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8.2
Partial Differential Equations
Plate, shell, and continuum problems can be governed by partial differential equations in
two or more independent variables. The problem of obtaining an approximate solution
using finite differences to such partial differential equations is similar in principle to the
one-dimensional problems considered heretofore. Suppose a problem is to be solved over
a rectangular
xy
region. Divide this surface into grids using equally spaced lines parallel
to the
x
axis and similar lines parallel to the
y
axis as shown in Fig. 8.8. The value of the
dependent variable
u
in a partial differential equation is to be calculated at the node points
x
i
,y
j
of this mesh, i.e.,
u
i, j
=
is to be determined. To accomplish this, we replace
the partial derivatives at
x
i
,y
j
with difference quotients and solve the resulting system of
algebraic equations.
u
(
x
i
,y
j
)
8.2.1 Difference Formulas for Partial Derivatives
Two-dimensional finite difference quotients can be built using one-dimensional expres-
sions. From Table 8.1, the simple central difference formula for
2
u
x
2
would be
∂
/∂
=
1
h
x
[
u
i
+
1
,j
−
=
1
h
x
1
2
u
x
2
∂
/∂
/
2
u
i, j
+
u
i
−
1
,j
]
/
−
2
1
u
(8.22a)
∂
/∂
while the expression for
u
y
is
1
0
−
∂
/∂
=
(
/
)
−
=
(
/
)
u
y
1
2
h
y
[
u
i, j
+
1
u
i, j
−
1
]
1
2
h
y
u
(8.22b)
1
where the
u
on the right-hand side symbolically represents appropriate nodal values of
u
.
FIGURE 8.8
Notation for finite differences in two dimensions.
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