Integral method of solving heat-conduction problems with boundary condition of the second-kind. 2. Analysis of accuracy

Abstract

An algorithm of finding polynomial solutions of boundary-value problems on nonstationary heat conduction with a time-dependent boundary condition of the secondary kind for bodies having a plane geometry, a cylindrical symmetry, or a spherical symmetry is presented. Thе algorithm is based on the introduction into consideration of the boundary characteristics in the form of a definite set of k-fold derivatives and n-fold integrals with respect to the time function of the heat flow on the surface of a body representing a boundary condition. Two stages of the heat-conduction process were considered separately: 1) the temperature front does not reach the center of a body and 2) the temperature front reaches the center of the body, and it is heated throughout its thickness. By the example of symmetric heating of a lengthy plate with a constant and variable heat flows, a very high accuracy of the proposed approach based on the integral method of boundary characteristics (BChIM) was demonstrated. As compared to the method of additional boundary characteristics, the BChIM makes it possible to de crease the relative approximation error (at one and the same polynomial degrees N) by three to five orders of magnitude and by larger values and brings it to a negligibly low level (0.00028 % at N = 11 and 0.000025 % at N = 14). It was established that, with each next approximation (with addition of three degrees into the polynomial), the approximation error decreases by an order of magnitude for the first stage of the process. An efficient algorithm of finding the eigenvalues of a boundary-value problem on heat conduction, based on the introduction into consideration of an additional function corresponding to the largest, in sequence order, boundary integral characteristic, is prtsented for the second stage of the process. Thе algorithm makes it possible to transform the integro-differential equation obtained on the basis of the BChIM into the ordinary differential equation with zero initial conditions. The calculations of the temperature at the center of the plate have shown that the approximation accuracy of the approach proposed is very high.