1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

424 Chapter 7 Numerical Methods

i

m 1234567 f(tm)

00000000 0

10000000 1. 0

20. 50. 25 0. 25 0. 50. 50. 25 0 2. 0

31. 22 0. 75 0. 69 0. 75 1. 22 0. 69 0. 25 3. 0

41. 99 1. 40 1. 19 1. 84 1. 98 1. 30 0. 69 4. 0

Table 10 Numerical solution of Eqs. (9)–(11). Entries are 100×ui(m)

As an example, let us consider the vibrations of a square membrane, as de-
scribed by the problem

∂^2 u

∂x^2 +

∂^2 u

∂y^2 =

∂^2 u

∂t^2 ,^0 <x<^1 ,^0 <y<^1 ,^0 <t, (14)

u(x, 0 ,t)= 0 , u(x, 1 ,t)= 0 , 0 <x< 1 , 0 <t, (15)

u( 0 ,y,t)= 0 , u( 1 ,y,t)= 0 , 0 <y< 1 , 0 <t, (16)

u(x,y, 0 )=f(x,y), 0 <x< 1 , 0 <y< 1 , (17)

∂u

∂t

(x,y, 0 )=g(x,y), 0 <x< 1 , 0 <y< 1. (18)

A typical replacement for the wave equation (14) is constructed using Eq. (1)
for the Laplacian and Eq. (13) for the time derivative:

ui(m+ 1 )− 2 ui(m)+ui(m− 1 )

(   t)^2

=uN(m)+uS(m)+uE(m)+uW(m)−^4 ui(m)

(   x)^2

. (19)

As usual we solve forui(m+ 1 ), using the abbreviationρ= t/ x.Theresult
is

ui(m+ 1 )=ρ^2

[

uE(m)+uW(m)+uN(m)+uS(m)

]

+( 2 − 4 ρ^2 )ui(m)−ui(m− 1 ). (20)

The stability rules given earlier still apply. Thus we must chooseρ^2 ≤ 1 /2in
order to get a sensible solution.
Let us now be specific. We shall take x= y= 1 /4,ρ^2 = 1 /2 (that is,
t= 1 / 4

√

2), and suppose that the initial data from Eqs. (11) and (12) are

f(x,y)=

{

1nearx=^14 , y=^14 ,

0elsewhere,

g(x,y)≡ 0.