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, but must turn in your own work. Be as neat and complete
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
2.0
Out[2]=
1.5
1.0
0.5
1.0
0.5
0.0
0.5
1.0
The number o
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 even on the interval a < x < a, then
g(x) a0 +
an cos
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 a Fourier Series.)
Consider the function f (x) = x for
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
2.0
Out[2]=
1.5
1.0
0.5
1.0
0.5
0.0
0.5
1.0
The number o
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,
PlotLabel "fx x on x ", PlotRange 0.5, 0.5
f x = x on -p
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 must turn in your own work. Be as neat and complete as
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 assignment, solve ALL problems by hand. You may also check
you
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 thermal diffusivity of copper; T0 and T1 are left and
right
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
0.5
Out[2]=
0.0
0.5
1.0
1.5
3
2
1
0
1
2
3
1
In[3]:=
Ou
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 , PlotStyle
RGBColor 1, 0, 0
1.5
1.0
Out[2]=
0.5
0.0
0.5
3
2
1
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 on [a, b], convergent at one point of [a, b], and if th