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
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