A REFINEMENT OF THE BOUNDARY ELEMENT COLLOCATION METHOD NEAR THE BOUNDARY OF DOMAIN IN THE CASE OF TWO-DIMENSIONAL PROBLEMS OF NON-STATIONARY HEAT CONDUCTION WITH BOUNDARY CONDITIONS OF THE SECOND AND THIRD KIND

In this paper, we consider initial-boundary value problems (IBVPs) for the equation partial derivative(t)u=a(2)Delta(2)u - p u with constants a, p > 0 in an open two-dimensional spatial domain Omega with boundary conditions of the second and third kind at a zero initial condition. A fully justified collocation boundary element method is proposed, which makes it possible to obtain uniformly convergent in the space-time domain Omega x [0, T] approximate solutions of the abovementioned IBVPs. The solutions are found in the form of the single-layer potential with unknown density functions determined from boundary integral equations of the second kind. To ensure the uniform convergence, integration on arc-length s when calculating the potential operator is carried out in two ways. If the distance r from the point x is an element of Omega at which the potential is calculated to the integration point x'is an element of partial derivative Omega does not exceed approximately one-third of the radius of the Lyapunov circle R-pi, then we use exact integration with respect to a certain component p of the distance r: p (r(2)- d(2))(1/2) (d is the distance from the point x is an element of Omega to the boundary partial derivative Omega). This exact integration is practically feasible for any analytically defined curve Omega. In this integration, functions of the variable p are taken as the weighting functions and the rest of the integrand is approximated by quadratic interpolation on p. The functions of p are generated by the fundamental solution of the heat equation. The integrals with respect to s for r > R-pi/3 are calculated using Gaussian quadrature with gamma points. Under the condition partial derivative Omega is an element of C-5 boolean AND C-2 gamma (gamma >= 2), it is proved that the approximate solutions converge to an exact one with a cubic velocity uniformly in the domain Omega x [0, T]. It is also proved that the approximate solutions are stable to perturbations of the boundary function uniformly in the domain Omega x [0, T]. The results of computational experiments on the solution of the IBVPs in a circular spatial domain are presented. These results show that the use of the exact integration with respect to p can substantially reduce the decrease in the accuracy of numerical solutions near the boundary partial derivative Omega, in comparison with the use of exclusively Gauss quadratures in calculating the potential.