506
8 Direct Methods of Variational Problems
Hint: Its approximate solution is searched in the form of y n (x) = (1−x)
2
(a 0 +
a 1 x + · · · + a n x
n
), and computing the value of n = 1.
8.4 Find the approximate solution of extremal problem of the functional J [y] =
1
0 (y
2
− y
2
− 2x y)dx, and do a comparison with the accurate solution, the
boundary conditions are y(0) = 0, y(1) = 0.
Hint: The approximate solution is searched in the form of y n (x) = x(1 −
x)(a 0 + a 1 x + · · · + a n x
n
), and calculating the values of n = 0 and n = 1.
8.5 Find the approximate solution of extremal problem of the functional J [y] =
2
1
x y
2
−
x
2 −1
x
y
2
− 2x
2 y
dx, and do a comparison with the accurate solution, the boundary conditions are y(1) = 0, y(2) = 0.
Hint: Its solution can be searched in the form of y(x) = a(x − 1)(x − 2).
8.6 Using the Ritz method to find the approximate solution of the minimum
problem of the functional J [y] =
2
0 (y
2
+ y
2
+ 2x y)dx, and do a comparison with the accurate solution, the boundary conditions are y(0) = 0,
y(2) = 0.
Hint: The approximate solution can be searched in the form of y n (x) = x(2 −
x)(a 0 + a 1 x + · · · + a n x
n
), and calculating the values of n = 0 and n = 1.
8.7 Using the Ritz method to find the following boundary value problem, and
comparing with the analytical solution at x = 0.5.
y
− y = x
(0 < x < 1)
y(0) = y(1) = 0
8.8 Using the Ritz method to find the approximate solution of the following
boundary value problem
y
+ (1 + x
2
)y + 1 = 0 (−1 < x < 1)
y(1) = 0, y(−1) = 0
8.9 Using the Ritz method to find the approximate solution of the boundary value
problem
−u = cos
πx
a
(x, y ∈ D)
∂u
∂n
Γ
= 0 Γ is the boundary of D
where the domain D is the rectangular area: 0 ≤ x ≤ a, 0 ≤ y ≤ b.
8.10 As shown in Fig. 8.3, Let D be the trapezoidal domain surrounded by y = ±kx
and x = a, x = b, Γ is the boundary of D, using Kantorovich method to solve
the following definite solution problem
8 Direct Methods of Variational Problems
Hint: Its approximate solution is searched in the form of y n (x) = (1−x)
2
(a 0 +
a 1 x + · · · + a n x
n
), and computing the value of n = 1.
8.4 Find the approximate solution of extremal problem of the functional J [y] =
1
0 (y
2
− y
2
− 2x y)dx, and do a comparison with the accurate solution, the
boundary conditions are y(0) = 0, y(1) = 0.
Hint: The approximate solution is searched in the form of y n (x) = x(1 −
x)(a 0 + a 1 x + · · · + a n x
n
), and calculating the values of n = 0 and n = 1.
8.5 Find the approximate solution of extremal problem of the functional J [y] =
2
1
x y
2
−
x
2 −1
x
y
2
− 2x
2 y
dx, and do a comparison with the accurate solution, the boundary conditions are y(1) = 0, y(2) = 0.
Hint: Its solution can be searched in the form of y(x) = a(x − 1)(x − 2).
8.6 Using the Ritz method to find the approximate solution of the minimum
problem of the functional J [y] =
2
0 (y
2
+ y
2
+ 2x y)dx, and do a comparison with the accurate solution, the boundary conditions are y(0) = 0,
y(2) = 0.
Hint: The approximate solution can be searched in the form of y n (x) = x(2 −
x)(a 0 + a 1 x + · · · + a n x
n
), and calculating the values of n = 0 and n = 1.
8.7 Using the Ritz method to find the following boundary value problem, and
comparing with the analytical solution at x = 0.5.
y
− y = x
(0 < x < 1)
y(0) = y(1) = 0
8.8 Using the Ritz method to find the approximate solution of the following
boundary value problem
y
+ (1 + x
2
)y + 1 = 0 (−1 < x < 1)
y(1) = 0, y(−1) = 0
8.9 Using the Ritz method to find the approximate solution of the boundary value
problem
−u = cos
πx
a
(x, y ∈ D)
∂u
∂n
Γ
= 0 Γ is the boundary of D
where the domain D is the rectangular area: 0 ≤ x ≤ a, 0 ≤ y ≤ b.
8.10 As shown in Fig. 8.3, Let D be the trapezoidal domain surrounded by y = ±kx
and x = a, x = b, Γ is the boundary of D, using Kantorovich method to solve
the following definite solution problem