Next steps

A text a bit above the level of this text is by Ascher and Petzold Ascher & Petzold (1998), 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 Ascher et al. (1995). 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 Quarteroni et al. (2007). An older and accessible treatment of Galerkin and finite element methods can be found in Strang and Fix Strang & Fix (1997).

For spectral methods, an introduction to BVPs may be found in Trefethen’s book Trefethen (2000), and a more theoretical take is in Quarteroni et al. Quarteroni et al. (2007).

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 Abramowitz & Stegun (2013). These special functions have come to be used for many things and are now available at https://dlmf.nist.gov Olver et al. (2010).