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Published 1973 | public
Journal Article

Variational Calculus Involving Singular Integral Equations

Abstract

A new class of optimization problems arising in fluid mechanics can be characterized mathematically as equivalent to extremizing a functional in which the two unknown argument functions are related by a singular Cauchy integral equation. Analysis of the first variation of the functional yields a set of dual, nonlinear, integral equations, as opposed to the Euler differential equation in classical theory. A necessary condition for the extremum to be a minimum is derived from consideration of the second variation. Analytical solutions by singular integral equation methods and by the Rayleigh‐Ritz method are discussed for the linearized theory. The general features of these solutions are demonstrated by numerical examples.

Additional Information

© 1973 WILEY‐VCH. Manuscript received: 9 January 1973. Funding Information: U. S. Navy. Grant Number: N 00014‐67‐A‐0094‐0011

Additional details

Created:
August 19, 2023
Modified:
October 19, 2023