1.1 Basic Concepts
1.1.1 The prevalence of partial differential equations
Partial differential equations are equations involving derivatives
with respect to more than one independent variable.
Partial differential equations are the basic equations in many areas
of science and engineering. Some examples:
- Fluid mechanics
- The basic equations that govern the inviscid
flow of simple idealized substances are called the Euler equations.
The basic equations that govern the viscous flow of simple
substances like air and water under normal conditions are called the
Navier-Stokes equations. Both are partial differential equations.
Nonlinear ones, unfortunately. However, they do become linear in
many special cases of great interest.
- Heat transfer
- The equations of heat conduction and convection
are partial differential equations. Radiation may be described by
Maxwell’s equations, which are also partial differential
equations. However, often radiation can be more simply described by
so-called boundary integral methods. In the most basic cases, the
partial differential equations describing convection are linear.
- Solid mechanics
- The equations that govern simple solids are
partial differential equations. Often the interest is in steady
problems. The equations for simple relatively stiff solids are
- The dynamics of flexible solids is governed by
partial differential equations. The equations for simple relatively
stiff solids are linear.
- Electromagnetics and optics
- The Maxwell equations that govern
basic electromagnetic phenomena are partial differential equations.
They are linear in vacuum. Simplified partial differential
equations govern the special cases of electrostatics and
- Many geometrical issues such as minimal surfaces and
developable surfaces are governed by partial differential equations.
The equation for minimal surfaces may be reduced to a very simple
That did not even touch on such areas as biology and economics that
are also awash in partial differential equations.
Here is a list of some basic definitions used for partial differential
- Partial differential equations
- are equations that involve
derivatives with respect to more than one independent variable. The
simplest partial differential equation that you can write is:
In that case would be unknown, or dependent variable. That
is the function you want to find. The independent variables would
be and . People usually think of a spatial coordinate when
they use , and time when they use .
It may be noted that equations that involve only derivatives with
respect to one independent variable are called “ordinary
differential equations.” Equations that also involve
integrals are called “integro-differential equations.”
- Partial derivatives
- are derivatives with respect to one
independent variable, keeping the other variables constant. The
figure below illustrates the definition of partial derivatives:
A simple numerical approximation of a partial derivative will just
eliminate the limit process. In that case and
are merely taken to be small compared to the typical
scale of the problem.
- The order of a partial differential equation is
the order of the highest derivative. Generally speaking,
the highest derivatives are most responsible for the nature
of the solutions.
- The degree of a partial differential equation is the
highest power to which the dependent variable appears in the
Consider for example the Burgers’ equation
It is the simplest nonlinear model for the equations of fluid
mechanics, and for other systems involving shock formation. It is
of order 1 and degree 2.
- partial differential equations are equations of first
degree. The terms that are of the first degree in the unknown are
called the homogeneous part. The remaining terms that do not
involve the unknown are called the inhomogeneous part.
Consider the following Poisson equation governing steady heat
conduction in a plate with external heat addition:
Here is the temperature, the heat conduction
coefficient, and the right hand side represents the external heat
added per unit area. This equation is first degree (in .) The
left hand side, linear in is the homogeneous part and the right
hand side, independent of , is the inhomogenous part.
The following equation describes steady heat conduction in a plate
without external heat addition, but with a heat conduction coefficient
that depends on temperature:
This equation is nonlinear. It is of infinite degree, not second
degree, because the Taylor series of intruduces all powers
of . However, this equation is still linear in terms of the
highest, second order, derivatives and . (Not
counting lower order derivatives.) Therefore it is called
- The domain
- is usually taken to be the spatial region
in which the partial differential equation applies. For unsteady
heat conduction in a sphere, the domain is the sphere and
time is an additional coordinate.
- The boundary
- is where the domain stops.
Boundary points are immediately adjacent to both points inside the
domain and points outside it. The boundary is also often indicated
by (for surface). Typically, indicates the domain
without the boundary points and
including the boundary points.
For unsteady heat conduction in a sphere of radius , the boundary
is all points with spherical coordinate equal to . It is the
surface of the sphere. For this example is all points
is all points .
- Boundary conditions
- are conditions on the solution that apply at
points on the boundary.
- Initial conditions
- are conditions on the solution that apply at
the starting time. Almost always, the starting time is taken to be
the zero of time.
- You cannot do much in partial differential
equations without having to deal with singularities. There might be
singularities in the solution, in the initial and boundary
conditions, or in the shape of the boundary. Some typical
singularities, in a rough order from relatively mild to more severe,
- Locations where all derivatives exist, but the function does
not have a Taylor series with a nonzero radius of convergence.
For example, that happens for the function at .
- Locations where higher order derivatives have singularities.
For example, the function has an infinite second order
derivative at =0. That means it has infinite radius of
curvature at . In general, the lower the order of the
derivative involved, the stronger the singularity.
- Kinks: Locations where the first derivative jumps from
one finite value to another. For example, the function
has a kink at .
- Cusps: where the function itself is still continuous,
but the derivative jumps from to or vice-versa.
For example, has a cusp at
- Jumps: where the function jumps from one finite value to
another. For example, the Heaviside function , which is 0
for negative and 1 for positive , has a jump at .
- Spikes: where the function is infinite at a single
point. An important example is the Dirac delta function
, which is the derivative of the Heaviside function.
The delta function is a single infinite spike at and the
area under the spike is 1. If you take derivatives of the delta
function, you get increasingly singular functions. For example,
the first derivative is the more singular “dipole.”
- Poles: where the function goes to infinity. For
example, the function has a “simple pole” at
. The higher the negative power of , the more singular
the function. For example is more singular than .
1.1.3 Typical boundary conditions
To get a meaningful solution to a partial differential equation, you
will need initial and/or boundary conditions. Initial conditions
are normally straightforward. Boundary conditions vary a lot,
There are some very simple types of boundary condition that you must
know by heart:
- The value of the unknown itself is given on the
Here is some given function of the position on the surface.
- The derivative
of the unknown
in the direction normal to the boundary is given. Here is the
coordinate normal to the boundary:
If the boundary is oblique, the following formula can be used to
relate the derivative in the direction normal to the boundary to the
partial derivatives of in Cartesian coordinates:
Here is a unit vector that is normal to the boundary at
the considered point.
Note that if itself is prescribed on the boundary, it already
implies the value of derivative along the boundary. That is
why only the derivative normal to the boundary is included in this
list of the most basic boundary conditions.
- A linear combination of and
is given on the boundary. This is also called a
“radiation” or “Robin” boundary
Here , , and are all given functions of the
position on the surface. Such boundary conditions are often used in
simple wave propagation problems to indicate that no waves enter the
domain from outside.