508
8 Direct Methods of Variational Problems
where, D = {(x, y)| − a < x < a, −b < y < b}; Γ is the boundary of the
rectangular domain D.
8.14 Soluting the following boundary value problem by the Galerkin method
[(x + 2)y
]
+ y = 3x (0 < x < 1)
y
(0) = y
(0) = 0, y(1) = y
(1) = 0
8.15 Find the first, second characteristic value λ and the characteristic function of
the boundary value problem by the Galerkin method
y
+ λ(1 + x
2
)y = 0
y(−1) = y(1) = 0
8.16 Given the circular thin plate as shown in Fig. 8.4, the radius is R, the periphery
is fixed, subject to the uniform distribution load q, let the deflection surface
function be w = a
1 −
r
2
R 2
2
, using the Galerkin method to find the deflection.
8.17 Using the Galerkin method to find the approximate solutions of the first, second
eigenvalues and the corresponding eigenfunction of eigenvalue problem
y
+ λy = 0
y(0) = y(1) = 0
and comparing with the exact solution of the problem.
8.18 Let the vibration equation of a string be
O
r
z
q
r
r
Fig. 8.4 The clamped-end circular plate