an XSeries on edX. Please visit the Differential Equations XSeries Program Page to learn more and to enroll in the modules.

Violinist photo by user: DeshaCAM. Copyright © 2018 Adobe Systems Incorporated. Used with permission.

课程大纲

Unit 1: Fourier Series

Introduction to Fourier series
Fourier series with arbitrary periods
Using Fourier series to solve differential equations

Unit 2: Partial Differential Equations

Boundary conditions and boundary value problems
The heat equation
The wave equation

预备知识

Introduction to Differential Equations (basic)

https://www.edx.org/course/introduction-differential-equations-mitx-18-031x

some linear algebra preferred

https://www.edx.org/course/differential-equations-2x2-systems-mitx-18-032x
https://www.edx.org/course/differential-equations-linear-algebra-and-nxn-systems-of-differential-equations

常见问题

Q: What are the pre-requisites for this course?

A: We assume familiarity with 18.031x Introduction to Differential Equations and 18.032x Differential Equations: 2x2 Systems, and 18.033x Differential Equations: Linear Algebra and NxN Systems. Knowledge from 18.03Lx is not required to succeed in this course.
 

Q: Do I need to purchase a MATLAB license?

A: No, registering for this course gives you access to a MATLAB license for the duration of this course!

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