Mathematical Physics Vol 1

7.8 Examples

403

Solution

∞ ∑ n = 1

2 h ℓ 2

n π x 0 ℓ

n π x ℓ

n π at ℓ

u ( x , t )=

sin

sin

cos

.

π 2 x

0 ( ℓ − x 0 )

Problem 265

Find the solution of equation a 2 u

xx = u t , 0 < x < l , 0 ≤ t ,

(7.311)

that satisfies initial

u ( x , 0 )= ϕ ( x )=   x ,

0 ≤ x ≤ l 2 ≤ x ≤ l

2 ;

(7.312)

l − x , l

and boundary conditions

u ( 0 , t )= 0 , u ( l , t )= 0 0 ≤ t .

(7.313)

Solution

2 l 

d x 

 l 2 Z 0

l Z l 2

n π x l

n π x l

 .

C n =

x sin

d x +

( l − x ) sin

(7.314)

For even values of n the constant C n is equal to zero, and for odd values B n =    4 l n 2 π 2 , za n = 1 , 5 , 9 ,... − 4 l n 2 π 2 , za n = 3 , 7 , 11 ,...

(7.315)

and the final solution is

4 l π 2

+ ... .

π x l

3 π x l

1 9

a π

3 a π

2 t

2 t

e − (

e − (

l )

l )

u ( x , t )=

sin

sin

(7.316)

−

Problem 266 Determine the type of PDE

4 u t = u xx , 0 ≤ x ≤ 2 , t > 0 ,

(7.317)