One multidimensional weakly singular integral equation (Q941211)

The authors establish sufficient conditions for the positive definiteness of the operator \[ A\varphi\equiv a(x)\varphi(x)+ \int_D \frac {h(x,y)\varphi(y)} {\rho^\alpha(x,y)}\,dy=f(x) \] where \(x\in D\), \(\alpha=\text{const}\), \(0\leq\alpha<2\), \(\rho(x,y)=|x-y|\) (Euclidean) in the space \(L_2\). A theorem on the existence and uniqueness of a solution to the singular integral equation is deduced. Numerical schemes of approximate solution methods are proposed.