Civil Engineering Reference
In-Depth Information
Evaluate the successive derivatives as follows:
()
===∴ =
′ ==−=−∴ =−
−
0
f
0
b
e
1
b
1
0
0
−
0
f
()
0
b
e
1
b
1
1
1
1
2
−
0
f
′′ ===∴
()
02 1
b
e
b
=
2
2
1
3
f
′′′
= −=−∴ =−
() !
03
b
e
−
0
1
b
3
3
!
1
4
−
0
f
′′ ′′
() !
04 1
=
b
= =∴ =
e
b
4
4
!
1
5
−
0
f
′′′ ′′ = −=−∴ =−
() !
05
b
e
1
b
5
5
!
The polynomial can then be written with the coefficients.
1
2
1
3
1
4
1
5
==−+ −
−
2
3
4
5
ye x
1
x
x
+
x
−
x
!
!
!
!
The value of the
e
-
1
can be evaluated to check the accuracy of the
approximation.
11
1
2
1
3
1
4
1
5
==−+
()
−
()
+
()
−
()
2
3
4
5
ye
−
1
1
1
1
!
!
!
!
y
=−+−
1105 0 16666
.
.
770041667 0 008333
+
.
−
.
y
=
0 366667
.
The exact solution to the same accuracy is
y =
0
.
367879. If the polynomial
was calculated up to the
x
9
term, the value would be
y =
0
.
3678791
versus
the exact value of
y =
0
.
3678794.
3.8
DiffEREncE OPERAtORS bY tAYLOR
SERiES EXPAnSiOn
The numerical differential equation relationships can be found using the
Taylor series expansion. Expanding the Taylor series for a function
y=f
(
x
)
at
x=
(
x
i
+h
) gives the following equation:
2
3
yx hyyh
yh yh
i
′′
′′′
(
)
=+′ +
i
i
+
+
+
(3.6)
i
i
2
!
3
!
