Engineering topics
Publications and source records attributed to Gingold, H..
A counterexample to the adiabatic approximation theorem is given when degeneracies are present. A formulation of an alternative version is proposed. A complete asymptotic decomposition for n dimensional self-adjoint Hamiltonian systems is restated and used.
A phenomena is described occuring in wave propagation for systems of conservation laws which are not hyperbolic. The nature of the wave propagation for linear systems of conservation laws which are degenerate in the sense that they are not equivalent to diagonal systems is examined. It is concluded that in the degenerate case, 'resonance' occurs, that is, the solutions are combinations of traveling waves and 'resonance waves'. Here resonance means that the solution may become unbounded even if the initial values are bounded. The solutions (waves) can be described as a superposition of 'packets' (groups) of 'resonance waves' traveling with the same speed. These are linear systems with constant coefficients which are self-exciting, which is what one would expect to encounter in nonlinear systems.
The oscillation or nonoscillation of a class of second-order linear differential equations is investigated analytically, with a focus on cases in which the functions phi(x) and y are complex-valued. Two linear transformations are introduced, and an asymptotic-decomposition procedure involving Shur triangularization is applied. The relationship of the present analysis to the nonoscillation criterion of Kneser (1896) and other more recent results is explored in two examples.
An invariant-matrix technique for the approximate solution of second-order ordinary differential equations (ODEs) of form y-double-prime = phi(x)y is developed analytically and demonstrated. A set of linear transformations for the companion matrix differential system is proposed; the diagonalization procedure employed in the final stage of the asymptotic decomposition is explained; and a scalar formulation of solutions for the ODEs is obtained. Several typical ODEs are analyzed, and it is shown that the Liouville-Green or WKB approximation is a special case of the present formula, which provides an approximation which is valid for the entire interval (0, infinity).