Numerical methods for integral equations of Mellin type

A survey of numerical methods for integral equations of Mellin type, including examples arising in boundary integral methods for partial differential equations on polygonal domains is presented. It is remarked that the numerical analysis of these 2D corner problems is still not as satisfactory as in the case of smooth boundaries where even fully discrete high-order methods of almost linear computational complexity are known. However, fast solution methods for classical first-kind integral equations on open arcs have been obtained applying the cosine transform and discrete trigonometric collocation. The development of analogous methods for more general problems on corner domains remains a challenge for the future.