Friday, June 6, 2008

The "Big Three" Partial Differential Equations of Physics

Often a mathematical physics class will focus on the "big three" partial differential equations of physics: the diffusion equation, the wave equation, and Laplace's equation. The 4th edition of Intermediate Physics for Medicine and Biology provides a gentle introduction to each of these equations. You won't find much mathematical theory; Russ Hobbie and I are concerned primarily with exploring their relevance to biology and medicine.

The diffusion equation (Eq. 4.24) is the topic of chapter 4. Diffusion is not covered in many physics classes, but is crucially important to biology. The diffusion equation is also known by another name: the heat equation. Problem 24 of chapter 4 explores an extension of this topic: a "reaction-diffusion" equation that governs the nonlinear propagation of calcium waves.

The wave equation (Eq. 13.5) was not discussed much in previous editions of Intermediate Physics for Medicine and Biology, but it is an essential topic in the 4th edition's new chapter 13 on sound and ultrasound. Here we meet classic wave behavior, such as the relationship between wavelength and frequency, the difference between propagating and standing waves, reflection, and the Doppler effect.

Laplace's equation (the displayed equation before Eq. 7.44a) is encountered in chapter 7 on the electrocardiogram. We don't talk as much about solutions to Laplace's equation as we do the diffusion and wave equations, but it plays a fundamental role in electrostatics, and appears when discussing steady-state solutions to the diffusion equation.

What other important partial differential equations are introduced in our book? The Navier-Stokes equation, governing fluid flow, is analyzed in problem 28 of chapter 1. The Helmholtz equation is presented under its pseudonyms: the cable equation (chapter 6) and the linearized Poisson-Boltzmann equation (chapter 9). Schrödinger's equation, the basic equation of quantum mechanics, is mentioned in the introduction to chapter 3, but is never written down.

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