Здесь находятся тезисы доклада, сделанного Богомоловым Яковом Леонидовичем на международной конференции «Некорректные и обратные задачи», проходившей 5–9 августа 2002 года в Новосибирске. В подготовке доклада также принимали участие Юнаковский Алексей Дмитриевич и Семёнов Евгений Сергеевич.
Siberian Branch of Russian Academy of Sciences
Sobolev Institute of Mathematics
Institute of Computational Mathematics and Mathematical Geophysics
Novosibirsk State University
Krasnoyarsk State University
ILL-POSED AND INVERSE PROBLEMS
DEDICATED TO PROF. M. M. LAVRENT'EV
on the occasion of his 70th anniversary
August 5–9, 2002
SOBOLEV INSTITUTE PRESS
ILL-POSED PROBLEMS FOR IRREGULAR CHANNELS
OF A SUPERCOLLIDER
Ya. L. Bogomolov, E. S. Semenov and A. D. Yunakovsky
Institute of Applied Physics, N. Novgorod, RUSSIA,
An acceleration part of a supercollider is considered. A separate section of this electrodynamical system represents a periodic set of metal rings irradiated by a convergent quasi-cylindrical wave flux. The profiles of metal rings are required to be optimized. The aim is to achieve a homogeneous (in radial coordinate) electric field of maximum amplitude in near axis domain. Moreover, a standing wave in longitudinal coordinate must be realized.
To investigate electromagnetic waves propagation in irregular waveguide channels of a supercollider, some model problems are considered. They are governed by the Helmholtz equation for the azimuth component of a magnetic field. Boundary conditions reflect equality to zero of tangential component of an electric field on a metal surface. The external problems for domains of several types are considered. The aim is to reach both resonance effect (to separate eigenfunction, corresponding to zero eigenvector) and a required electric field in near axis domain. To find an approximate solution, the method of discrete sources (MDS) is exploited. An unknown wave is sought as a sum of an incident wave (regular part of a solution) and a linear combination of Green (source) functions (irregular part of a solution). To obtain unknown coefficients of this combination, the boundary conditions at some discrete points are verified. As a result we arrive at a set of linear algebraic equations (SLAE).
The problems considered have a property of being ill-posed. Sometimes, very small displacements of source points yield very large changes in a numerical solution. That is why, we use the singular value decomposition technique for SLAE, which provides a constructive algorithm for source placement. Along with a classical regularization procedure this algorithm permits us to obtain required solutions for some model tests. Influence of a source point placement on a numerical solution is investigated.
This work was partially supported by the RFBR grant 01-01-00577.