Approximate solution of Cauchy's problem for laplace's equation applicable to the problem of shaping of dense spatially inhomogeneous beams of charged particles: PMM vol. 35, no. 4, pp. 656–668

Abstract

A solution of the problem of shaping (Sect. 1) of a number of three-dimensional beams is given in the form of asymptotic expansions. The results are compared with exact expressions which determine the shaping electrodes for a plane flow along circular trajectories (Sect. 2). From the paraxial approximation for the electrostatic beams, cases which satisfy the conditions of a full spatial charge on the emitter, without disturbing the regularity of this approximation (Sect. 3) are selected. Quasi-axially symmetric beams (Sect. 4) and a quasi-cylindrical domain of arbitrary section (Sect. 5) are considered. The hydrodynamic theory of intense beams of charged particles represents one of the branches of the mechanics of continuous medium. However, the asymptotic methods, although used widely and for a long time in other branches of mechanics, began to find application in this field only recently. The inverse problem or the problem of synthesis which appears when a system with desired characteristics is constructed, consists of two parts; the internal problem, which deals with solutions of the equations of the beam, and the external problem, connected with determining the shaping electrodes which provide the realization of the computed flow. The Cauchy problem for the Laplace equation represents the mathematical expression for the latter. For the solution of the internal problem for narrow beams the asymptotic method of the extension or the narrow strip type [4,5] have been successfully used in [1–3], although the study of the problem of shaping was complicated by the existence of singularities at the flow boundary. An approximate solution of this problem with singularities present in the Cauchy conditions based on the multiscale and factorization method, is given below.