Mathematical Physics Vol 1

Chapter 7. Partial differential equations

402

Solution

2 n

1 +( − 1 ) n + 1 1 − l

π

∞ ∑ n = 1

an π t

n π x l

u ( x , t )=

cos

sin

l ·

.

Problem 263

Find the solution of equation

2 u

u tt = a

xx ,

that satisfies boundary

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

and initial conditions

u ( x , 0 )= x + a , u t ( x , 0 )= 0 , where a = const .

Solution

∞ ∑ n = 1

n π x l

an π t

2 n π

( − 1 ) n + 1 ( l + a )+ a cos

u ( x , t )=

sin

l ·

.

Problem 264

Find the solution of equation

2 u

u tt = a

xx ,

that satisfies boundary

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

and initial conditions

u ( x , 0 )=    h x 0

x ,

0 ≤ x ≤ x 0 ,

h ( ℓ − x ) ℓ − x 0

, x 0 ≤ x ≤ ℓ

u t ( ℓ, t )= 0 , 0 ≤ x ≤ ℓ.