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