Partial differential equations

The primary partial differential equation treated in this text is the heat equation $T_t = \kappa T_{xx}$ and its relatives in other geometries. It is derived in section 9.4, then solved by separation of variables and Fourier series in section 9.5 and by the method of lines in section 9.6. Separating this equation in a circle provides a key example in section 11.5, Regular Singular Points.

The equilibrium solutions of this equation are defined by ordinary boundary-value problems, providing a direct tie to the material of section 9.2, Boundary-value Problems: Analytic Tools, and section 9.3, Boundary-value Problems: Numerical Methods. Furthermore, analyzing rates of approach to equilibrium makes use of the analytic methods of section 9.5, Fourier Methods, and the numerical methods of section 9.6, Initial-Boundary-value Problems: Numerical Methods. Such qualitative analyses are less common in the usual elementary study of partial differential equations, but they are consistent with the spirit of model-analyze-interpret that runs through the text.