In contrast to ODEs, PDEs are the governing equations for mathematical models in which the system has spatial dependence as well as time dependence (think of a vibrating guitar string, whose displacement depends on position, compared to an idealized point mass suspended by a spring and undergoing oscillations).

You
may
be
wondering
why
we
use
a
different
symbol
for
derivatives
(∂
instead
of
d)
when
working
with
PDEs.
The
symbol
d
indicates
an **ordinary
derivative** and
is
used
for
the
derivative
of
a
function
of
one
variable,
y
=
y(t).
The
symbol
∂
indicates
a **partial
derivative**,
and
is
used
when
differentiating
a
function
of
two
or
more
variables,
u
=
u(x,t).
For
example means
differentiate
u(x,t)
with
respect
to
t,
treating
x
as
a
constant.
Partial
derivatives
are
as
easy
as
ordinary
derivatives!

There
are **three
famous
PDEs** that
you
will
encounter
in
our
foundational
course
on
PDEs.
We
have
already
see
the **wave
equation**.
The
second
of
the
"big
three"
is
the **diffusion
equations**:

If
a
drop
of
dye
falls
into
a
container
of
clear
water
it
will
gradually
diffuse
throughout
the
container.
This
process
is
described
by
the
diffusion
equation,
with
u
=
u(x,t)
representing
the
concentration
of
dye.
This
PDE
also
describes
other
processes
of
diffusion,
for
example
the **diffusion
of
heat**.
Think
of
holding
a
metal
toasting
fork
in
a
camp
fire.
Eventually
your
hand
will
feel
the
heat.
In
this
situation
u
=
u(x,t)
represents
the
temperature
of
the
fork.

The
applications
of
the
diffusion
equation
are
not
confined
to
science,
however.
Surprisingly,
in
recent
years
the
diffusion
equation
has
also
played
an
important
role
in
the
area
of **mathematical
finance**.

The
third
member
of
the
"big
three"
is **Laplace's
equation**:

**state of equilibrium**, and here Laplace's equation comes into play.

As a simple example, think of a square sheet of metal, with three edges in contact with ice (0° Celsius) and the fourth edge in contact with steam (100° C). Heat will diffuse through the sheet, but eventually the temperature of the sheet will reach a steady state u(x,y), that depends on position (x,y) but not on time. This function will satisfy Laplace's equation. What would you expect the temperature to be at the center of the plate in this example?

The preceding examples merely illustrate the "tip of the iceberg" as regards the subject of PDEs. Mathematical models based on partial differential equations (PDEs) are ubiquitous these days, arising in all areas of science and engineering, and also in finance and economics.