x partial-differential y EndFraction plus c StartFraction partial-differential squared u Over partial-differential y squared EndFraction equals upper F left-parenthesis x comma y comma u comma StartFraction partial-differential u Over partial-differential x EndFraction comma StartFraction partial-differential u Over partial-differential y EndFraction right-parenthesis period"/>
Here,
The highest-order terms determine the classification. Thediscriminant of Eq. (1.1) is
hyperbolic at any point of the ‐plane where ;
parabolic at any point of the ‐plane where ;
elliptic at any point of the ‐plane where .
Extending this terminology, we say that a first‐order PDE of the form
is hyperbolic at any point
Exercise 1.1 Verify the following classifications, where
Mathematicians associate the wave equation with time‐dependent processes that exhibit wave‐like behavior, the heat equation with time‐dependent processes that exhibit diffusive behavior, and the Laplace equation with steady‐state processes. These associations arise from applications, some of which this book explores, reinforced by theoretical analyses of the three exemplars in Exercise 1.1. For more information about the classification of PDEs, see [65, Section 2‐6].
1.3 Dimensions and Units
In contrast to most texts on pure mathematics, in this book physical dimensions play an important role. We adopt the basic physical quantities length, mass, and time, having physical dimensions
For example, the physical dimension of force
Analyzing the physical dimensions of quantities that arise in physical laws can yield surprisingly powerful mathematical results. Subsequent chapters exploit this concept many times.
Physical laws such as
