A text a bit above the level of this text is 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 ascherNumericalSolution1995. 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 et al. (2007). An older and accessible treatment of Galerkin and finite element methods can be found in Strang & Fix (1997).
For spectral methods, an introduction to BVPs may be found in Trefethen (2000), and a more theoretical take is in 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://
- Ascher, U. M., & Petzold, L. R. (1998). Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. Society for Industrial and Applied Mathematics. 10.1137/1.9781611971392
- Quarteroni, A., Sacco, R., & Saleri, F. (2007). Numerical Mathematics. Springer.
- Strang, G., & Fix, G. J. (1997). An Analysis of the Finite Element Method. Wellesley-Cambridge Press.
- Trefethen, L. N. (2000). Spectral Methods in MATLAB. SIAM.
- Abramowitz, M., & Stegun, I. A. (Eds.). (2013). Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables. Dover Publ.
- Olver, F. W. J., Lozier, D. W., Boisvert, R. F., & Clark, C. W. (2010). NIST Handbook of Mathematical Functions Hardback and CD-ROM. Cambridge University Press.