Vladislav V. Kravchenko Boeken

Focusing on direct and inverse Sturm-Liouville problems, this book explores recent advancements in both theory and practice across finite and infinite intervals. It introduces a universal method for solving spectral and scattering problems using transmutation operators, detailing their efficient construction. The text derives analytical solutions for Sturm-Liouville equations and integral kernels in functional series form, highlighting unique characteristics that facilitate straightforward numerical solutions for a diverse range of problems.

Pseudoanalytic function theory generalizes and preserves many crucial features of complex analytic function theory. The Cauchy-Riemann system is replaced by a much more general first-order system with variable coefficients which turns out to be closely related to important equations of mathematical physics. This relation supplies powerful tools for studying and solving Schrödinger, Dirac, Maxwell, Klein-Gordon and other equations with the aid of complex-analytic methods. The book is dedicated to these recent developments in pseudoanalytic function theory and their applications as well as to multidimensional generalizations. It is directed to undergraduates, graduate students and researchers interested in complex-analytic methods, solution techniques for equations of mathematical physics, partial and ordinary differential equations.

Quaternionic analysis is the most natural and close generalization of complex analysis that preserves many of its important features. The present book is meant as an introduction and invitation to this theory and its applications (in fact it was inspired by a course given by the author to graduate engineering students). Restricting ourselves to Maxwell's equations and the Dirac equation only we show the progress achieved in applied quaternionic analysis during the last five years, emphasising results which can not so easily be obtained by other methods. Thus, the main objective of this work is to introduce the reader to some topics of quaternionic analysis whose selection is motivated by particular models from the theory of electromagnetic and spinor fields, and to show the usefulness and necessity of applying the tools of quaternionic analysis to these kinds of problems.