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. 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.
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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
This course is part of the 18.03x Differential Equations XSeries Program.
What you'll learn
- 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
Syllabus
- 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
Prerequisites:
- Introduction to Differential Equations (basic)
Introduction to Differential Equations
- some linear algebra preferred
Differential Equations: 2x2 Systems
Differential Equations: Linear Algebra and NxN Systems of Differential Equations
