Partial differential equations

An ordinary differential equation is one with (ordinary) derivatives of functions of a single variable each—time, in many applications. These typically describe quantities in some sort of lumped-parameter way: mass as a “point particle,” a spring’s force as a function of time-varying displacement across it, a resistor’s current as a function of time-varying voltage across it. Given the simplicity of such models in comparison to the wildness of nature, it is quite surprising how well they work for a great many phenomena. For instance, electronics, rigid body mechanics, population dynamics, bulk fluid mechanics, and bulk heat transfer can be lumped-parameter modeled.

However, as we saw in (vecs?), there are many phenomena of which we require more detailed models. These include:

• detailed fluid mechanics,

• detailed heat transfer,

• solid mechanics,

• electromagnetism, and

• quantum mechanics.

In many cases, what is required to account for is the time-varying spatial distribution of a quantity. In fluid mechanics, we treat a fluid as having quantities such as density and velocity that vary continuously over space and time. Deriving the governing equations for such phenomena typically involves vector calculus; we observed in (vecs?) that statements about quantities like the divergence (e.g., continuity) can be made about certain scalar and vector fields. Such statements are governing equations (e.g., the continuity equation) and they are partial differential equations (PDEs) because the quantities of interest, called dependent variables (e.g., density and velocity), are both temporally and spatially varying (temporal and spatial variables are therefore called independent variables).

In this chapter, we explore the analytic solution of PDEs. This is related to but distinct from the numeric solution (i.e., simulation) of PDEs, which is another important topic. Many PDEs have no known analytic solution, so for these numeric solution is the best available option.¹ However, it is important to note that the insight one can gain from an analytic solution is often much greater than that from a numeric solution. This is easily understood when one considers that a numeric solution is an approximation for a specific set of initial and boundary conditions. Typically, very little can be said of what would happen in general, although this is often what we seek to know. So, despite the importance of numeric solution, one should always prefer an analytic solution.

Three good texts on PDEs for further study are Kreyszig (2011, Ch. 12), Strauss (2007), and Haberman (2018).