10.7. Next steps#

A text a bit above the level of this text is by Ascher and Petzold [AP98], which covers shooting and finite-difference collocation methods for linear and nonlinear BVPs, with a number of theoretical and applications problems. A graduate-level text solely on numerical solution of BVPs is by Ascher, Mattheij, and Russell [AMR95]. Besides the shooting and finite-difference methods, it briefly discusses Galerkin and spline-based methods, and it goes into more depth on theoretical issues. A more detailed treatment of the Galerkin method can be found in Quarteroni, Sacco, and Saleri [QSS07]. An older and accessible treatment of Galerkin and finite element methods can be found in Strang and Fix [SF97].

For spectral methods, an introduction to BVPs may be found in Trefethen’s book [Tre00], and a more theoretical take is in Quarteroni et al. [QSS07].

In this chapter, a number of linear variable-coefficient BVPs for so-called special functions were mentioned: Bessel’s equation, Laguerre’s equation, etc. These ODEs and their solutions arise in the solution of partial differential equations of mathematical physics, and were extensively characterized before prior to wide use of computers [AS13]. These special functions have come to be used for many things and are now available at https://dlmf.nist.gov [OLBC10].