An Introduction to Partial Differential Equations in the Undergraduate Curriculum

LECTURE 1
What is a Partial Differential Equation?

1.1. Out line of Lecture

• What is a Partial Differential Equation?
• Classifying PDE’s: Order, Linear vs . Nonlinear
• Homogeneous PDE’s and Superposition
• The Transport Equation

1.2. What is a Partial Differential Equation?

You’ve probably all seen an ordinary differential equation (ODE); for
example the pendulum equation,

describes the angle, , a pendulum makes with the vertical as a
function of time, t. Here g and L are constants (the accele ration due to
gravity and length of the pendulum respectively), t is the independent
variable and is the dependent variable. This is an ODE because
there is only one independent variable, here t which re presents time .

A partial differential equation (PDE) relates the partial derivatives
of a function of two or more independent variables together . For example,
Laplace’s equation for ,

arises in many places in mathematics and physics. For simplicity , we
will use subscript notation for partial derivatives, so this equation can
also be written .

We say a function is a solution to a PDE if it satisfy the equation
and any side conditions given. Mathematicians are often interested in
if a solution exists and when it is unique.

Exercise 1. Show that and are solutions to
Laplace’s equation (1.2). How can you combine them to create a new
solution?

Exercise 2. Show that

is a solution to the minimal surface equation,



in the region 0 < x <π , 0 < y < π. What happens on the boundary
of this region? Suppose we consider a constant multiple of Z (x, y) – is
it still a solution of the PDE?

1.3. Classifying PDE’s: Order, Linear vs. Nonlinear

When studying ODEs we classify them in an attempt to group similar
equations which might share certain properties, such as methods of
solution . We classify PDE’s in a similar way. The order of the differential
equation is the highest partial derivative that appears in the
equation. So, for example Laplace’s Equation (1.2) is second-order.

Some other examples are the convection equation for u(x, t),

which is first-order. Here C is the wave speed. The minimal surface
equation,

describes an area minimizing surface, Z(x, y), and is second-order. Finally,
the Korteweg-deVries equation (sometimes called KdV),

is a model of the amplitude of a wave, h(x, t), on the surface of a fluid
and is third-order.

We also define linear PDE’s as equations for which the dependent
variable (and its derivatives) appear in terms with degree at most one.
Anything else is called nonlinear. So, for example, the most general
first- order linear PDE for u(x, t) would be

where a, b, c and d are known functions (called coefficients).

Exercise 3. Which of Laplace’s equation (1.2), the convection equation
(1.4), the minimal surface equation (1.5) and the Korteweg-deVries
equation (1.6) are linear?

Exercise 4. Write down the most general constant coefficient linear
second-order equation for .

1.4. Homogeneous PDE’s and Superposition

Linear equations can further be classified as homogeneous for which
the dependent variable (and it derivatives) appear in terms with degree
exactly one, and non-homogeneous which may contain terms which
only depend on the independent variable. So, the convection equation



is homogeneous, but its cousin, the general first-order linear PDE for
u(x, t), is non-homogeneous



unless d(x, t) = 0.

Because partial differentiation is distributive , you can quickly convince
yourself that if two solutions, say and, satisfy a linear
homogeneous PDE, that any linear combination of them



is also a solution. So, for example, since



both satisfy Laplace’s equation,, so does any linear
combination of them



This property is extremely useful for constructing solutions which satisfy
certain initial conditions and boundary conditions.