Partial Differential Equation (PDE)

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Partial Differential Equation (PDE)
An ordinary differential equation is a
differential equation that has only one
independent variable. For example, the angular
position of a swinging pendulum as a function of
time qq(t). However, most physical systems
cannot be modeled by an ordinary differential
equation because they usually depends on more
than one variables. A differential equation
involving more than one independent variables is
called a partial differential equation. For
example, the equations governing tidal waves
should deal with the description of wave
propagation varying both in time and space.
WfrontWfront(x,y,z,t).
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The Wave Equation
Mechanical vibrations of a guitar string, or in
the membrane of a drum, or a cantilever beam are
governed by a partial differential equation,
called wave equation, since they deal with
variations taking place both in time and space
taking a form of wave propagation. To derive the
wave equation we consider an elastic string
vibrating in a plane, as the string on a guitar.
Assume u(x,t) is the displacement of the string
away from its equilibrium position u0. We can
derive a partial differential equation governing
the behavior of u(x,t) by applying the Newtons
second law and several simple assumptions (see
chapter 11.2 in textbook)
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u
u(x,t)
x
x
xDx
T(xDx,t)
b
a
T(x,t)
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