On approximation of the normal derivative of the single layer heat potential near the boundary of a two-dimensional domain

Within the framework of the collocation boundary element method, approximations of the normal derivative (ND) of the single layer heat potential (SLHP) are obtained. The approximations uniformly converge with a cubic rate near the boundary of a two-dimensional spatial domain. For this purpose, exact integration with respect to the variable rho equivalent to(r(2) - d(2))1/2 is used, where d and r are the distances from the observed point to the boundary of the spatial domain and to the boundary point of integration, respectively. Namely, the integrals on the boundary elements (BEs) arising after piecewise quadratic interpolation of the density function are calculated using exact integration over the variable. if the values d and. do not exceed approximately the value D-a third of the radius of the Lyapunov circle. In other cases, the integrals on the BEs are calculated using simple Gaussian quadrature formulas (SGQF). To make exact integration over. possible, the smooth parts of the integrands are approximated by quadratic interpolation in the variable rho. In particular, in this way the direct value of the ND of the SLHP is calculated when d = 0 and, on the basis of this, the boundary integral equation is approximated, which makes it possible to obtain a solution of the initial-Neumann problem for the heat equation (INPHE) at a zero initial condition. It is proved that the corresponding approximations of the ND of the INPHE's solution converge with a cubic rate uniformly in the nearboundary domain, where d is an element of (0,D] With some simplifications, it's proved that the calculation of integrals on the BEs exclusively with the help of the SGQF entails a violation of the uniform convergence of the ND SLHP approximations near the boundary of the spatial domain. The results of calculating the ND of the INPHE solution in the unit circle are presented, confirming the theoretical conclusions.