Nonlinear finite-part singular integral equations arising in two-dimensional fluid mechanics
The authors study a nonlinear Volterra integral equation of the type \[ u(x) = F\{x,u(x)\Phi(u),B_1(u),\ldots,B_k(u)\} + v(x), \] where \(\Phi\) is a finite-part integral \[ \Phi(u) = {1 \over 2\pi i} \int_L {u(t) \over (t-x)^\mu} \] (here \(\mu\) is a positive integer and \(L\) a contour) and \(B_j(u) = \int_L K_j(x,t)u(t) dt\). First an approximation method is used to prove the existence of a solution, and then the same approximation method is studied further with the purpose of getting numerical approximations of the solution. The paper ends with an application to a two-dimensional fluid mechanics problem.
nonlinear finite-part singular integral equations
scientific article; zbMATH DE number 1491652