Download Wave equation

Higher Order PD
Example-1
Find the second partial derivatives of
f(x, y) = x3 + x2y3 – 2y2
Mixed Partial Derivatives
(Clairaut’s Theorem )
Clairaut’s Theorem sometimes called
Schwarz's theorem
• Partial derivatives of order 3 or higher
can also be defined. For instance,
and using Clairaut’s Theorem we can
show that fxyy = fyxy = fyyx if these
functions are continuous.
Example-2
Calculate
fxxyz
if
f(x, y, z) = sin(3x + yz).
Partial Differential Equations
• Partial derivatives occur in partial
differential equations that express certain
physical laws.
• For
the partial differential equation
2 instance,
2
u u
 2 0
2
x y
is called Laplace’s equation.
• Solutions of this equation are called
harmonic functions and play a role in
problems of heat conduction, fluid flow,
and electric potential.
Example-3
Show that the function u(x, y) = ex sin y is
a solution of Laplace’s equation:
Wave equation
u
2  u

a
2
2
t
x
2
2
describes the motion of a waveform,
which could be an ocean wave, sound
wave, light wave, or wave traveling along
a string.
A pulse traveling through a string with
fixed endpoints as modeled by the wave
equation
Spherical waves coming from a
point source
A solution to the 2D wave equation
Example-4
Verify that the function
u(x, t) = sin(x – at)
satisfies the wave equation.
Class work
Is there a case when mixed PD
are not equal?
-Yes, where the continuity condition dropped
Example: