888
Appendix: All Solutions to the Problems
y 2 =
3969 − 4200x
2
+ 231x
4
4252
8.9 Using the Ritz method to find the approximate solution of the boundary value
problem
−u = cos
π x
a
(x, y ∈ D)
∂u
∂n
Γ
= 0, Γ is the boundary ofD
where the domain D is the rectangular area: 0 ≤ x ≤ a, 0 ≤ y ≤ b.
Solution Let the form of approximate solution be
u mn =
m
i=1
n
j=0
c i j cos
iπ x
a
cos
jπ y
b
Taking i = 1, j = 0, m = 1, n = 0, we have
u 10 = c 10 cos
π x
a
It follows from Example 7.8.1 that the functional corresponding to the differential
equation is
J [u] =
¨
D
[(∇u)
2
− 2u f ]dxdy
where, f = cos
π x
a
, substituting u 10 = c 10 cos
π x
a
into the functional, and taking
note that
(∇u 10 )
2
− 2u 10 f =
−
π
a
c 10 sin
π x
a
2 − 2c 10 cos
2 πx
a
we get
J [c 10 ] =
a
0
b
0
−
π
a
c 10 sin
π x
a
2 − 2c 10 cos
2 π x
a
dxdy
= b
a
0
π
2
a 2 c
2
10
1
2
1 − cos
2π x
a
− c 10
1 + cos
2π x
a
dx
=
1
2
π
2 b
a
c
2
10 − c 10 ab