Semi-analytical approximation of the normal derivative of the heat simple layer potential near the boundary of a two-dimensional domain

A semi-analytical approximation of the normal derivative of the simple layer heat potential near the boundary of a two-dimensional domain with $C^{5} $ smoothness is proposed. The calculation of the integrals that arise after piecewise quadratic interpolation of the density function with respect to the variable of arc length $s$, is carried out using analytical integration over the variable $\rho =\sqrt{r^{2} -d^{2} } $, where $r$ and $d$ are the distances from the observation point to the integration point and to the boundary of the domain, respectively. To do this, the integrand is represented as the sum of two products, each of which consists of two factors, namely: a function smooth in а near-boundary domain containing the Jacobian of the transition from the integration variable $s$ to the variable $\rho $, and a weight function containing a singularity at $r=0$ and uniformly absolutely integrable in the near-boundary region. Smooth functions are approximated with the help of the piecewise quadratic interpolation over the variable $\rho $, and then analytical integration becomes possible. Analytical integration over $\rho $ is carried out on a section of the boundary fixed in width, containing the projection of the observation point, and on the rest of the boundary, the integrals over $s$ are calculated using the Gauss formulas. Integration over the parameter of $C_{0} $-semigroup formed by time shift operators is also  carried out analytically. To do this, the $C_{0} $-semigroup is approximated using the piecewise quadratic interpolation over its parameter. It is proved that the proposed approximations have stable cubic convergence in the Banach space of continuous functions with the uniform norm, and such convergence is uniform in the closed near-boundary region. The results of computational experiments on finding of  the normal derivative of solutions of the second initial-boundary problem of heat conduction in a unit circle with a zero initial condition are presented, confirming the uniform cubic convergence of the proposed approximations of  the normal derivative of the simple layer heat potential.