The Method of Fischer-Riesz Equations for Elliptic Boundary Value Problems

We develop the method of Fischer-Riesz equations for general boundary value problems elliptic in the sense of Douglis-Nirenberg. To this end we reduce them to a boundary problem for a (possibly overdetermined) first-order system whose classical symbol has a left inverse. For such a problem there is a uniquely determined boundary value problem which is adjoint to the given one with respect to the Green formula. On using a well-elaborated theory of approximation by solutions of the adjoint problem, we find the Cauchy data of solutions of our problem.

Integral Equations of the Linear Sloshing in an Infinite Chute and Their Discretization

In this work we propose a new method for solving linear elliptic equations in an unbounded domain. The method is based on the representation of the exact solution as the sum of two functions. The former is the solution of some auxiliary problem and the latter can be found using the Green formula. Using finite-difference schemes, this method has a quadratic order of accuracy independent of the size of the computational domain, and in the 2D case requires O(N³) operations to find the solution, where N³ is the number of nodes within the computational domain. In the 3D case the method requires O(N^4) operations. Test computational examples showing the method's efficiency are given.