1.2. EXAMPLES 3
NOTATION. We write “PDE” as an abbreviation for both the singular
“partial diﬀerential equation” and the plural “partial diﬀerential equations”.
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
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
uxixi = 0.
2. Helmholtz’s (or eigenvalue) equation
−Δu = λu.
3. Linear transport equation
4. Liouville’s equation