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Background on Partial Differential Equations
There are many phenomena in nature, which, even though occurring over
finite regions of space and time, can be described in terms of
properties that prevail at each point of space and time
separately. This description originated with Newton, who with the aid
of his differential calculus showed how to grasp a global
phenomenonfor example, the elliptic orbit of a planet, by means of
a locally applied law, for example .
This approach has been extended from the motion of single point
particles to the behavior of other forms of matter and energy,
including fluids, light, heat, electricity, signals traveling along
optical fibers, neurons, and even gravitation.
This extension consists of formulating or stating a partial
differential equation governing the phenomenon, and then solving that
differential equation for the purpose of predicting measurable
properties of the phenomenon.
PDEs are classified into three categories, hyperbolic (e.g. wave)
equations, parabolic (e.g. diffusion) equations, and elliptic (e.g.
Laplace) equations, on the basis of their characteristics, or the
curves of information propagation.
Suppose there exists an unknown function of . We
can denote 's partial derivative with respect to as follows;
Then,
 Laplace Equation  An equation for unknown function , of
, , and , as follows;
Solutions to this equation, known as harmonic functions, serve as the
potentials of vector field in physics, such as the gravitational or
electrostatic fields.
A generalization of Laplace's equation is Poisson's equation:

(3) 
where
is a given function. The solutions to this equation
describe potentials of gravitational and electrostatic fields in the
presence of masses or electrical charges, respectively.
 Diffusion Equation  An equation describing the temperature in a
given region over time for unknown function , with
respect to , , and , as follows;

(4) 
Solutions will typically ``even out'' over time. The number describes
the heat conductivity of the material.
 Wave Equation  An equation for unknown function , with
respect to , , , and , where is a time variable, as follows;

(5) 
Its solutions describe waves such as sound or light waves; is a
number which represents the speed of the wave. In lower dimensions,
this equation describes the vibration of a string or drum. Solutions
will typically be combinations of oscillating sine waves.
Next: Laplace Equation Using RedBlack
Up: Partial Differential Equation
Previous: Partial Differential Equation
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Bryan Carpenter
20040609