Graphics Programs Reference
In-Depth Information
plot(xa(:, 1), xa(:, 2))
end
hold off
10
8
6
4
2
0
−
2
−
4
−
6
−
2
0
2
4
6
8
10
12
14
16
We conclude that the minimum velocity needed is somewhere between 7.25
and 7.3.
Numerical Solution of the
Heat Equation
In this section we will use MATLAB to numerically solve the
heat equation
(also known as the
diffusion equation
), a partial differential equation that
describes many physical processes including conductive heat flow or the
diffusion of an impurity in a motionless fluid. You can picture the process of
diffusion as a drop of dye spreading in a glass of water. (To a certain extent
you could also picture cream in a cup of coffee, but in that case the mixing is
generally complicated by the fluid motion caused by pouring the cream into
the coffee and is further accelerated by stirring the coffee.) The dye consists
of a large number of individual particles, eachof whichrepeatedly bounces
off of the surrounding water molecules, following an essentially random
path. There are so many dye particles that their individual random motions
form an essentially deterministic overall pattern as the dye spreads evenly
in all directions (we ignore here the possible effect of gravity). In a similar
way, you can imagine heat energy spreading through random interactions of
nearby particles.
In a three-dimensional medium, the heat equation is
∂
t
=
k
∂
2
u
2
u
2
u
∂
z
2
∂
u
∂
x
2
+
∂
∂
y
2
+
∂
.
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