Degenerate kernels of polyharmonic and poly-Helmholtz operators in polar and spherical coordinates

Abstract

This paper presents the degenerate kernels for the polyharmonic and poly-Helmholtz partial differential operators in the polar and spherical coordinates. These degenerate kernels are essential if the analytical/semi-analytical solutions and the mathematical degeneracies involving these operators are conducted by the boundary integral equations. In addition, a two-dimensional creeping flow problem is considered to illustrate the use of the degenerate kernels for problems governed by coupled partial differential equations. The coupled governing equations of the creeping flow problems are converted into a biharmonic equation by using the Hörmander linear partial differential operator theory and the Cartesian partial derivatives on the harmonic and biharmonic degenerate kernels are used to obtain the degenerate kernels of two-dimensional creeping flow problems for BIE analyses. Finally, numerical experiments are conducted to study the convergence of the polyharmonic and poly-Helmholtz degenerate kernels.