The most important tools for this purpose are the method of separation of variables, or the Fourier method, and the Wiener–Hopf method. the solutions are given in the form of series or integrals containing special functions. For domains of special type the problems in the mathematical theory of diffraction may be solved in an explicit form, i.e. variational methods and numerical solution of the integral equations of potential theory - are also applicable in this case. Ordinary numerical methods of mathematical physics - e.g. The solution of the Laplace equation may be taken as the first approximation. In equation (5) may be regarded as a small parameter, which makes it possible to apply in this case the different variants of perturbation theory. In the former cases the quantity $ \omega $ If the wavelength is large (small) as compared with the characteristic dimensions of the body, one speaks of the diffraction of long (short) waves. Of major importance in problems of the mathematical theory of diffraction is the ratio between the wavelength and the characteristic dimensions of the body in the vicinity of which the wave process is studied. Methods for solving problems in the mathematical theory of diffraction. In proving the solvability of the Dirichlet and the Neumann problem the difficulties encountered are completely similar to the classical difficulties involved in solving the exterior Dirichlet problem for the Laplace equation by methods of potential theory. Here $ F, U _ \mu ( N)Ī Fredholm integral equation of the second kind is obtained.
problems involving initial and boundary conditions for the wave equation
Transient problems of diffraction are, essentially, mixed problems (cf. These problems may be discussed taking the wave equation as an example problems for other equations describing wave processes are formulated in a similar manner. Problem formulation in the mathematical theory of diffraction.