Biomedical Engineering Reference
In-Depth Information
Backward difference formula:
h
(
u
i,j
−
u
x
(
x, y
)
≈
u
i−
1
,j
)
,
(11)
k
(
u
i,j
−
u
y
(
x, y
)
≈
u
i,j
+1
)
.
Centered-difference formula:
1
2
h
(
u
i
+1
,j
−
≈
u
x
(
x, y
)
u
i−
1
,j
)
,
(12)
1
2
k
(
u
i,j
+1
−
≈
u
y
(
x, y
)
u
i,j−
1
)
.
Similarly, the three second-order partial derivatives are given by
hk
(
u
i
+1
,j
−
2
u
i,j
+
u
i−
1
,j
)
,
u
xx
(
x, y
)
≈
(13)
hk
(
u
i,j
+1
−
2
u
i,j
+
u
,j−
1
)
,
u
yy
(
x, y
)
≈
hk
(
u
i
+1
,j
+1
−
2
u
i,j
+
u
i−
1
,j−
1
)
.
≈
u
xy
(
x, y
)
For clarity, we now demonstrate the idea of FDM using the following exam-
ples.
Example 1
As an example, we consider using FDM to solve the following differ-
ential equation:
y
(
x
)=
y
(
x
)+5
.
Applying the finite-difference scheme, we know that
y
(
x
+
h
)
−
y
(
x
)
y
(
x
+
h
)
−
y
(
x
)
y
(
x
) = lim
h→
0
≈
for h
→
0
.
(14)
h
h
Therefore, we get
y
(
x
+
h
)=
y
(
x
)+
hy
(
x
)+5
h,
(15)
and the problem can be solved iteratively. The error between the approximate
solution and the real solution is called the discretization error or truncate error,
whichwill be bigger as h increases. In practical applications, h should be chosen to
be small in order to avoid big discretization errors introduced by the approximation
of derivatives in (14).
Again, we will not provide a comprehensive description of the FDM. We refer
the readers to [23] for details.
