**1.1. Generic and Standard Forms of 2 ^{nd} Order Linear PDEs.** The generic form of a
second

order linear PDE in two variables is

We shall see latter that by a suitable change of coordinates
we can cast any PDE of

the form (??) into one of the following three (standard) forms.

**(1) Parabolic Equations:**

**(2) Elliptic Equations:**

**(3) Hyperbolic Equations:**

Associated to each of these standard forms are prototypical examples, each of
which also corresponds to a

fundamental PDE occuring in physical applications. For the next few weeks we
shall discuss the solutions

or each of these equations extensively.

**1.2. The Heat Equation.**

This equation arises in studies of heat flow. For example, if a 1-dimensional
wire is heated at one end, then

the function describing the temperature of
the wire at position x and time t will satisfy (??). The

heat equation is the prototypical example of a parabolic PDE.

**1.3. Laplace's Equation.**

This equation arises in a variety of physical situations: the function
might be interpretable as the

electric potential at a point (x, y) in the plane, or the steady state
temperature of a point in the plane.

Laplace's equation is the prototypical example of an elliptic PDE.

**1.4. The Wave Equation.**

This equation governs the propagation of waves in a medium, such as the
vibrations of a taunt string,

pressure fluctuations in a compressible fluid, or electromagnetic waves. The
wave equation is the prototypical

example of a hyperbolic PDE. The coordinate transformation that casts (7) into
the form (??) is

In stark constrast to the theory of ordinary differential equations where
boundary conditions play a

relatively innocuous role in the construction of solutions, the nature of the
boundary conditions imposed on a

partial differential equation can have a dramatic effect on whether or not the
PDE/BVP (partial differential

equation / boundary value problem) is solvable.

**2.1. Cauchy Conditions.** The specification of the function and its
normal derivative along the boundary curve.

Cauchy boundary conditions are commonly applicable in dynamical situations (where the system is interpreted as evolving with respect to a time parameter t:

**2.2. Dirichlet Conditions.** The specification of the function on the
boundary curve.

As an example of a PDE/BVP with Dirichlet boundary conditions, consider the
problem of nding the

equilibrium temperature distribution of a rectangular sheet whose edges are
maintained at some prescribed

(but non-constant) temperature.

**2.3. Neumann Conditions.** The specification of the normal derivative of
the function along the

boundary curve.

As an example of a PDE/BVP with Neumann boundary conditions, consider the
problem of determining

the electric potential inside a superconducting cylinder.

In order to get a feel for the general nature of partial differential
equations, we shall now look for simple

solutions for the heat equation

We shall construct solutions of this equation by presuming the existence of
solutions of a particularly simple

(but sufficiently general) form. Our initial assumptions will be justified by
the fact that we obtain in this

manner lots of solutions.

Let us then suppose that there exist solutions of (8) of the form

where F is a function of x alone and G is a function of t alone. Substituting this ansatz for into (8) yields

or

Now this equation should hold for all x and t. However, the left hand side
depends only on t while the right

hand side depends only on x. Consequently, if we vary t but keep x fixed, we
must have
equal to the

fixed number
equals some constant, say C. Similarly, by varying x and keeping t fixed

we can conclude that
is a constant as well, say D. Equation (10) then becomes

Thus, when we presume the existence of solutions of the form (9), the
diffusion equation (8) is equivalent

to the following pair of ordinary differential equations

Therefore, if we can construct solutions G and F of the ordinary differential
equations (4.1) and (4.1), then

(9) will be a solution of the partial differential equation (8). Rewriting (4.1)
and (4.1, respectively, as

We see that both of these ordinary differential equations are linear with
constant coefficients. The general

solution of (12a) will be

and the general solution of (12b) will have the form

Thus, any function of the general form

will be solutions of (8). Note that there are 3 undetermined parameters here,
C, c_{1} and c_{2}. For fixed values

of , we obtain a two dimensional space of
solutions, since

are linearly independent. However, if then the functions
are all linearly

independent.

If we take the separation constant C = k^2, with k real, we obtain

Varying c we thus obtain two 1-parameter families of
linear independent solutions whose magnitudes grow

exponentially in time:

If we take, with λ real constant, we have

and so

and

In the second step we have used Euler's formula

to replace the exponential functions by sine and cosine functions:

Varying λ we obtain two more 1-parameter families of linearly independent solutions that decay exponentially as , and oscillate sinusoidally as one varies x.

In summary, the method of separation of variables (i.e.,
the ansatz produces four

1-parameter sets of linearly independent, real-valued solutions

Given this plethora of linearly independent solutions, it
is appropriate to ask under what additional conditions

can we expect to find a unique solution. Clearly, specifying the value of
Φ at a single point will be

insufficient. We shall see latter that in order to obtain a unique solution we
will have to specify the values

of and its partial derivatives at every point along some curve in order to
completely determine a solution.