Chapters 10–12 described continuum models of engineered membranes that involved partial differential equations (PDEs). This appendix gives a brief introduction to the classification of PDEs, linear PDEs, nondimensionalization of PDEs, and numerical methods for solving PDEs. Uniqueness and existence results are not discussed here since they involve advanced results in functional analysis that are outside the scope of this book. For a comprehensive advanced treatment of PDEs at a graduate mathematics level see [115].

Linear, Semilinear, and Nonlinear Partial Differential Equations

PDEs are typically classified into linear, semilinear, and nonlinear. Linear PDEs are important as several methods exist to obtain closed-form solutions of these PDEs (under simple boundary conditions), including separation of variables, superposition, Fourier series, Laplace transform, and Fourier transform. The solution to some semilinear PDEs can be obtained using the symmetry method or method of characteristics. When a linear or semilinear PDE has more sophisticated boundary conditions (as is the case with engineered membranes), the PDE needs to be solved numerically. Similarly, nonlinear PDEs typically do not have an exact solution and must also be solved numerically.

Consider a generic PDE of two scalar variables (space x and time t), where u(x, t) is a function of the variables x and t. A linear PDE has the form

where a(x, t), b(x, t), c(x, t), and d(x, t) are generic functions of x and t but not u. A PDE is semilinear if the coefficient function a(x, t) of the highest partial derivative is dependent on x and t but not u. Therefore, (A.1) is a semilinear PDE if the coefficient functions can be expressed as a(x, t), b(x, t, u), c(x, t, u), and d(x, t, u). If the PDE is neither linear nor semilinear, it is nonlinear. Linear, semilinear, and nonlinear PDEs are all used to model the dynamics of engineered membranes. For example:

Linear. Poisson's equation ((10.3) on page 180), the Nernst–Planck equation ((10.9) on page 182), the Poisson–Fermi equation ((11.23) on page 233), and the Fokker– Planck equation ((14.28) on page 318).