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Unformatted text preview: Chapter 4 Separation of Variables There are three paradigmatic linear second order partial differential equations that have collectively driven the development of the entire subject. The first two we have already encountered: The wave equation describes vibrations and waves in continuous me- dia, including sound waves, water waves, elastic waves, electromagnetic waves, and so on. The heat equation models diffusion processes, including thermal energy in solids, solutes in liquids, and biological populations. Third, and most important of all, is the Laplace equation and its inhomogeneous counterpart, the Poisson equation , that govern equilib- rium mechanics. These two equations arise in an astonishing variety of mathematical and physical contexts, ranging through elasticity and solid mechanics, fluid mechanics, electro- magnetism, potential theory, thermomechanics, geometry, probability, number theory, and many more. The solutions to the Laplace equation are known as harmonic functions, and the discovery of their many remarkable properties forms one of the most celebrated chap- ters in the history of mathematics. All three equations, along with their multi-dimensional kin, appear repeatedly throughout this text. The aim of the current chapter is to develop the method of separation of variables for solving these key partial differential equations in their two independent variable incar- nations. For the wave and heat equations, the variables are time, t , and a single space coordinate, x , leading to initial-boundary value problems modeling the dynamical behav- ior of a one-dimensional medium. For the Laplace and Poisson equations, both variables represent space coordinates, x and y , and the associated boundary value problems model the equilibrium configuration of a planar body, e.g., the deformations of a membrane. Sep- aration of variables seeks special solutions that can be written as the product of functions of the individual variables, thereby reducing the partial differential equation to a pair of ordinary differential equations. More general solutions can then be expressed as infinite series in the appropriate separable solutions. For the two-variable equations considered here, this results in a Fourier series representation of the solution. In the case of the wave equation, separation of variables serves to focus attention on the vibrational character of the solution, whereas the earlier dAlembert approach emphasizes its particle-like aspects. Unfortunately, for the Laplace equation, separation of variables only applies to boundary value problems in very special geometries, e.g., rectangles and disks. Further development of the separation of variables method for solving partial differential equations in three or more variables can be found in Chapters 12 and 13....
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