The Chemistry Maths Book, Second Edition

15.7 Exercises 443

12.A function with period 2 lis defined by

(i)Draw the graph of the function in the interval−l 1 ≤ 1 x 1 ≤ 1 l. (ii)Find the Fourier series of

the function.

13.A function with period 2 lis defined by

(i)Draw the graph of the function in the interval−l 1 ≤ 1 x 1 ≤ 1 l. (ii)Find the Fourier series of

the function.

Section 15.5

14.Find the solution of the diffusion equation

in the interval 01 ≤ 1 x 1 ≤ 1 lthat satisfies the boundary conditionsT(0,t) 1 = 1 T(l,t) 1 = 10 and

initial conditionT(x,0) 1 = 1 x(l 1 − 1 x), and that decreases exponentially with time (See

Exercise 17 of Chapter 14).

Section 15.6

Find the Fourier transform:

15. 
    

16.e

−a|x|

(a 1 > 1 0)

17.Show that the Fourier transform ofe

−ax

2

(a 1 > 1 0)is

Hint change variable to:=+and use

−

tax

iy

2 a

Z

∞

∞∞

edt

−t

=

















α

α

2

π

1

2

15 14

2

4

a

e

−/ya

()Figure. b

fx

axb

()=

,<<

,











1

0

if

otherwise

∂

∂

=

∂

∂

T

t

D

T

x

2

2

fx

xl x x l

xl x

()

()

()

=

−<<

−+

if

if

0

−< <









 lx 0

fx

xl x x l

xl x

()

()

()

=

−<<

• 
  

if

if

0

−< <









 lx 0