468
8 Direct Methods of Variational Problems
the boundary condition is
u(S) = f (S) (the boundary of S ∈ Ω)
(8.4.2)
The general steps of solving the variational problem of the functional by the
Kantorovich method is:
(1) Properly choosing the coordinate function sequence
ϕ 1 (x 1 , x 2 , . . . , x n ), ϕ 2 (x 1 , x 2 , . . . , x n ), . . . , ϕ m (x 1 , x 2 , . . . , x n ), . . .
Making the function
u m =
m
k=1
a k (x i )ϕ k (x 1 , x 2 , . . . , x n )
(8.4.3)
where, a k (x i ) is the function of a independent variable.
(2) Substituting u m into the functional J, such that the functional J is converted
into the functional J with the function a 1 (x i ), a 2 (x i ), …, a m (x i ), and the choice
of a 1 (x i ), a 2 (x i ), …, a m (x i ) should make the functional J reach an extremum,
thus the m-th approximate solution of the original functional J is obtained.
(3) Putting m → ∞, if the limit
u(x 1 , x 2 , . . . , x n ) =
∞
k=1
a k (x i )ϕ k (x 1 , x 2 , . . . , x n )
(8.4.4)
exists, then It is the accurate solution of the variational problem.
The difference of the Kantorovich method and Ritz method is that a k in the Ritz
method are undetermined constants, but a k (x i ) in the Kantorovich method are the
functions of a independent variable. In general, under the conditions of the same
coordinate functions and the same degree of approximate solutions, the approximate
solution obtained by the Kantorovich method is more accurately than the approximate
solution obtained by the Ritz method, it is that the function class of coefficients with
the variables a k (x i ) is wider than the function class of coefficients with the constants
a k in the choice of function range.
Example 8.4.1 Find the first approximate solution of the extremal problem of the
functional
J [u(x, y)] =
a
−a
b
−b
(u
2
x + u
2
y − 2u)dxdy
8 Direct Methods of Variational Problems
the boundary condition is
u(S) = f (S) (the boundary of S ∈ Ω)
(8.4.2)
The general steps of solving the variational problem of the functional by the
Kantorovich method is:
(1) Properly choosing the coordinate function sequence
ϕ 1 (x 1 , x 2 , . . . , x n ), ϕ 2 (x 1 , x 2 , . . . , x n ), . . . , ϕ m (x 1 , x 2 , . . . , x n ), . . .
Making the function
u m =
m
k=1
a k (x i )ϕ k (x 1 , x 2 , . . . , x n )
(8.4.3)
where, a k (x i ) is the function of a independent variable.
(2) Substituting u m into the functional J, such that the functional J is converted
into the functional J with the function a 1 (x i ), a 2 (x i ), …, a m (x i ), and the choice
of a 1 (x i ), a 2 (x i ), …, a m (x i ) should make the functional J reach an extremum,
thus the m-th approximate solution of the original functional J is obtained.
(3) Putting m → ∞, if the limit
u(x 1 , x 2 , . . . , x n ) =
∞
k=1
a k (x i )ϕ k (x 1 , x 2 , . . . , x n )
(8.4.4)
exists, then It is the accurate solution of the variational problem.
The difference of the Kantorovich method and Ritz method is that a k in the Ritz
method are undetermined constants, but a k (x i ) in the Kantorovich method are the
functions of a independent variable. In general, under the conditions of the same
coordinate functions and the same degree of approximate solutions, the approximate
solution obtained by the Kantorovich method is more accurately than the approximate
solution obtained by the Ritz method, it is that the function class of coefficients with
the variables a k (x i ) is wider than the function class of coefficients with the constants
a k in the choice of function range.
Example 8.4.1 Find the first approximate solution of the extremal problem of the
functional
J [u(x, y)] =
a
−a
b
−b
(u
2
x + u
2
y − 2u)dxdy