November 30 Math 2306 sec 51 Fall 2015Section 11.3: Fourier Cosine and Sine Series

Half Range Sine and Half Range Cosine Series: For f defined on0 < x < p.

Half range cosine series f (x) =a0

2+

∞∑n=1

an cos(

nπxp

)

where a0 =2p

∫ p

0f (x)dx and an =

2p

∫ p

0f (x) cos

(nπx

p

)dx .

Half range sine series f (x) =∞∑

n=1

bn sin(

nπxp

)where bn =

2p

∫ p

0f (x) sin

(nπx

p

)dx .

November 27, 2015 1 / 34

Half Range SeriesFor the given function, plot the graph of the function along with threefull periods on the interval (−3p,3p) of (a) the half range cosine seriesand (b) the half range sine series.

f (x) ={

x , 0 ≤ x < 32

3 − x , 32 ≤ x < 3

Figure: Plot of f alone. November 27, 2015 2 / 34

(a) Even Extension

Figure: Half range sine series.November 27, 2015 3 / 34

(b) Odd Extension

Figure: Half range cosine series.November 27, 2015 4 / 34

Solution of a Differential Equation

An undamped spring mass system has a mass of 2 kg attached to aspring with spring constant 128 N/m. The mass is driven by anexternal force f (t) = 2t for −1 < t < 1 that is 2-periodic so thatf (t + 2) = f (t) for all t > 0. Determine a particular solution xp for thedisplacement for t > 0.

November 27, 2015 5 / 34

November 27, 2015 6 / 34

November 27, 2015 7 / 34

November 27, 2015 8 / 34

November 27, 2015 9 / 34

November 27, 2015 10 / 34

An Eigenvalue Problem

An eigenvalue problem consists of a Boundary Value Problem whichincludes an unknown parameter λ. The task is to determine values ofthe parameter and corresponding nonzero functions (i.e. nontrivial)that solve the BVP. The values are called eigenvalues and thecorresponding functions are called eigenfunctions.

Example: Solve −u′′ = λu for 0 < x < 1 subject to u(0) = 0 andu(1) = 0.

November 27, 2015 18 / 34

November 27, 2015 19 / 34

November 27, 2015 20 / 34

November 27, 2015 21 / 34

November 27, 2015 22 / 34