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Variational Calculus Involving Singular Integral Equations
Author(s) -
Wu T. YaoTsu,
Whitney A. K.
Publication year - 1973
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.19730531103
Subject(s) - mathematics , integral equation , mathematical analysis , singular integral , singular solution , calculus of variations , cauchy distribution , nonlinear system , euler equations , euler's formula , physics , quantum mechanics
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.