ABSTRACT
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 11.1 Turning Up the Heat-Variants of the Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . 304
11.1.1 One-Dimensional Diffusion with Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 11.1.2 Two-Dimensional Diffusion with Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 11.1.3 Abstract Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
11.2 Seepage of Fluid Through Fissured Rocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 11.2.1 One-Dimensional Fissured Rock Model with Forcing . . . . . . . . . . . . . . . . 332 11.2.2 Higher-Dimensional Fissured Rock Model with Forcing . . . . . . . . . . . . . 342 11.2.3 Abstract Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
11.3 The Classical Wave Equation and its Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 11.3.1 One-Dimensional Wave Equation with Source Effects . . . . . . . . . . . . . . . . 352 11.3.2 Two-Dimensional Wave Equation with Source Effects . . . . . . . . . . . . . . . . 373 11.3.3 Abstract Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384
As in the ODE models explored in Part I, nonnegligible external forces are often present and not negligible and so they need to be incorporated, when forming a mathematical model of a phenomenon. The nature of such external forces can be rather complex and their description can be very “nonlinear” in the sense of depending not only on time and position, but also on the solution itself and its partial derivatives. As before, we start off relatively tame and consider only forcing terms that are independent of the unknown function we seek to describe. More precisely, we only consider forcing terms of the form f(x, t).
