1.2. EXAMPLES 3
NOTATION. We write “PDE” as an abbreviation for both the singular
“partial differential equation” and the plural “partial differential equations”.
1.2. EXAMPLES
There is no general theory known concerning the solvability of all partial
differential equations. Such a theory is extremely unlikely to exist, given
the rich variety of physical, geometric, and probabilistic phenomena which
can be modeled by PDE. Instead, research focuses on various particular
partial differential equations that are important for applications within and
outside of mathematics, with the hope that insight from the origins of these
PDE can give clues as to their solutions.
Following is a list of many specific partial differential equations of in-
terest in current research. This listing is intended merely to familiarize the
reader with the names and forms of various famous PDE. To display most
clearly the mathematical structure of these equations, we have mostly set
relevant physical constants to unity. We will later discuss the origin and
interpretation of many of these PDE.
Throughout x U, where U is an open subset of
Rn,
and t 0. Also
Du = Dxu = (ux1 , . . . , uxn ) denotes the gradient of u with respect to the
spatial variable x = (x1, . . . , xn). The variable t always denotes time.
1.2.1. Single partial differential equations.
a. Linear equations.
1. Laplace’s equation
Δu =
n
i=1
uxixi = 0.
2. Helmholtz’s (or eigenvalue) equation
−Δu = λu.
3. Linear transport equation
ut +
n
i=1
biuxi
= 0.
4. Liouville’s equation
ut
n
i=1
(biu)xi
= 0.
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