Graphics Programs Reference
In-Depth Information
EXAMPLE 8.6
Write out Eqs. (8.11)for the following linear boundary value problemusing
n
=
11:
y
=−
y
(
4
y
+
4
x
y
(0)
=
0
π/
2)
=
0
Solve these equations with a computerprogram.
Solution
In thiscase
α
=
0 applicable
to
y
),
β
=
0 applicable
to
y
) and
y
)
f
(
x
,
y
,
=−
4
y
+
4
x
. Hence Eqs. (8.11) are
y
1
=
0
h
2
(
y
i
−
1
−
2
y
i
+
y
i
+
1
−
−
4
y
i
+
4
x
i
)
=
0,
i
=
2
,
3
,...,
10
h
2
(
2
y
10
−
2
y
11
−
−
4
y
11
+
4
x
11
)
=
0
or, using matrix notation
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
1
0
y
1
y
2
.
y
10
y
11
0
4
h
2
x
2
.
4
h
2
x
10
4
h
2
x
11
4
h
2
1
−
2
+
1
.
.
.
.
.
.
.
.
.
=
1
−
2
+
4
h
2
1
4
h
2
2
−
2
+
Note that the coefficient matrix istridiagonal, so that the equationscan be solved
efficientlybythe functions
LUdec3
and
LUsol3
describedinArt.2.4.Recalling that
these functions store the diagonals of the coefficient matrix in vectors
c
,
d
and
e
, we
arrive at the following program:
function fDiff6
% Finite difference method for the second-order,
% linear boundary value problem in Example 8.6.
xStart = 0; xStop = pi/2;
% Range of integration.
n=11;
%Numberofmeshpoints.
freq = 1;
% Printout frequency.
h = (xStop - xStart)/(n-1);
x = linspace(xStart,xStop,n)';
[c,d,e,b] = fDiffEqs(x,h,n);
[c,d,e] = LUdec3(c,d,e);
printSol(x,LUsol3(c,d,e,b),freq)
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