Fourier Series-Differential Equations-Lecture Slides, Slides of Differential Equations and Transforms

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Reminder:

• TA’s Review session
• Date: July 17 (Tuesday, for all students)
• Time: 10 - 11:40 am
• Room: 304 BH

Final Exam

• Date: July 19 (Thursday)
• Time: 10:30 - 12:30 pm
• Room: LC-C
• Covers: all materials
• I will have a review session on Wednesday

Examples

• Compute the Fourier series for

g x x - x.

x, x

, π x ,

f(x



Convergence of Fourier Series

• Pointwise Convegence
• Theorem. If f and f  are piecewise continuous on

[ -T, T ], then for any x in (-T, T), we have

• where the an,s and bn,s are given by the previous

fomulas. It converges to the average value of the

left and right hand limits of f(x). Remark on x =

T, or -T.

 ( ) ( ,

cos sin

0  





 (^)   f x f x

T

n x

b

T

n x

a

a

n

n n

 

Consider the heat flow problem:







 



 

 



 

   

  



π -x , x π

x , x ,

( ) u(x, )

( ) u( , t) u , t ,

, x t , x

u

t

u ( )

2

2

0

3 0

2 0 ( , t) 0

1 2 0 0 2

2









Solution

• Since the boundary condition forces us to

consider sine waves, we shall expand f(x)

into its Fourier Sine Series with T = .

Thus





 0

( ) sin

bn f x nxdx