Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: ON EXPLICIT AND NUMERICAL SOLVABILITY OF PARABOLIC INITIAL-BOUNDARY VALUE PROBLEMS ALEXANDER KOZHEVNIKOV AND OLGA LEPSKY | ON EXPLICIT AND NUMERICAL SOLVABILITY OF PARABOLIC INITIAL-BOUNDARY VALUE PROBLEMS ALEXANDER KOZHEVNIKOV AND OLGA LEPSKY Received 26 July 2005 Revised 15 January 2006 Accepted 22 March 2006 A homogeneous boundary condition is constructed for the parabolic equation dt 1 -A u f in an arbitrary cylindrical domain o X R O c R being a bounded domain I and A being the identity operator and the Laplacian which generates an initial-boundary value problem with an explicit formula of the solution u. In the paper the result is obtained not just for the operator dt 1 A but also for an arbitrary parabolic differential operator dt A where A is an elliptic operator in R of an even order with constant coefficients. As an application the usual Cauchy-Dirichlet boundary value problem for the homogeneous equation dt 1 A u 0 in o X R is reduced to an integral equation in a thin lateral boundary layer. An approximate solution to the integral equation generates a rather simple numerical algorithm called boundary layer element method which solves the 3D Cauchy-Dirichlet problem with three spatial variables . Copyright 2006 A. Kozhevnikov and O. Lepsky. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction It is well known that the initial-boundary value problem with the Dirichlet Neumann boundary condition for the parabolic equation dt 1 A u f can be solved using the Green function. But the Green function can be found explicitly just for a few very specific domains o such as balls and half-spaces. Unfortunately in the case of an arbitrary domain o there is no explicit formula for the solution. In this paper the following question is investigated. How can one define boundary conditions for an arbitrary domain o in order to obtain an explicitly solvable initial boundary value problem An answer is obtained not just for .
