On the solution of the generalized airfoil equation (Q1366817)

The paper is devoted to the solution of the integral equations \[ {1\over \pi}\int^1_{-1} {f(y)\over x-y} dy+ {m(x)\over\pi} \int^1_{-1} f(y)\ln|x- y|dy+ {1\over\pi} \int^1_{-1} k(x,y)f(y)dy= g(x) \] which describes an oscillating two-dimensional motion around a wing of infinite amplitude and has applications in the two-dimensional elasticity (here \(f\) is the unknown function, and \(m\), \(k\) and \(g\) are given functions). The solution can be represented as a circulation around the wing. It has singularities of Cauchy's and logarithmic type and was studied by many authors in particular cases (\(m=0\) or \(k=0\)). Here the authors extend earlier results for \(k=0\), using functional analytic and operator methods. They examine the conditions under which the unperturbed generalized airfoil equation (\(f=0\)) has solutions and solve the equation in these cases. Finally, a numerical procedure is given for solving the equation in general case, together with the stability study and error estimates.