Some direct numerical approaches to the solution of the singular nonlinear transonic integral equation

Abstract

The nonlinear singular integral equation which is of interest to researchers in transonic flow is critically examined, and reasons are advanced to indicate why standard numerical techniques are not satisfactory in solving this type of equation. The inherent difficulties in the approximation of the integral term of this equation are pointed out, and special techniques are proposed to mitigate the inaccuracies resulting from the standard approximations. The basic principle for each of these techniques is the same, but the orders of approximation differ; for each such approach, numerical results are given together with the computational time involved. Wherever possible, estimates for the error have been included; it is also shown how the integral may be bounded from above and below. Though the essence of the paper is numerical, it also contains certain analytic features; it has been shown, for example, how the infinite domain of integration may be reduced to a finite one, and how the singular kernel may be reduced to a non-singular one. Numerical results have been plotted and compared with those of earlier investigators and it is seen that the methods described here are an improvement over the preceding works, either in numerical accuracy or computational economy.