Math 124A – September 28, 2009
Viktor Grigoryan
1
Introduction
Recall that an ordinary differential equation (ODE) contains an independent variable
x
and a dependent
variable
u
, which is the unknown in the equation. The defining property of an ODE is that derivatives
of the unknown function
u
0
=
du
dx
enter the equation. Thus, an equation that relates the independent
variable
x
, the dependent variable
u
and derivatives of
u
is called an
ordinary differential equation
. Some
examples of ODEs are:
u
0
(
x
) =
u
u
00
+ 2
xu
=
e
x
u
00
+
x
(
u
0
)
2
+ sin
u
= ln
x
In general, and ODE can be written as
F
(
x, u, u
0
, u
00
, . . .
) = 0.
In contrast to ODEs, a partial differential equation (PDE) contains partial derivatives of the depen
dent variable, which is an unknown function in more than one variable
x, y, . . .
. Denoting the partial
derivative of
∂u
∂x
=
u
x
, and
∂u
∂y
=
u
y
, we can write the general first order PDE for
u
(
x, y
) as
F
(
x, y, u
(
x, y
)
, u
x
(
x, y
)
, u
y
(
x, y
)) =
F
(
x, y, u, u
x
, u
y
) = 0
.
(1)
Although one can study PDEs with as many independent variables as one wishes, we will be primarily
concerned with PDEs in two independent variables. A
solution
to the PDE (1) is a function
u
(
x, y
) which
satisfies (1) for all values of the variables
x
and
y
. Some examples of PDEs (of physical significance) are:
u
x
+
u
y
= 0
transport equation
(2)
u
t
+
uu
x
= 0
inviscid Burger’s equation
(3)
u
xx
+
u
yy
= 0
Laplace’s equation
(4)
u
tt

u
xx
= 0
wave equation
(5)
u
t

u
xx
= 0
heat equation
(6)
u
t
+
uu
x
+
u
xxx
= 0
KdV equation
(7)
iu
t

u
xx
= 0
Shr¨
odinger’s equation
(8)
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 Fall '08
 Ponce
 Math, Differential Equations, Equations, Derivative, Partial Differential Equations, Ode, Partial differential equation, PDEs

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