Null-space distributions. -- A new approach to finite convolution equations with a Hankel kernel (Q1818823)

Continuing earlier research on the mixed problem for the Helmholtz equation, the author studies the boundary integral equation \(\int^1_{-1} H_0 (k|x-y|) u(y) dy = f(x)\), \(-1 \leq x \leq 1\), where \(H_0\) is the first kind Hankel function of order zero. Using a distributional solution of the homogeneous equation (the null space distribution), the integral equation can be solved for arbitrary right-hand sides. The approach is analogous to the method of variation of parameters in the theory of ordinary differential equations. The singularities of the null space distribution are studied, and its relation to the associated ordinary differential equation is discussed.