473/573 Boundary Value Problems, Spring 2014, Homework 2
This homework assignment is due Friday, February 28, 2014 at the beginning of class. You may discuss problems
with other people in our class, b
473 Periodic Extension Example
Creating a periodic extension of a function via Mathematica's Piecewise command
: x3
x2
In[1]:=
f x
1
In[2]:=
Plot f x , x,
1, 1 , Frame
True, PlotRange
0.5, 3
3.0
2.5
2
473/573 Boundary Value Problems, Spring 2014, Fourier Series Theorems
In this handout, Theorem #.% corresponds to Theorem % in Section 1.# Class Notes!
Theorem 2.1: (Even and Odd Functions)
If g(x) is
473/573 Boundary Value Problems, Spring 2014, Why we need all these theorems!
Sometimes things that seem like they should be valid intuitively are not true. Here are two examples.
1. (Dierentiation of
473 Periodic Extension Example
Creating a periodic extension of a function via Mathematica's Piecewise command
: x3
x2
In[1]:=
f x
1
In[2]:=
Plot f x , x,
1, 1 , Frame
True, PlotRange
0.5, 3
3.0
2.5
2
Why we need all these theorems!
1. Differentiation of a Fourier Series
Find Fourier Series for f(x) = x on -p < x < p.
In[1]:=
In[2]:=
fx_ : x
Plotfx, x, , , PlotStyle RGBColor1, 0, 0, Frame True,
Pl
473/573 Boundary Value Problems, Spring 2014, Homework 3
This homework assignment is due Friday, March 21, 2014 at the beginning of class. You may discuss
problems with other people in our class, but
473/573 Boundary Value Problems, Spring 2014, Homework 1
This homework assignment is due Friday, February 7, 2014 at the beginning of class. Be as neat and
complete as possible. Note: For this assignm
473 2.3: Heat Equation - Fixed-End Temperature Example: 80 cm
copper rod, with insulated lateral surface, left end fixed at 0 C,
right end fixed at 100 C.
Find Solution (a is length of rod; k is ther
473 1.6: Mean Square Error Example
Motivation for the idea of Mean Square Error
In[1]:=
f x
: If x
In[2]:=
Plot f x , x,
0,
1, 1
, , Frame
True, PlotRange
1.5, 1.5 , PlotStyle
RGBColor 1, 0, 0
1.5
1.0
473 1.1 and 1.2: Fourier Series Examples
Example 1: Here is a way to draw a square wave with Fourier Series:
In[1]:=
h x
: If x
In[2]:=
Plot h x , x,
0, 0, 1
, , Frame
True, PlotRange
0.5, 1.5 , PlotS
473/573 Boundary Value Problems, Spring 2014, Note on Theorem 5 in our Section 1.5 notes
From Olmstead, Advanced Calculus, p. 452 we have this theorem:
If un (x) is a series of dierentiable functions