Spline Collocation Method for Integral Equations of the First Kind with Logarithmic Singularity

摘要

Integral equations of the first kind with a logarithmic singularity are encountered when solving a wide class of two-dimensional elliptic boundary value problems. Generally, integral equations of the first kind may be solved by means of methods of regularization. However, if the kernel contains a logarithmic singularity, then the equation can be solved by a direct numerical method. In the present paper the authors present a numerical method for solving integral equations of the first kind with weak singularity. The method is a spline collocation method. The solution of the integral equation is approximated by means of piecewise polynomials, in particular, B-splines of second and third order are chosen as a basis. Using the basis function such as the B-spline allows analytical calculation of the singular integral. Hence, the element matrix of the system of the linear equations is obtained analytically.