On solving plane problems of non-stationary heat conduction by the collocation boundary element method

In this paper, we propose a fully justified collocation boundary element method allowing one to obtain numerical solutions of internal and external initial-boundary value problems (IBVPs) with boundary conditions of the first, second, and third kind for the equation d_tu = a ²A₂u - pu with constants a, p > 0 in a plane spatial domain Q (in a bounded one Q⁺ or in its exterior Q^- ) on a finite time interval I_T = [0,T] at a zero initial condition. The solutions are found in the form of the double-layer potential for the Dirichlet IBVP and in the form of the simple layer potential for the Neumann-Robin IBVP with unknown density functions determined from the boundary integral equations (BIEs) of the second kind. In this paper, instead of the usual piecewise-polynomial interpolation of the density function on time variable t, the BIEs are approximated by the piecewise-quadratic interpolation (PQI) of the C₀ -semigroup of right shifts on time. Also, on the basis of the PQI, the approximation of the multiplier e~^pT in kernels of the integral operators is carried out. In addition, the PQI of density functions is performed: for the BIE, only on arc-length s; for the potentials, on both variables s and t. Then, the integration with respect to the variable t on the boundary elements (BEs) is performed exactly. The integration with respect to the variable s on the BE for the potentials is performed approximately by using the Gaussian quadrature with y> 2 points. For the BIE, the integration with respect to the arc-length s is carried out in two ways. On singular BEs and on nearby singular BEs, adjacent to a singular BE in some fixed arc-length region, an exact integration with respect to the variable r is carried out (r is the distance from the boundary point at which the integral is calculated as a function of parameter to the current boundary point of the integration). In this integration, functions of the variable r are taken as the weighting functions. The functions of r are generated by the fundamental solution of the heat equation and the rest of the integrand is approximated by quadratic interpolation on r. The integrals with respect to s on the remaining BEs are calculated using the Gaussian quadrature with у points. The cubic convergence of approximate solutions of the IBVP at any point of the set QxI_T is proved under conditions dQ e C⁵ n C ^2y and w e C^dQ). It is also proved that such solutions are resistant to perturbations of the boundary function w in the norm of the space Cj⁰ (dQ). Here, C^k m„ (dQ) = C^k m (dQ) n C°_m+_n (dQ) and C^k m(dQ) is the Banach space of k times continuously differentiable on dQ vector functions with values in Sobolev's space which is the domain of definition of the operator B^m ((Bf ) (t) = f ' (t), f (t = 0) = 0 ). In conclusion, results of the numerical experiments are presented. They confirm the cubic convergence of approximate solutions for all three IBVPs in a circular domain.

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