Theory of equations in mathematical physics. (Teoriya rivnyan' matematichnoï fiziki). (Q2744680)

The book covers the most important topics of a graduate course in mathematical physics. NEWLINENEWLINENEWLINEIn the first chapter, after introducing basic notions and definitions, the authors expose traditional topics concerning classification of PDEs and their reduction to canonical forms.NEWLINENEWLINENEWLINEThe second chapter deals with the equations of hyperbolic type. First, the equations of wave processes in infinite domains are studied. The authors derive classical d'Alembert, Kirchhoff, and Poisson formulas for the solutions of 1-D, 3-D, and 2-D Cauchy problems. Next, they discuss the questions of correctness for these problems. By means of the successive approximations method, a solution of the Goursat problem is constructed. This result is then used to justify the Riemann's method of finding solutions for the Cauchy problem. The 3-D Cauchy problem with initial data on a hyperplane is studied by means of the Lorentz transforms. The chapter is ended by traditional topics concerning the method of characteristics and the Fourier method for solving mixed problems of wave equations.NEWLINENEWLINENEWLINEIn the third chapter, the theory of heat equation is exposed. Here, the main topics are: the maximum principle, statement of the mixed problem, its correctness; uniqueness and continuous dependence of solutions of the mixed problems; method of separation of variables; statement of the Cauchy problem, existence of solutions and correctness; studies of heat propagation in semi-infinite rod.NEWLINENEWLINENEWLINEThe fourth chapter begins with examples of physical problems which can be described by elliptic PDEs. Next, some basic facts from the theory of harmonic functions are exposed: maximum principle and its applications to the Dirichlet problem; construction of the solution to the Dirichlet problem in a disk by means of the Fourier method, Poisson's integral; 3-D and 2-D Green formulae; the mean value theorem; the Harnack theorem on sequences of harmonic functions and the Liouville theorem; the Green function for the Laplace operator; solving the Dirichlet problem for a sphere; uniqueness properties of 2-D Neumann problems; solution of the spherical internal and external Neumann problems. In concluding sections of this chapter, elements of the potential theory are exposed.NEWLINENEWLINENEWLINEThe theoretical results containing in the book are successfully supplemented by examples of physical applications (oscillations of membranes, hydrodynamic problems, wave processes, etc.). Each chapter is completed by a collection of interesting problems.