On the role of singular and hypersingular BIEs for the BVPs containing a degenerate boundary

Abstract

ABSTR A C T In this paper, we find that using either the singular boundary integral equation (BIE) alone or the hypersingular BIE alone may solve the symmetric and anti-symmetric problems containing a degenerate boundary subject to the remote anti-plane shear, respectively. Although the rank of the influence matrix is deficient due to the degenerate boundary, it is still solvable. The mechanism was analytically demonstrated by using the degenerate kernel and numerically performed by using the singular value decomposition (SVD) technique. Besides, solv-ability was also discussed by using the Fredholm alternative theorem in both the boundary integral equation and boundary element method (BEM). We can analytically prove and numerically explain that the boundary density solved by using the singular BIE alone is different from the exact solution but the field solution is acceptable. The solvability of the linear algebraic system was discussed from the mapping table of the vector space of influence matrix where the domain is composed by the right singular vectors and the vector space of the range is composed of left singular vectors. The corresponding null space and the left null space due to the zero singular value in-dicates the homogeneous solution and the range deficiency, respectively. Several approaches, the singular or hypersingular BIE alone, adding symmetric or anti-symmetric boundary constraints, the self-regularized approach provided us alternative ways to solve the degenerate-boundary problem without using the dual BEM.