Methods for Solving Mathematical Physics Problems

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The aim of the book is to present to a wide range of readers (students, postgraduates, scientists, engineers, etc.) basic information on one of the directions of mathematics, methods for solving mathematical physics problems. The authors have tried to select for the book methods that have become classical and generally accepted. However, some of the current versions of these methods may be missing from the book because they require special knowledge. The book is of the handbook-teaching type. On the one hand, the book describes the main definitions, the concepts of the examined methods and approaches used in them, and also the results and claims obtained in every specific case. On the other hand, proofs of the majority of these results are not presented and they are given only in the simplest (methodological) cases. Another special feature of the book is the inclusion of many examples of application of the methods for solving specific mathematical physics problems of applied nature used in various areas of science and social activity, such as power engineering, environmental protection, hydrodynamics, elasticity theory, etc. This should provide additional information on possible applications of these methods. To provide complete information, the book includes a chapter dealing with the main problems of mathematical physics, together with the results obtained in functional analysis and boundary-value theory for equations with partial derivatives.

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Valeri Agoshkov is a Doctor of Physical and Mathematical Sciences, Professor, Leading Researcher of the Institute of Numerical Mathematics, Russian Academy of Sciences. Pavel Dubovski received a Ph.D in Theoretical and Mathematical Physics from Moscow State University and Dr. Sc degree in Mathematical Physics & Mathematical Modelling from the Institute of Numerical Mathematics, Russian Academy of Sciences. Table of Contents Main Problems of Mathematical Physics Main concepts and notations Concepts and assumptions from the theory of functions and functional analysis Main equations and problems of mathematical physics Methods of Potential Theory Fundamentals of potential theory Using the potential theory in classic problems of mathematical physics Other applications of the potential method Eigenfunction Methods Eigenvalue problems Special functions Eigenfunction method Eigenfunction method for problems of the theory of electromagnets phenomenon Eigenfunction method for problems in the theory of oscillations Methods of Integral Transforms Main integral transformations Using integral transforms in problems of oscillation theory Using integral transforms in heat conductivity problems Using integral transformations in the theory of neutron diffusion Application of integral transformations to hydrodynamic problems Using integral transforms in elasticity theory Using integral transforms in coagulation equation Methods of Discretisation of Mathematical Physics Problems Finite-difference methods Variational methods Projection methods Splitting Methods for applied problems of mathematical physics Methods for Solving Non-Linear Equations References Continuity and differentiability of nonlinear mappings Adjoint non-linear operators Convex functionals and monotonic operators Variational method of examining nonlinear equations Minimising sequences The method of the steepest descent The Ritz method The Newton-Kantorovich method The Galerkin-Petrov method for non-linear equations Perturbation method Applications to some problem of mathematical physics Index.