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Linear Differential Operators (Classics in Applied Mathematics): Cornelius Lanczos

Don't let the title fool you! If you are interested in numerical analysis, applied mathematics, or the solution procedures for differential equations, you will find this book useful. Because of Lanczos' unique style of describing mathematical facts in nonmathematical language, Linear Differential Operators also will be helpful to nonmathematicians interested in applying the methods and techniques described. Originally published in 1961, this Classics edition continues to be appealing because it describes a large number of techniques still useful today. Although the primary focus is on the analytical theory, concrete cases are cited to forge the link between theory and practice. Considerable manipulative skill in the practice of differential equations is to be developed by solving the 350 problems in the text. The problems are intended as stimulating corollaries linking theory with application and providing the reader with the foundation for tackling more difficult problems.

Review:

As the other reviewers have said, this is a master piece for various reasons. Lanczos is famous for his work on linear operators (and efficient algorithms to find a subset of eigenvalues). Moreover, he has an "atomistic" (his words) view of differential equations, very close to the founding father's one (Euler, Lagrange,...).

A modern book on linear operators begins with the abstract concept of function space as a vector space, of scalar product as integrals,... The approach is powerful but somehow we loose our good intuition about differential operators.

Lanczos begins with the simplest of differential equations and use a discretization scheme (very natural to anybody who has used a computer to solve differential equations) to show how a differential equation transforms into a system a linear algebraic equation. It is then obvious that this system is undetermined and has to be supplemented by enough boundary condition to be solvable. From here, during the third chapters, Lanczos develops the concept of linear systems and general (n x m) matrices, the case of over and under determination, the compatibility conditions, ...

It is only after these discussions that he returns (chapter 4) to the function space and develops the operator approach and the role of boundary conditions in over and under-determination of solutions and the place of the adjoint operators. The remaining of the book develops these concepts : chp5 is devoted to Green's function and hermitian problems, chap7 to Sturm-Liouville,... The last chapter is devoted to numerical techniques, amazing if one think that the book was written at the very beginning of computers, which is a gem by itself.

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