Biomedical Engineering Reference
In-Depth Information
the variational problem for this type of function is solved by the following Euler-
Lagrange differential equation:
∂g
∂f
∂g
∂f
−
d
dx
=0
.
(7)
For a more general case with a functional defined as
b
g
(
x, f
(
x
)
,f
(
x
)
,f
(
x
)
, ..., f
(
n
)
(
x
))
dx,
I
(
f
(
x
)) =
(8)
a
the variational problem is solved by the following Euler-Lagrange differential
equation:
∂g
∂f
∂g
∂f
∂g
∂f
(
n
)
d
2
dx
2
d
n
dx
n
∂g
∂f
−
d
dx
n
+
−
...
+(
−
1)
=0
.
(9)
The formal proof of the Euler-Lagrange equation is beyond the scope of this
chapter. We refer the reader to [22] for further details.
2.2. Finite-Difference Method (FDM)
The-finite difference method (FDM) consists of two steps: (1) replacing the
(partial) derivatives by some numerical differentiation formulas to get a difference
equation, in other words, derivatives are discretized by using the “difference”; and
(2) solving the derived difference equation by using either an iterative or a direct
method.
We first partition the domain Ω by a mesh grid. For example, we use a uniform
mesh grid with grid lines
x
j
=
x
0
+
jh, j
=0
,
1
, ..., J,
y
l
=
y
0
+
lk, l
=0
,
1
, ..., L,
where
h
=
x
i
+1
−
y
i
, are the mesh size in the
x
and
y
directions,
respectively. For simplicity, we write
f
j,l
=
f
(
x
j
,y
l
), where the function values
are the nodes of the mesh.
Using Taylor expansion and the intermediate value theorem, we can derive
the following numerical differentiation formulas:
x
i
, and
k
=
y
i
+1
−
Forward difference formula:
h
(
u
i
+1
,j
−
u
x
(
x, y
)
≈
u
i,j
)
,
(10)
k
(
u
i,j
+1
−
u
y
(
x, y
)
≈
u
i,j
)
.
