8
1 Idea and Derivation of the Method
Fig. 1.5 Typical subregion for the collocation by subregion method and appropriate weight function, adapted from [3]
−
d
2 u
dx 2 W dX = 0 .
(1.23)
Integrating by parts gives the following expression
−
d
2 u
dX 2 W dX =
du
dX
W
−
−
−
du
dX
dW
dX
dX = 0 ,
(1.24)
or rearranged as:
−
du
dX
dW
dX
dX =
du
dX
W
−
.
(1.25)
It holds for the step function that W = 1 and
dW
dX
= 0 and the last equation can be
simplified to:
0 =
du
dX
+
−
du
dX
−
.
(1.26)
Replacing the differentials ’d’ by incremental differences, Eq. (1.26) can be written
as
1 Idea and Derivation of the Method
Fig. 1.5 Typical subregion for the collocation by subregion method and appropriate weight function, adapted from [3]
−
d
2 u
dx 2 W dX = 0 .
(1.23)
Integrating by parts gives the following expression
−
d
2 u
dX 2 W dX =
du
dX
W
−
−
−
du
dX
dW
dX
dX = 0 ,
(1.24)
or rearranged as:
−
du
dX
dW
dX
dX =
du
dX
W
−
.
(1.25)
It holds for the step function that W = 1 and
dW
dX
= 0 and the last equation can be
simplified to:
0 =
du
dX
+
−
du
dX
−
.
(1.26)
Replacing the differentials ’d’ by incremental differences, Eq. (1.26) can be written
as