Other numerical methods 263
where f is an unknown function and the prime denotes the order of differentiation. Then the relationship, whenever it exists, between the functional
and its Euler equation is given as
I f
F x f f f dx stationary
F
d
dx
F
d
D
f
f
[ ]
( , , , )
()
=
′ ′′
=
⇔ −
+
∫
′
2 2
2
0
dx
F f
( )
′′ =
(9.33)
where F
F
f
f
n
n =
∂
∂
and n is the order of differentiation.
Example 9.2
Let us consider the functional
I f
F f f dx
df
dx
gf dx
D
D
[ ]
( , )
=
′
=
+
∫
∫
1
2
2
(9.34)
where g is a known function of x. The function that makes the functional stationary must satisfy the Euler equation
d f
dx
g x
2
2
0
−
=
( )
(9.35)
Let g(x) = –x and the boundary conditions for Equation 9.35 are taken
as f(0) = f(1) = 0, (0 ≤ x ≤ 1). An approximate solution of the problem
can be accomplished by assuming a third-order power series function.
Then, the approximate function that also satisfies the boundary conditions reads
f x x
x
=
−
+
(
)(
)
1 1
2
α α
(9.36)
Substituting Equation 9.36 into Equation 9.34 and integrating from 0
to 1 yields
I[ , ]
α α
α
α
α α
α
α
1
2
1
2
2
2
1 2
1
2
6 15
6
12 20
=
+
+
+
+
(9.37)
A stationary solution of Equation 9.34 can be obtained according
to the conditions
∂
∂
=
+
+
=
I
α
α
α
1
1
2
2
6
6
1
12
0
(9.38)
where f is an unknown function and the prime denotes the order of differentiation. Then the relationship, whenever it exists, between the functional
and its Euler equation is given as
I f
F x f f f dx stationary
F
d
dx
F
d
D
f
f
[ ]
( , , , )
()
=
′ ′′
=
⇔ −
+
∫
′
2 2
2
0
dx
F f
( )
′′ =
(9.33)
where F
F
f
f
n
n =
∂
∂
and n is the order of differentiation.
Example 9.2
Let us consider the functional
I f
F f f dx
df
dx
gf dx
D
D
[ ]
( , )
=
′
=
+
∫
∫
1
2
2
(9.34)
where g is a known function of x. The function that makes the functional stationary must satisfy the Euler equation
d f
dx
g x
2
2
0
−
=
( )
(9.35)
Let g(x) = –x and the boundary conditions for Equation 9.35 are taken
as f(0) = f(1) = 0, (0 ≤ x ≤ 1). An approximate solution of the problem
can be accomplished by assuming a third-order power series function.
Then, the approximate function that also satisfies the boundary conditions reads
f x x
x
=
−
+
(
)(
)
1 1
2
α α
(9.36)
Substituting Equation 9.36 into Equation 9.34 and integrating from 0
to 1 yields
I[ , ]
α α
α
α
α α
α
α
1
2
1
2
2
2
1 2
1
2
6 15
6
12 20
=
+
+
+
+
(9.37)
A stationary solution of Equation 9.34 can be obtained according
to the conditions
∂
∂
=
+
+
=
I
α
α
α
1
1
2
2
6
6
1
12
0
(9.38)