Abstract
The aim of the paper is to investigate the solvability of an infinite system of nonlinear integral equations on the real half-axis. The considerations will be located in the space of function sequences which are bounded at every point of the half-axis. The main tool used in the investigations is the technique associated with measures of noncompactness in the space of functions defined, continuous and bounded on the real half-axis with values in the space l∞ consisting of real bounded sequences endowed with the standard supremum norm. The essential role in our considerations is played by the fact that we will use a measure of noncompactness constructed on the basis of a measure of noncompactness in the mentioned sequence space l∞. An example illustrating our result will be included.
An efficient numerical method, using integral equations, is developed to calculate precisely the acoustic eigenfrequencies and their associated eigenvectors, located in a given high frequency interval. It is currently known that the real symmetric matrices are well adapted to numerical treatment. However, we show that this is not the case when using integral representations to determine with high accuracy the spectrum of elliptic, and other related operators. Functions are evaluated only in the boundary of the domain, so very fine discretizations may be chosen to obtain high eigenfrequencies. We discuss the stability and convergence of the proposed method. Finally we show some examples.
The purpose of this paper is to find out fixed point results for the family of multivalued mappings fulfilling a generalized rational type F-contractive conditions on a closed ball in complete dislocated b-metric space. An application to the system of integral equations is presented to show the novelty of our results. Our results extend several comparable results in the existing literature.
In this work, first, the partially cavitating hydrofoil problem is formulated in linear theory in terms of vorticity and source distributions on the projection of the hydrofoil to the free-stream direction. The resulting system of integral equations is inverted and the solution is expressed in terms of integrals of the horizontal perturbation velocity in fully wetted flow, multiplied by weighting functions that are independent of the shape of the hydrofoil. Second, the linearized dynamic boundary condition on the cavity is modified so that the total velocity on the cavity as predicted by applying Lighthill's leading-edge corrections is a constant. This results in a varying horizontal perturbation velocity on the cavity rather than a constant as required by conventional linear theory. The modified system of integral equations is inverted and the solution is expressed in terms of integrals of known quantities. The present linearized theory with the leading-edge corrections included, predicts a finite cavitation inception number as well as the correct effect of foil thickness on cavity size.
Measure of noncompactness and a generalized Darbo’s fixed point theorem and its applications to a system of integral equations