Analytical integration of weakly singular integrals in boundary element analysis of Helmholtz and advection–diffusion equations Original Research Article

This paper presents analytical evaluation of weakly singular integrals arising in the boundary element analysis of Helmholtz, modified Helmholtz and advection–diffusion equations. Expressions for these boundary integrals are presented in terms of elementary integrals for straight line elements of arbitrary order, which are applicable not only to the singular integrals but also to the regular integrals when the collocation point is collinear with the integration element. Analytical expressions have been derived for the elementary integrals using integrals of Bessel functions. Analytical approximations of the zeroth order elementary integrals for the modified Helmholtz and advection–diffusion equations suitable for single as well as double precision computations have been proposed, which would be specially useful for large wave numbers or Peclét numbers for the respective problems.