Mathematics / Matematik

Permanent URI for this collectionhttps://hdl.handle.net/11147/8

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Access Right: Open AccessSubject: search.filters.subject.Fourier series

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  • Article
    Citation - WoS: 1
    Citation - Scopus: 1
    Initial Stages of Gravity-Driven Flow of Two Fluids of Equal Depth
    (American Institute of Physics, 2023) Korobkin, Alexander; Yılmaz, Oğuz; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
    Short-time behavior of gravity-driven free surface flow of two fluids of equal depth and different densities is studied. Initially, the fluids are at rest and separated with a vertical rigid plate of negligible thickness. Then, the plate disappears suddenly and a gravity-driven flow of the fluids starts. The flow in an early stage is described by the potential theory. The initial flow in the leading order is described by a linear problem, which is solved by the Fourier series method. The motions of the interface between the fluids and their free surfaces are investigated. The singular behaviors of the velocity field at the bottom point, where the interface meets the rigid bottom, and the top point, where the interface meets both free surfaces, are analyzed in detail. The flow velocity is shown to be log-singular at the bottom point. The leading-order inner asymptotic solution is constructed in a small vicinity of this point. It is shown that the flow close to the bottom point is self-similar. The motion of the interface is independent of any parameters, including the density ratio, of the problem in specially stretched variables. In the limiting case of negligible density of one of the fluids, the results of the classical dam break problem are recovered. The Lagrangian representation is employed to capture the behavior of the interface and the free surfaces at the top, where the fluid interface meets the free surfaces. The shapes of the free surfaces and the interface in the leading order computed by using the Lagrangian variables show a jump discontinuity of the free surface near the top point where the free surfaces and the interface meet. Inner region formulation is derived near the top point.
  • Article
    Citation - WoS: 1
    Citation - Scopus: 1
    Computing the Electric and Magnetic Matrix Green's Functions in a Rectangular Parallelepiped With a Perfect Conducting Boundary
    (Hindawi Publishing Corporation, 2014) Yakhno, Valery; Ersoy, Şengül; 01. Izmir Institute of Technology
    A method for the approximate computation of frequency-dependent magnetic and electric matrix Green's functions in a rectangular parallelepiped with a perfect conducting boundary is suggested in the paper. This method is based on approximation (regularization) of the Dirac delta function and its derivatives, which appear in the differential equations for magnetic and electric Green's functions, and the Fourier series expansion meta-approach for solving the elliptic boundary value problems. The elements of approximate Green's functions are found explicitly in the form of the Fourier series with a finite number of terms. The convergence analysis for finding the number of the terms is given. The computational experiments have confirmed the robustness of the method.