Study of approximate solution to integral equation associated with mixed boundary value problem for Laplace equation

Abstract

We consider an approximate solution of the integral equation arising after reduction of a mixed problem for the Laplace equation. The main advantage of applying the method of integral equations to studying external boundary value problems is that such approach allows one to reduce the problem posed in an unbounded domain to a problem in a domain of a smaller dimension. In the work we study an approximate solution to the integral equation, to which the mixed problem for the Laplace equation is reduced. We seek its solution as a combination of logarithmic single layer potentials and double layer potential, we reduce the problem to an integral equations depending not only on the operators generated by the logarithmic potentials but also on the composition of such operators. We prove that the obtained integral equation has the unique solution in the space of continuous functions. Since the integral equations can be solved in the closed form only in very rare cases, it is of a high importance to develop approximate methods for solving integral equations and give their appropriate theoretical justification. We partition a curve into elementary parts and by certain nodes we construct quadrature formulae for a class of curvilinear potentials and for the composition of the integrals generated by logarithmic potentials and we also estimate the errors of these formulae. Employing these quadrature formulae, the obtained integral equation is replaced by the system of algebraic equations. Then by means of Vainikko's convergence theorem for linear operator equations, we establish the existence and uniqueness of solutions to this system. We prove the convergence of the obtained system of algebraic equations to the values of the exact solution of the integral equation at the chosen nodes. Moreover, we find the convergence rate of this method. As a result, we find a sequence converging to the solution of the mixed boundary value problem for the Laplace equation and its convergence rate is known.