Methods of applied mathematics with a software overview (Q303901)

The book under review is a lucidly written introduction to advanced methods in applied mathematics with a clear focus on harmonic analysis. The reader should already be familiar with elementary notions from linear algebra, calculus and ordinary diffential equations. A very appealing feature is its closeness to implementation issues. The author thereby focuses on free tools, in particular Octave, Python and Maxima, in contrast to a former edition, where MATLAB has been used. The outline of the presentation follows a general scheme: Physically motivated boundary-/inital value problems lead via diagonalization of the associated differential operators to series expansions of the solutions. The author nicely describes how this approach, applied to boundary value problems (bv-problems) of the heat equation, leads to the advent of Fourier series. Here he is following the historical route of \textit{J. Fourier} [The analytical theory of heat. New York: Dover Publications, Inc. (1955; Zbl 0066.07801)]. Subtle convergence issues are discussed and illustrated with numerical examples and it is demonstrated that Fourier series also apply to bv-problems associated with hyperbolic (wave equation) and elliptic (Laplace equation) PDE's. The book continues with the discussion of Sturm-Liouville problems and eventually leads to ``continuous versions'' of the series approaches. Thus Laplace- and Fourier transforms are introduced and discussed in large detail. Since the respective reconstruction formulae require integration techniques on the complex plane, a section on complex analysis is also included. The material covered there turns out to be useful also in a subsequent section on discrete transforms such as the Z-Transform, where inversion formulae also rely on Cauchy's integral formula. In a final section more generalizations are adressed. These generalizations concern orthogonal function systems like the Walsh system, which has an inherent multiscale structure, as well as bv-problems with associated differential operators, which are diagonalized, e.g., via the Mellin transform. It is very appealing that the material is not presented in a theorem-proof-example manner. Rather, all topics discussed above, are motivated by physical or engineering tasks. Moreover, the book contains an immense stock of problems providing the reader with the opportunity to work with the mathematical material and to deepen her or his understanding. Concluding, the book is an extremely valuable textbook covering a wide range of applied and engineering mathematics. For future editions two suggestions could be useful: Instead of including in the text body rather lengthy and specific source code snippets, it could be helpful to provide a software download site accessible to the interested reader. This would allow for reducing those snippets to the real essentials relevant to the adressed problem. Furthermore, as mentioned above, the harmonic analysis topics presented in the book are mainly motvated by ''classical'' mathematical physics dealing with initial-/boundary value problems and the associated differential operators. In the book it is mentioned, that applications of Fourier analysis and related techniques now have spread nearly everywhere, in particular to problems in signal processing. Therefore I suggest, that some more examples also from this discipline could be included in further editions of the book. As a reference, see, e.g. [\textit{S. B. Damelin} and \textit{W. Miller jun.}, The mathematics of signal processing. Cambridge: Cambridge University Press (2012; Zbl 1257.94001)].