Mathematical Physics Vol 1

Chapter 7. Partial differential equations

424

y

b

a

x

Figure 7.7: Rectangular membrane.

and initial conditions

u ( x , y , 0 )= f ( x , y ) ,

(7.398)

∂ u ∂ t t = 0

= g ( x , y ) .

(7.399)

In this case we will look for the solution by applying the Fourier method

u ( x , y , t )= F ( x , y ) T ( t ) ,

(7.400)

and thus the equation (7.396) is decomposed on two

¨ T T

1 c 2

1 F

2 ⇒

( F xx + F yy )= − ν ¨ T + λ 2 T = 0 i

=

(7.401) (7.402)

2 F = 0 .

F xx + F yy + ν

We have introduced here the notation c 2 ν 2 = λ 2 . Further, assume the equation (7.402) can also be represented in the from

F ( x , y )= X ( x ) Y ( y ) .

(7.403)

With this assumption, the equation (7.402) becomes

2 XY = 0 ⇒ + ν 2 Y = − k 2 .

Y ( y ) X xx ( x )+ X ( x ) Y yy ( y )+ ν

(7.404)

1 Y

d 2 X d x 2

d 2 Y d y 2

1 X

(7.405)

= −

From here we obtain:

d 2 X d x 2

+ k 2 X = 0 ,

(7.406)

d 2 Y d y 2

+( ν 2 − k 2 ) Y = 0 .

(7.407)

The solutions of this system are

X ′′ + k 2 X = 0 ⇒ X = A cos kx + B sin kx , Y ′′ + p 2 Y = 0 ⇒ Y = C cos py + D sin py .

(7.408) (7.409)