Equations in Mathematical Physics: A practical course

This handbook is addressed to students of technology institutf's where a course on mathematical physics of relatively reduced volume is offered, as well as to engineers and scientists. The aim of the handbook is to treat (demonstrate) the basic methods for solving the simplest problems of classical mathematical physics. The most basic among the methods considered hrre i8 the superposition method. It allows one, based on particular linearly indepmdent HolutionH (solution "atoms"), to obtain the solution of a given problem. To that end the "Hupply" of solution atoms must be complete. This method is a development of the well-known method of particular solutions from the theory of ordinar~' differelltial equations. In contrast to the case of ordinary differential equations, where the number of linearly independent 80lutions is always finite, for a linear partial differrntial equation a complete "supply" of solution atoms is always infinite. This infinite set of Holutions may be discrete (for example, for regular boundary vahlP problems in a bounded domain), or form a continuum (for example, in the case of problems in the whole space). In the first case the superposition method reduces to tlH' construction of a series in the indicated solution atoms with unknown coefficipnts, while in the second case the series is replaced by an integral with respect to the corm:iponding parameters (variables). This first step leads us to the general solution of the associated hOlllogeneous equation under the assumption that the set of solution atoms i;; "complete.

Table of Contents:
1. Elliptic problems.- 1.1 The Dirichlet problem for the Laplace equation in an annulus.- 1.2 Examples of Dirichlet problems in an annulus.- 1.3 The interior and exterior Dirichlet problems.- 1.4 The Poisson integral for the disc. Complex form. Solution of the Dirichlet problem when the boundary condition is a rational function R(sin ?, cos ?).- 1.5 The interior and exterior Dirichlet problems.- 1.6 Boundary value problems for the Poisson equation in a disc and in an annulus.- 1.7 Boundary value problems for the Laplace and Poisson equations in a rectangle.- 1.8 Boundary value problems for the Laplace and Poisson equations in a bounded cylinder.- 1.9 Boundary value problems for the Laplace and Poisson equations in a ball.- 1.10 Boundary value problems for the Helmholtz equations.- 1.11 Boundary value problem for the Helmoltz equation in a cylinder.- 1.12 Boundary value problems for the Helmoltz equation in a disc.- 1.13 Boundary value problems for the Helmoltz equation in a ball.- 1.14 Guided electromagnetic waves.- 1.15 The method of conformal mappings (for the solution of boundary value problems in the plane).- 1.16 The Green function method.- 1.17 Other methods.- 1.18 Problems for independent study.- 1.19 Answers.- 2. Hyperbolic problems.- 2.1 The travelling-wave method.- 2.2 The method of selection of particular solutions.- 2.3 The Fourier integral transform method.- 2.4 The Laplace integral transform met hod.- 2.5 The Hankel integral transform method.- 2.6 The method of standing waves. Oscillations of a bounded string.- 2.7 Some examples of mixed problems for the equation of oscillations of a string.- 2.8 The Fourier method. Oscillations of a rectangular membrane.- 2.9 The Fourier method. Oscillations of a circular membrane.- 2.10 The Fourier method. Oscillationsof a beam.- 2.11 The perturbation method.- 2.12 Problems for independent study.- 2.13 Answers.- Chaper 3. Parabolic problems.- 3.1 The Fourier integral transform method.- 3.2 The Lapalce integral transform method.- 3.3 The Fourier method (method of separation of variables).- 3.4 A modification of the method of separation of variables for solving the Cauchy problem.- 3.5 Problems for independent study.- 3.6 Answers.- References.