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This concise volume presents an overview of equations of mathematical physics and generalized functions. While intended for advanced readers, the accessible introduction and text structure allows beginners to study at their own pace as the material gradually increases in difficulty. The text introduces the concept of generalized Sobolev functions and L. Schwartz distributions briefly in the opening section, gradually approaching a more in-depth study of the generalized differential equation (also known as integral equality). In contrast to the traditional presentation of generalized Sobolev functions and L. Schwartz distributions, this volume derives the topology from two natural requirements (which are equivalent to it). The text applies the same approach to the theory of the canonical Maslov operator. It also features illustrative drawings and helpful supplementary reading in the footnotes concerning historical and bibliographic information related to the subject of the book. Additionally, the book devotes a special chapter to the application of the theory of pseudodifferential operators and Sobolev spaces to the inverse magneto/electroencephalography problem.
Explicit numerically realizable formulas related to the Cauchy problem for elliptic equations (including quasilinear ones) and also to the Poincaré--Steklov operators are presented. The book is completed by three additions, which were written by famous mathematicians Yu. V. Egorov, A. B. Antonevich, and S. N. Samborski.
List of contents
Introduction. - Introduction to problems of mathmatical physics.- The spaces D , D and D'. Elements of the distribution theory. - Pseudodifferential operators and Fourier operators. - New approach to the theory of generalized functions (Yu.V. Egorov). - Algebras of mnemonic functions (A. B. Antonevich).- Extensions first-order partial differential operators (S.N. Samborskii). - References-. Index.
Report
"The book is highly recommended for readers interested in mathematical physics and generalized functions, as well as in the historical aspects of the research eld. It serves as an inspiring text for lecturers due to its pertinent methodological approach. As a particular virtue of the book, we emphasize numerous footnotes that contain valuable historical comments, anecdotes, and additional facts about the topic and the scientists who signicantly contributed to it." (Nenad M. Teofanov, Mathematical Reviews, February, 2025)
