Partial Differentials Notes_Part_1

Partial Differentials Notes_Part_1 - Chapter 12 Dynamics of...

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Unformatted text preview: Chapter 12 Dynamics of Planar Media In previous chapters we studied the equilibrium configurations of planar media — plates and membranes — governed by the two-dimensional Laplace and Poisson equations. In this chapter, we analyze their dynamics, modeled by the two-dimensional heat and wave equations. The heat equation describes diffusion of, say, heat energy in a metal plate, an animal population dispersing over a region, or pollutants in a shallow lake. The wave equation models small vibrations of a two-dimensional membrane such as a drum. Since both equations fit into the general framework for dynamics that we established in Section 10.5, their solutions share many of the general qualitative and analytic properties possessed by their respective one-dimensional counterparts. Although the increase in dimension may challenge our analytical prowess, we have, in fact, already mastered the primary solution techniques: separation of variables, eigen- function series, and fundamental solutions. (Disappointingly, conformal mappings are not of much help in the dynamical universe.) When applied to partial differential equations in higher dimensions, separation of variables often leads to new linear, but non-constant coefficient, ordinary differential equations, whose solutions are no longer elementary func- tions. Rather, they are expressed in terms of a variety of important special functions , which include the error and Airy functions we have already encountered, the Bessel func- tions playing a starring role in the present chapter, and the Legendre functions, spherical harmonics, and spherical Bessel functions arising in three-dimensional problems. Special functions are ubiquitous in more advanced applications in physics, chemistry, mechanics, and mathematics, and, over the last two hundred and fifty years, many prominent math- ematicians have devoted significant effort to establishing their fundamental properties....
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Partial Differentials Notes_Part_1 - Chapter 12 Dynamics of...

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