• The Transactions of The Korean Institute of Electrical Engineers
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• v.65 no.11
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• pp.1854-1859
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• 2016

Consider a linear fourth-order system with no zero that is represented in terms of four specific parameters: two damping ratios and two natural frequencies. We investigate several interesting questions about the maximum overshoot of the system with respect to the four-tuple parameters. Some remarkable results are presented.

• Journal of applied mathematics & informatics
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• v.36 no.5_6
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• pp.491-504
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• 2018

In this article, we consider a fourth order nonlinear difference system. By making use of the critical point theory, we obtain some new existence theorems of at least one periodic solution with minimal period. Our main approach used in this article is the variational technique and the Saddle Point Theorem.

• Journal of applied mathematics & informatics
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• v.8 no.3
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• pp.935-942
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• 2001

A finite element approximation of a fourth-order nonlinear boundary value problem is given. In the direct implementation, a nonlinear system will be obtained and also a full size matrix will be introduced when Newton’s method is adopted to solve the system. To avoid this difficulty we introduce an iterative scheme which can be shown to converge the positive solution of the system lying between 0 and $sin{\pi}x$.

• Structural Engineering and Mechanics
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• v.89 no.2
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• pp.199-211
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• 2024

The loading capacity of engineering structures/components reduces after the initiation and propagation of crack eventually leads to the final failure. Hence, it becomes essential to deal with the crack and its effects at the design and simulation stages itself, by detecting the prone area of the fracture. The phase-field (PF) method has been accepted widely in simulating fracture problems in complex geometries. However, most of the PF methods are formulated with second order continuity theoryinvolving C⁰ continuity. In the present study, PF method based on fourth-order (i.e., higher order) theory, maintaining C¹ continuity has been proposed for ductile fracture simulation. The formulation includes fourth-order derivative terms of phase field variable, varying between 0 and 1. Applications of fourth-order PF theory to ductile fracture simulation resulted in novelty in this area. The proposed formulation is numerically solved using a two-dimensional finite element (FE) framework in 3-layered manner system. The solutions thus obtained from the proposed fourth order theory for different benchmark problems portray the improvement in the accuracy of the numerical results and are well matched with experimental results available in the literature. These results are also compared with second-order PF theory and a comparison study demonstrated the robustness of the proposed model in capturing ductile behaviour close to experimental observations.

• Journal of the Chungcheong Mathematical Society
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• v.21 no.3
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• pp.413-426
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• 2008

An asymptotic solution of a fourth order critically damped nonlinear differential system has been found by means of extended Krylov-Bogoliubov-Mitropolskii (KBM) method. The solutions obtained by this method agree with those obtained by numerical method. The method is illustrated by an example.

• Transactions of the Korean Society of Mechanical Engineers
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• v.11 no.5
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• pp.781-788
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• 1987

The state variables estimated in an observer were useed in feedback control of a hydraulic servosystem to increase the system stability and to enhance the system performance. The nonlinear hydraulic servosystem with the inherent nonlinearities due to the square root function of flow equation, the Coulomb friction and so on, was modelled as a fourth order linear hydraulic servosystem. Also, a second order linear system was derived for the observer-controller design. For these models, a fourth order linear observer and a second order linear observer were constructed respectively to evaluate the performance of the observer-based hydraulic servosystem. The results obtained from series of simulation showed that the system which had shown oscillatory phenomenon under proportional control became stable with the same maximum acceleration and velocity that it had started under proportional control.

• Journal of applied mathematics & informatics
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• v.34 no.5_6
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• pp.495-508
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• 2016

A singularly perturbed reaction-diffusion fourth-order ordinary differential equation(ODE) with discontinuous source term is considered. Due to the discontinuity, interior layers also exist. The considered problem is converted into a system of weakly coupled system of two second-order ODEs, one without parameter and another with parameter ε multiplying highest derivatives and suitable boundary conditions. In this paper a computational method for solving this system is presented. A zero-order asymptotic approximation expansion is applied in the second equation. Then, the resulting equation is solved by the numerical method which is constructed. This involves non-overlapping Schwarz method using Shishkin mesh. The computation shows quick convergence and results presented numerically support the theoretical results.

• ETRI Journal
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• v.43 no.5
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• pp.869-880
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• 2021

Herein, we estimate the direction of arrival (DOA) of non-Gaussian signals for nested arrays (NAs) by implementing the fourth-order difference co-array (FODC) and successive methods. In particular, considering the property of the fourth-order cumulant (FOC), we first construct the FODC of the NA, which can obtain O(N⁴) virtual elements using N physical sensors, whereas conventional FOC methods can only obtain O(N²) virtual elements. In addition, the closed-form expression of FODC is presented to verify the enhanced degrees of freedom (DOFs). Subsequently, we exploit the vectorized FOC (VFOC) matrix to match the FODC of the NA. Notably, the VFOC matrix is a single snapshot vector, and the initial DOA estimates can be obtained via the discrete Fourier transform method under the underdetermined correlation matrix condition, which utilizes the complete DOFs of the FODC. Finally, fine estimates are obtained through the spatial smoothing-Capon method with partial spectrum searching. Numerical simulation verifies the effectiveness and superiority of the proposed method.

• Bulletin of the Korean Mathematical Society
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• v.54 no.3
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• pp.895-909
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• 2017

This paper is concerned with the following fourth-order elliptic equations $${\Delta}^2u-{\Delta}u+V(x)u-{\frac{k}{2}}{\Delta}(u^2)u=f(x,u),\text{ in }{\mathbb{R}}^N$$, where $N{\leq}6$, ${\kappa}{\geq}0$. Under some appropriate assumptions on V(x) and f(x, u), we prove the existence of infinitely many negative-energy solutions for the above system via the genus properties in critical point theory. Some recent results from the literature are extended.

• Journal of the Korean Mathematical Society
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• v.59 no.5
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• pp.891-909
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• 2022

The main objective of this paper is to develop a concrete inverse formula of the system induced by the fourth-order finite difference method for two-point boundary value problems with Robin boundary conditions. This inverse formula facilitates to make a fast algorithm for solving the problems. Our numerical results show the efficiency and accuracy of the proposed method, which is implemented by the Thomas algorithm.