Introduction to Linear Algebra with Mathematica

# Preface

Inthe previous section, we discyssed regular Sturm--Liouville problems on some finite interval |α, β| for the second order self-adjoint differential operator
$$\label{EqSingular.1} L \left[ x, \texttt{D} \right] = - \texttt{D}\,p(x)\, \texttt{D} - q(x)\,\texttt{I} = - \frac{\text d}{{\text d}x}\,p(x)\,\frac{\text d}{{\text d}x} - q(x) , \qquad \texttt{D} = \frac{\text d}{{\text d}x} \quad \texttt{I} = \texttt{D}^0 .$$
In its definition, we assumed that coefficients q(x), p(x), and its derivative dp(x)/dx are continuous and bounded functions on a finite closed interval [α, β].

1. Everitt, W.N., A Catalogue of Sturm-Liouville Differential Equations, In: Sturm-Liouville Theory: Past and Present, pp. 271–331. Ed. W.O. Amrein, A.M. Hinz, D.B. Pearson, 2005, Birkhauser Verlag, Basel/Switzerland.
2. Herron, I.H., Solving singular boundary value problems for ordinary differential equations, Caribbean Journal of Mathematical and Computing Sciences, 2013, 15, pp. 1--30.
3. Krall, A.M., (1986). The Singular Sturm--Liouville Problem. In: Applied Analysis. Mathematics and Its Applications, vol 31. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-4748-1_12
4. Lützen, J. Sturm and Liouville’s work on ordinary linear differential equations. The emergence of Sturm-Liouville theory, Archive for History of Exact Sciences, 1984, volume 29, pages 309–376. https://doi.org/10.1007/BF00348405
5. Philip, P., The Green's Function for a Singular Boundary Value Problem, M.S. Thesis, Department of Mathematics, Howard University, Washington D.C., 1975.
6. Stakgold, I., Green's Functions and Boundary Value Problems, 1998, Second edition, Wiley Interscience, New York.
7. Titchmarsh, E.C., Eigenfunction Expansions Associated with Second-order Differential Equations, 1946, Clarendon, Oxford.
8. Zettl, A., Sturm--Liouville Theory, 2005, American Mathematical Sosiety, Providence, RI.