1.4. OVERVIEW 9
to deriving formulas for these solutions. This may seem wasted or misguided
effort, but in fact mathematicians are like theologians: we regard existence
as the prime attribute of what we study. But unlike theologians, we need
not always rely upon faith alone.
1.3.3. Typical difficulties.
Following are some vague but general principles, which may be useful to
keep in mind:
(i) Nonlinear equations are more difficult than linear equations; and,
indeed, the more the nonlinearity affects the higher derivatives, the
more difficult the PDE is.
(ii) Higher-order PDE are more difficult than lower-order PDE.
(iii) Systems are harder than single equations.
(iv) Partial differential equations entailing many independent variables
are harder than PDE entailing few independent variables.
(v) For most partial differential equations it is not possible to write out
explicit formulas for solutions.
None of these assertions is without important exceptions.
1.4. OVERVIEW
This textbook is divided into three major Parts.
PART I: Representation Formulas for Solutions
Here we identify those important partial differential equations for which
in certain circumstances explicit or more-or-less explicit formulas can be had
for solutions. The general progression of the exposition is from direct formu-
las for certain linear equations to far less concrete representation formulas,
of a sort, for various nonlinear PDE.
Chapter 2 is a detailed study of four exactly solvable partial differen-
tial equations: the linear transport equation, Laplace’s equation, the heat
equation, and the wave equation. These PDE, which serve as archetypes for
the more complicated equations introduced later, admit directly computable
solutions, at least in the case that there is no domain whose boundary geom-
etry complicates matters. The explicit formulas are augmented by various
indirect, but easy and attractive, “energy”-type arguments, which serve as
motivation for the developments in Chapters 6, 7 and thereafter.
Chapter 3 continues the theme of searching for explicit formulas, now
for general first-order nonlinear PDE. The key insight is that such PDE
Previous Page Next Page