• Journal of the Chungcheong Mathematical Society
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• v.34 no.3
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• pp.189-207
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• 2021

In this paper, we consider the approximate controllability for a class of semilinear integro-differential functional control equations in which nonlinear terms of given equations satisfy quasi-homogeneous properties. The main method used is to make use of the surjective theorems that is similar to Fredholm alternative in the nonlinear case under restrictive assumptions. The sufficient conditions for the approximate controllability is obtain which is different from previous results on the system operator, controller and nonlinear terms. Finally, a simple example to which our main result can be applied is given.

• Korean Journal of Mathematics
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• v.30 no.4
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• pp.603-614
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• 2022

The main intent of this paper is to prove an existence and uniqueness fixed point result under Geraghty contractions in b-metric-like spaces, which remains an extended version of corresponding results in b-metric spaces and metriclike spaces. Using two types of Geraghty contractions, an approach is adopted to verify some fixed point results in b-metric-like spaces. Our main result is an extension of an earlier result given by Geraghty in b-metric-like-space. An example is also provided to demonstrate the validity of our main result. Moreover, as an application of our main result, the existence of solution of a Fredholm integral equation is established which may further be utilized to study the seismic response of dams during earthquakes.

• The Korean Journal of Applied Statistics
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• v.18 no.1
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• pp.57-66
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• 2005

In this paper, we propose a method to evaluate the solutions of the renewal equations related to SPRT for Erlang distribution. In SPRT, the Average Sample Number(ASN) and type I or type II error probabilities are shown in Fredholm type integral equations. The integral equations are generally solved by the approximation method using Gaussian quadrature. For Erlang distribution, it has been known that the exact solutions of the equations exist. We propose the algorithm to solve the equations.

• Bulletin of the Society of Naval Architects of Korea
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• v.19 no.1
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• pp.23-32
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• 1982

Herein the motion of a freely-floating sphere in a water of finite depth is analysed within the framework of a linear potential theory. A velocity potential describing fluid motion is generated by distributing pulsating sources and dipoles on the immersed surface of the sphere, without introducing an inner flow model. The potential becomes the solution of an integral equation of Fredholm's second type. In the light of the vertical axisymmetry of the flow, surface integrals reduce to line integrals, which are approximated by summation of the products of the integrand and the length of segments along the contour. Following this computational scheme the diffraction potential and the radiation potential are determined from the same algorithm of solving a set of simultaneous linear equations. Upon knowing values of the potentials hydrodynamic forces such as added mass, hydrodynamic damping and wave exciting forces are evaluated by the integrating pressure over the immersed surface of the sphere. It is found in the case of finite water depth that the hydrodynamic forces are much different from the corresponding ones in deep water. Accordingly motion response of the sphere in a water of finite depth displays a particular behavior both in a amplitude and phase.

• Structural Engineering and Mechanics
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• v.85 no.5
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• pp.621-633
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• 2023

The major objective of this paper is to study the receding contact problem between two functional graded layers under a flat indenter. The gravity is assumed negligible, and the shear moduli of both layers are assumed to vary exponentially along the thickness direction. In the absence of body forces, the problem is reduced to a system of Fredholm singular integral equations with the contact pressure and contact size as unknowns via Fourier integral transform, which is transformed into an algebraic one by the Gauss-Chebyshev quadratures and polynomials of both the first and second kinds. Then, an iterative speediest descending algorithm is proposed to numerically solve the system of algebraic equations. Both semi-analytical and finite element method, FEM solutions for the presented problem validate each other. To improve the accuracy of the numerical result of FEM, a graded FEM solution is performed to simulate the FGM mechanical characteristics. The results reveal the potential links between the contact stress/size and the indenter size, the thickness, as well as some other material properties of FGM.

• The Journal of the Acoustical Society of Korea
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• v.39 no.3
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• pp.151-162
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• 2020

This paper presents an acoustic inversion method for estimating the bubble size distribution. The estimation error of the attenuation coefficient represented by a Fredholm integral equation of the first kind is defined as an objective function, and an optimal solution is found by applying the Levenberg-Marquardt (LM) method. In order to validate the effectiveness of the inversion method, numerical simulations using two types of bubble distribution are performed. In addition, a series of experiments are carried out in a water tank (1.0 m × 0.54 m × 0.6 m), using bubbles generated by three different generators. Images of the distributed bubbles are obtained by a high-speed camera, and the insertion losses of the bubble layer are measured using a source and a hydrophone. The image is post-processed to glance a distribution characteristics of each bubble generator. Finally, the size distribution of bubbles is estimated by applying the inversion method to the measured insertion loss. From the inversion results, it was observed that the number of bubbles increases exponentially as the bubble size decreases, and then increases again after the local peak at 70 ㎛ - 120 ㎛.

• Bulletin of the Society of Naval Architects of Korea
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• v.23 no.4
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• pp.25-34
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• 1986

A two-dimensional linear method has been developed for the motion and the second-order steady force arising from the hydrodynamic coupling between waves and currents in the presence of a body of arbitrary shape. Interaction between the incident wave and current in the absence of the body lies in the realm beyond our interest. A Fredholm integral equation of the second kind is employed in association with the Haskind's potential for a steadily moving source of pulsating strength located in or below the free surface. The numerical calculations at the preliminary stage showed a significant fluctuation of the hydrodynamic forces on the surface-piercing body. The problem is approximately solved by using the asymptotic Green function for $U^2{\rightarrow}0$. The original Green function, however, is applied for the fully submerged body. Numerical calculations are made for a submerged and for a half-immersed circular cylinder and extensively for the mid-ship section of a Lewis-form. Some of the results are compared with other analytical results without any available experimental data. The current has strong influence on roll motion near resonance. When the current opposes the waves, the roll response are generally negligible in the low frequency region. The current has strong influence on roll motion near resonance. When the current opposes the wave, the roll response decreases. When the current and wave come from the same direction, the roll response increases significantly, as the current speed increases. The mean drift forces and moment on the submerged body are more affected by current than those on the semi-immersed circular cylinder or on the ship-like section in the encounter frequency domain.

• Geomechanics and Engineering
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• v.31 no.5
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• pp.477-488
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• 2022

Pile composite foundation (PCF) has been commonly applied in practice. Existing research has focused primarily on semi-infinite media having equal pile lengths with little attention given to the effects of inclined bedrock and dissimilar pile lengths. This investigation considers the effects of inclined bedrock on vertical loaded PCF with dissimilar pile lengths. The pile-soil system is decomposed into fictitious piles and extended soil. The Fredholm integral equation about the axial force along fictitious piles is then established based on the compatibility of axial strain between fictitious piles and extended soil. Then, an iterative procedure is induced to calculate the PCF characteristics with a rigid cap. The results agree well with two field load tests of a single pile and numerical simulation case. The settlement and load transfer behaviors of dissimilar 3-pile PCFs and the effects of inclined bedrock are analyzed, which shows that the embedded depth of the inclined bedrock significantly affects the pile-soil load sharing ratios, non-dimensional vertical stiffness N₀/wdE_s, and differential settlement for different length-diameter ratios of the pile l/d and pile-soil stiffness ratio k conditions. The differential settlement and pile-soil load sharing ratios are also influenced by the inclined angle of the bedrock for different k and l/d. The developed model helps better understand the PCF characteristics over inclined bedrock under vertical loading.

• Journal of the Society of Naval Architects of Korea
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• v.28 no.1
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• pp.12-18
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• 1991

Three-dimensional nonlinear sloshing effects due to tank motions are simulated by solving boundary value problem using the panel method based on boundary integral technique. While Shinkai used boundary elements on which source strengths vary linearly between nodes, the source of constant strength is distributed on each triangular panel in the present study. The source strength at each time step is determined by solving the Fredholm integral equation of the second kind obtained from Green's theorem. To avoid cumulative numerical errors as time elapses, Adam-Bashforth-Moulton method is employed. Numerical examples for the case of partially filled spherical tank on board oscillating in harmonic sway mode or pitch mode are solved. The elevation of the free surface is compared with the result by Shinkai and confirmed in good agreement during early time. The input and the output energy are comparatively evaluated to check the overall accuracy of the present numerical scheme. Although some leakage of energy are found as time marches, it is plausible when we take into account nonlinearities of the problem and the number of panels of the model.