Summary: The notion of spaces of fractional quotients of periodic functions was introduced earlier, where discrete operators in these spaces were studied. In the present paper, we apply this theory to the analysis of the scheme (based on the method of discrete vortex pairs) for solving the hypersingular and singular integral equations on the circle to which the Neumann problem for the Laplace equation with a closed contour can be reduced with the use of a double layer potential.
We analyze the properties, including stability, of discrete operators approximating hypersingular and singular integrals operators in the entire scale of spaces of fractional quotients and present theorems on the stability of a hypersingular discrete operator in the uniform metric. We prove the convergence of quadrature formulas in the upper part of the scale of spaces of fractional quotients. We prove the existence and uniqueness of the solutions of these equations under natural conditions, the convergence of approximate solutions to the exact solution in spaces of fractional quotients, and estimates of the convergence rate in spaces of fractional quotients and in the uniform metric with the use of a discrete analog of the Sobolev embedding theorem.
Moreover, these results are obtained in the entire scale of Sobolev-Slobodetskii spaces (and respective spaces of fractional quotients). In particular, we prove the convergence of approximate solutions to the exact solution for arbitrarily “bad” right-hand sides. This result has applications to ejection problems in gas dynamics. We point out that these results depend on the smoothness of the kernels of additional integral terms in the hypersingular (singular) operator.
We obtain convergence rate estimates for the approximate solutions in a uniform metric.
By using the Weyl operators of a fractional integration and differentiation and their discrete analogs in spaces of fractional quotients, we solve the problem on the interpolation of discrete information obtained in the solution of a system of linear algebraic equations. We prove that the resulting interpolants of approximate solutions converge in the Sobolev-Slobodetskii spaces to the exact solution at the same rate as the discrete solutions converge in spaces of fractional quotients.