Lecture Notes

User Tools

Site Tools

You are here: Course Notes » Math for physics Sciences » Homework 8
math121a-f23:hw_8

Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Next revision 		Previous revision
math121a-f23:hw_8 [2023/10/20 21:57]
pzhou created 		math121a-f23:hw_8 [2023/10/24 21:43] (current)
pzhou
Line 1: Line 1:
 ====== Homework 8 ====== ====== Homework 8 ======
  
-1. Sine wave decomposition. +1. Sine and Cosine decomposition. 
  
-Suppose you are given a function on an interval, $f(x): [0, 1] \to \R,suchthat, such that f(x)vanishesonbothendpoints vanishes on both end points f(0)=f(1)=0.Suchfunction. Such function f(x)$ can be expressed as a sum of 'sine waves'.  +Suppose you are given a function on an interval, f(x):[0,1]→Rf(x): [0, 1] \to \R. Such function f(x)f(x) can be expressed as a sum of 'sine waves' and cosine waves and constant
-f(x)=∑n=1∞cnsin⁡(nπx). f(x) = \sum_{n=1}^\infty c_n \sin(n \pi x). +
  
-Can you figure out a way to determine the coefficients $c_n$?  +f(x)=a0+∑n=1∞ancos⁡(2nπx)+bnsin⁡(2nπx). f(x) = a_0 + \sum_{n=1}^\infty a_n \cos(2n \pi x) + b_n \sin(2n \pi x).  
 + 
 +Can you figure out a way to determine the coefficients $a_nand and b_n$?  
 + 
 +Test out your method for the following function 
 + f(x) = \begin{cases} 1 & 0 < x < 1/2 \cr 
 +0 & 1/2 \leq x \leq 1  
 +\end{cases} 
 + 
 + 
 +find a0,a1,b1a_0, a_1, b_1 and plot the truncated Fourier series 
 + a_0 + a_1 \cos(2 \pi x) + b_1 \sin(2 \pi x). $$ 
 +How does this resemble your original given function?  
 + 
 + 
  
  
Line 12: Line 25:
 f′(t)+ f(t)=0 f'(t) +  f(t) = 0  f′(t)+ f(t)=0 f'(t) +  f(t) = 0
 And suppose f(0)=1f(0) = 1. Can you solve f(t)f(t) for t>0t > 0?  And suppose f(0)=1f(0) = 1. Can you solve f(t)f(t) for t>0t > 0? 
 +
  
 3. Consider the following equation, for t>0t>0, 3. Consider the following equation, for t>0t>0,
Line 18: Line 32:
  
 4. Consider the following equation, for t>0t>0, 4. Consider the following equation, for t>0t>0,
 +[(d/dt)2+1]f(t)=0 [(d/dt)^2 + 1] f(t) = 0
 +And suppose $f(0) = 1, f'(0)=0.Canyousolve. Can you solve f(t)for for t > 0$? 
 +
 +5 (bonus, optional). Consider the following equation, for t>0t>0,
 (d/dt+1)(d/dt+1)f(t)=0 (d/dt + 1) (d/dt + 1) f(t) = 0  (d/dt+1)(d/dt+1)f(t)=0 (d/dt + 1) (d/dt + 1) f(t) = 0
 And suppose $f(0) = 1, f'(0)=0.Canyousolve. Can you solve f(t)for for t > 0$?  And suppose $f(0) = 1, f'(0)=0.Canyousolve. Can you solve f(t)for for t > 0$? 
  
  
math121a-f23/hw_8.1697864276.txt.gz · Last modified: 2023/10/20 21:57 by pzhou

Page Tools

Except where otherwise noted, content on this wiki is licensed under the following license: CC Attribution-Share Alike 4.0 International
CC Attribution-Share Alike 4.0 International Donate Powered by PHP Valid HTML5 Valid CSS Driven by DokuWiki