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 ) y h = 0. 2. Guess the form of a particular solution y p ( x ). 3. Add these together: y ( x ) = y h ( x ) + y p ( x ). f ( x ) guess e kx y p ( x ) = ae kx c k x k + · · · + c 1 x + c 0 y p ( x ) = a k x k + · · · + a 1 x + a 0 sin kx or cos kx y p ( x ) = a cos kx + b sin kx . Question What if the forcing term is a piecewise function like a square wave? f ( x ) guess ∞ y p ( x ) = a 0 � a n cos n π x + b n sin n π x square wave 2 + L L n =1 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 � 1 0 < x < 1 Solve y ′′ + 3 y ′ + 2 y = f ( x ), for the square wave of period 2: f ( x ) = − 1 − 1 < x < 0 M. Macauley (Clemson) Lecture 3.3: Solving ODEs with Fourier series Advanced Engineering Mathematics 3 / 4

Example 2 � 1 0 < x < 1 Solve y ′′ + ω 2 y = f ( x ), ω � = n π , for the square wave of period 2: f ( x ) = − 1 − 1 < x < 0 M. Macauley (Clemson) Lecture 3.3: Solving ODEs with Fourier series Advanced Engineering Mathematics 4 / 4