Properties and Numerical Solution of an Integral Equation to Minimize Airplane Drag
In this paper, we consider an (open) airplane wing, not necessarily symmetric, for which the optimal circulation distribution has to be determined. This latter is the solution of a constraint minimization problem, whose (Cauchy singular integral) Euler-Lagrange equation is known. By following an approach different from a more classical one applied in previous papers, we obtain existence and uniqueness results for the solution of this equation in suitable weighted Sobolev type spaces. Then, for the collocation-quadrature method we propose to solve the equation, we prove stability and convergence and derive error estimates. Some numerical examples, which confirm the previous error estimates, are also presented. These results apply, in particular, to the Euler-Lagrange equation and the numerical method used to solve it in the case of a symmetric wing, which were considered in the above mentioned previous papers.
- 1.Berthold, D., Hoppe, W., Silbermann, B.: A fast algorithm for solving the generalized airfoil equation. J. Comput. Appl. Math. 43(1–2), 185–219 (1992). In: Monegato, G. (ed.) Orthogonal Polynomials and Numerical MethodsGoogle Scholar
- 2.Capobianco, M.R., Criscuolo, G., Junghanns, P.: A fast algorithm for Prandtl’s integro-differential equation. ROLLS Symposium (Leipzig, 1996). J. Comput. Appl. Math. 77(1–2), 103–128 (1997)Google Scholar
- 8.Saff, E.B., Totik, V.: Logarithmic Potentials with External Fields. Grundlehren der Mathematischen Wissenschaften, vol. 316. Springer, Berlin/Heidelberg (1997)Google Scholar