High-Accuracy Numerical Schemes for Solving Plane Boundary Problem for a Polyharmonic Equation and Their Application to Problems of Hydrodynamics

Boundary value problems are considered for harmonic, biharmonic equations, as well as the general polyharmonic equation for multiply connected domains on the plane. The problems are reduced to solving linear integral equations on boundary contours, which are assumed to be smooth. An algorithm for deriving an approximation of integral equations by a linear system is presented, taking into account the logarithmic singularities of the kernels of integral operators, through which integral equations are expressed. The algorithm uses the periodicity of functions defined on closed boundary contours. As the number of grid points increases, the approximation error decreases faster than the grid spacing to any fixed degree. Applications to solving problems of hydrodynamics, filtration and other problems of theoretical physics are considered.