Lecture 3.3: Solving differential equations with Fourier series - - PowerPoint PPT Presentation

Lecture 3.3: Solving differential equations with Fourier series

Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4340, Advanced Engineering Mathematics

• M. Macauley (Clemson)

Lecture 3.3: Solving ODEs with Fourier series Advanced Engineering Mathematics 1 / 4

Motivation

Recall the method of undetermined coefficients to solve a 2nd order linear inhomogeneous ODE y′′ + a(x)y′ + b(x)y = f (x):

• 1. Solve the related homogeneous equation: y′′

h + a(x)y′ h + b(x)yh = 0.

• 2. Guess the form of a particular solution yp(x).
• 3. Add these together: y(x) = yh(x) + yp(x).

f (x) guess ekx yp(x) = aekx ckxk + · · · + c1x + c0 yp(x) = akxk + · · · + a1x + a0 sin kx or cos kx yp(x) = a cos kx + b sin kx.

Question

What if the forcing term is a piecewise function like a square wave? f (x) guess square wave yp(x) = a0 2 +

∞

• n=1

an cos nπx

L

+ bn sin nπx

L

This is generally much easier than using Laplace transforms!

• M. Macauley (Clemson)

Lecture 3.3: Solving ODEs with Fourier series Advanced Engineering Mathematics 2 / 4

Example 1

Solve y′′ + 3y′ + 2y = f (x), for the square wave of period 2: f (x) =

• 1

0 < x < 1 −1 −1 < x < 0

• M. Macauley (Clemson)

Lecture 3.3: Solving ODEs with Fourier series Advanced Engineering Mathematics 3 / 4

Example 2

Solve y′′ + ω2y = f (x), ω =nπ, for the square wave of period 2: f (x) =

• 1

0 < x < 1 −1 −1 < x < 0

• M. Macauley (Clemson)

Lecture 3.3: Solving ODEs with Fourier series Advanced Engineering Mathematics 4 / 4