A
differential equation is a
mathematical equation
for an unknown
function of
one or several
variables that
relates the values of the function itself and its
derivatives of various orders. Differential
equations play a prominent role in
engineering,
physics,
economics and other disciplines.
Differential equations arise in many areas of science and
technology: whenever a
deterministic
relationship involving some continuously varying quantities
(modelled by functions) and their rates of change in space and/or
time (expressed as derivatives) is known or postulated. This is
illustrated in
classical
mechanics, where the motion of a body is described by its
position and velocity as the time varies.
Newton's Laws allow one to relate the
position, velocity, acceleration and various forces acting on the
body and state this relation as a differential equation for the
unknown position of the body as a function of time. In some cases,
this differential equation (called an
equation of motion) may be solved
explicitly.
An example of modelling a real world problem using differential
equations is determination of the velocity of a ball falling
through the air, considering only gravity and air resistance. The
ball's acceleration towards the ground is the acceleration due to
gravity minus the deceleration due to air resistance. Gravity is
constant but air resistance may be modelled as proportional to the
ball's velocity. This means the ball's acceleration, which is the
derivative of its velocity, depends on the velocity. Finding the
velocity as a function of time requires solving a differential
equation.
Differential equations are mathematically studied from several
different perspectives, mostly concerned with their
solutions, the set of functions that satisfy the
equation. Only the simplest differential equations admit solutions
given by explicit formulas; however, some properties of solutions
of a given differential equation may be determined without finding
their exact form. If a self-contained formula for the solution is
not available, the solution may be numerically approximated using
computers. The theory of
dynamical
systems puts emphasis on qualitative analysis of systems
described by differential equations, while many
numerical methods have been developed to
determine solutions with a given degree of accuracy.
Directions of study
The study of differential equations is a wide field in
pure and
applied mathematics,
physics, and
engineering.
All of these disciplines are concerned with the properties of
differential equations of various types. Pure mathematics focuses
on the existence and uniqueness of solutions, while applied
mathematics emphasizes the rigorous justification of the methods
for approximating solutions. Differential equations play an
important role in modelling virtually every physical, technical, or
biological process, from celestial motion, to bridge design, to
interactions between neurons. Differential equations such as those
used to solve real-life problems may not necessarily be directly
solvable, i.e. do not have
closed
form solutions. Instead, solutions can be approximated using
numerical
methods.
Mathematicians also study
weak
solutions (relying on
weak
derivatives), which are types of solutions that do not have to
be differentiable everywhere. This extension is often necessary for
solutions to exist, and it also results in more physically
reasonable properties of solutions, such as possible presence of
shocks for equations of hyperbolic type.
The study of the stability of solutions of differential equations
is known as
stability theory.
Nomenclature
The theory of differential equations is quite developed and the
methodsused to study them vary significantly with the type of the
equation.
- An ordinary
differential equation (ODE) is a differential equation in which
the unknown function (also known as the dependent
variable) is a function of a single independent
variable. In the simplest form, the unknown function is a real or
complex valued function, but more generally, it may be vector-valued or matrix-valued: this corresponds to
considering a system of ordinary differential equations for a
single function. Ordinary differential equations are further
classified according to the order of the highest
derivative with respect to the dependent variable appearing in the
equation. The most important cases for applications are first order
and second order differential equations. In the classical
literature also distinction is made between differential equations
explicitly solved with respect to the highest derivative and
differential equations in an implicit form.
- A partial differential
equation (PDE) is a differential equation in which the unknown
function is a function of multiple independent variables
and the equation involves its partial derivatives. The order is
defined similarly to the case of ordinary differential equations,
but further classification into elliptic, hyperbolic, and parabolic
equations, especially for second order linear equations, is of
utmost importance. Some partial differential equations do not fall
into any of these categories over the whole domain of the
independent variables and they are said to be of mixed
type.
Both ordinary and partial differential equations are broadly
classified as
linear and
nonlinear. A differential equation is
linear if the unknown function and its derivatives
appear to the power 1 (products are not allowed) and
nonlinear otherwise. The characteristic property
of linear equations is that their solutions form an affine subspace
of an appropriate function space, which results in much more
developed theory of linear differential equations.
Homogeneous linear differential equations are a
further subclass for which the space of solutions is a linear
subspace i.e. the sum of any set of solutions or multiples of
solutions is also a solution. The coefficients of the unknown
function and its derivatives in a linear differential equation are
allowed to be (known) functions of the independent variable or
variables; if these coefficients are constants then one speaks of a
constant coefficient linear differential
equation.
There are very few methods of explicitly solving nonlinear
differential equations; those that are known typically depend on
the equation having particular
symmetries. Nonlinear differential equations can
exhibit very complicated behavior over extended time intervals,
characteristic of
chaos. Even the
fundamental questions of existence, uniqueness, and extendability
of solutions for nonlinear differential equations, and
well-posedness of initial and boundary value problems for nonlinear
PDEs are hard problems and their resolution in special cases is
considered to be a significant advance in the mathematical theory
(cf
Navier–Stokes
existence and smoothness).
Linear differential equations frequently appear as
approximations to nonlinear equations. These
approximations are only valid under restricted conditions. For
example, the harmonic oscillator equation is an approximation to
the nonlinear pendulum equation that is valid for small amplitude
oscillations (see below).
Examples
In the first group of examples, let
u be an unknown
function of
x, and
c and
ω are known
constants.
- Inhomogeneous first order linear constant coefficient ordinary
differential equation:
- \frac{du}{dx} = cu+x^2.
- Homogeneous second order linear ordinary differential
equation:
- \frac{d^2u}{dx^2} - x\frac{du}{dx} + u = 0.
- Homogeneous second order constant coefficient linear ordinary
differential equation describing the harmonic oscillator:
- \frac{d^2u}{dx^2} + \omega^2u = 0.
- First order nonlinear ordinary differential equation:
- \frac{du}{dx} = u^2 + 1.
- Second order nonlinear ordinary differential equation
describing the motion of a pendulum of
length L:
- g\frac{d^2u}{dx^2} + L\sin u = 0.
In the next group of examples, the unknown function
u
depends on two variables
x and
t or
x
and
y.
- Homogeneous first order linear partial differential
equation:
- \frac{\partial u}{\partial t} + t\frac{\partial u}{\partial x}
= 0.
- Homogeneous second order linear constant coefficient partial
differential equation of elliptic type, the Laplace equation:
- \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2
u}{\partial y^2} = 0.
- \frac{\partial u}{\partial t} = 6u\frac{\partial u}{\partial x}
- \frac{\partial^3 u}{\partial x^3}.
Related concepts
- A delay differential
equation (DDE) is an equation for a function of a single
variable, usually called time, in which the
derivative of the function at a certain time is given in terms of
the values of the function at earlier times.
Connection to difference equations
The theory of differential equations is closely related to the
theory of
difference equations,
in which the coordinates assume only discrete values, and the
relationship involves values of the unknown function or functions
and values at nearby coordinates. Many methods to compute numerical
solutions of differential equations or study the properties of
differential equations involve approximation of the solution of a
differential equation by the solution of a corresponding difference
equation.
Universality of mathematical description
Many fundamental laws of
physics and
chemistry can be formulated as
differential equations. In
biology and
economics differential equations are used
to
model the behavior of
complex systems. The mathematical theory of differential equations
first developed, together with the sciences, where the equations
had originated and where the results found application. However,
diverse problems, sometimes originating in quite distinct
scientific fields, may give rise to identical differential
equations. Whenever this happens, mathematical theory behind the
equations can be viewed as a unifying principle behind diverse
phenomena. As an example, consider propagation of light and sound
in the atmosphere, and of waves on the surface of a pond. All of
them may be described by the same second order
partial differential equation,
the
wave equation, which allows us to
think of light and sound as forms of waves, much like familiar
waves in the water. Conduction of heat, whose theory was developed
by
Joseph Fourier, is governed by
another second order partial differential equation, the
heat equation. It turned out that many
diffusion processes, while seemingly
different, are described by the same equation;
Black-Scholes equation in finance is for
instance, related to the heat equation.
Notable differential equations
Biology
Economics
See also
References
- D. Zwillinger, Handbook of Differential Equations (3rd
edition), Academic Press, Boston, 1997.
- A. D. Polyanin and V. F. Zaitsev, Handbook of Exact
Solutions for Ordinary Differential Equations (2nd edition),
Chapman & Hall/CRC Press, Boca Raton, 2003. ISBN
1-58488-297-2.
- W. Johnson, A Treatise on Ordinary and Partial Differential
Equations, John Wiley and Sons, 1913, in University of
Michigan Historical Math Collection
- E.L. Ince, Ordinary Differential Equations, Dover
Publications, 1956
- E.A. Coddington and N. Levinson, Theory of Ordinary
Differential Equations, McGraw-Hill, 1955
- P. Blanchard, R.L. Devaney, G.R. Hall, Differential
Equations, Thompson, 2006
External links