Chapter 6
Answers to Supplementary Problems
6.1 Answers for Problems from Chap. 1
1.1 Forward difference approximation of the first order derivative
Intermediate result:
du
dX
i
=
u i+1 − u i
X
−
1
2
d
2 u
dX 2
i
X −
1
6
d
3 u
dX 3
i
X
2
− · · ·
O((X 2 )
.
(6.1)
1.2 Centered difference approximation of the third order derivative
Starting point:
du
dX
i
=
u i+1 − u i−1
2X
.
(6.2)
Introducing in the above equation the expressions for the second order derivatives at
nodes i + 1 and i − 1, i.e.
d
2 u
dX 2
i+1
≈
u i+2 − 2u i+1 + u i
X 2
→ u i+1 ,
(6.3)
d
2 u
dX 2
i−1
≈
u i − 2u i−1 + u i−2
X 2
→ u i−1 ,
(6.4)
gives the derivative of the second order derivative, i.e. the third order derivative, as
stated in Table 1.1.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
A. Öchsner, Structural Mechanics with a Pen,
https://doi.org/10.1007/978-3-030-65892-2_6
115
Answers to Supplementary Problems
6.1 Answers for Problems from Chap. 1
1.1 Forward difference approximation of the first order derivative
Intermediate result:
du
dX
i
=
u i+1 − u i
X
−
1
2
d
2 u
dX 2
i
X −
1
6
d
3 u
dX 3
i
X
2
− · · ·
O((X 2 )
.
(6.1)
1.2 Centered difference approximation of the third order derivative
Starting point:
du
dX
i
=
u i+1 − u i−1
2X
.
(6.2)
Introducing in the above equation the expressions for the second order derivatives at
nodes i + 1 and i − 1, i.e.
d
2 u
dX 2
i+1
≈
u i+2 − 2u i+1 + u i
X 2
→ u i+1 ,
(6.3)
d
2 u
dX 2
i−1
≈
u i − 2u i−1 + u i−2
X 2
→ u i−1 ,
(6.4)
gives the derivative of the second order derivative, i.e. the third order derivative, as
stated in Table 1.1.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
A. Öchsner, Structural Mechanics with a Pen,
https://doi.org/10.1007/978-3-030-65892-2_6
115