• Journal for History of Mathematics
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• v.28 no.4
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• pp.167-179
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• 2015

We study the solution of truncated series of Lee Sang-hyeog with the aspect of visualization. Lee Sang-hyeog solved a problem of truncated series by 4 ways: Shen Kuo' series method, splitting method, difference sequence method, and Ban Chu Cha method. As the structure and solution of truncated series in tertiary number is already clarified with algebraic symbols in some previous research, we express and explain it by visual representation. The explanation and proof of algebraic symbols about truncated series is clear in mathematical aspects; however, it has a lot of difficulties in the aspects of understanding. In other words, it is more effective in the educational situations to provide algebraic symbols after the intuitive understanding of structure and solution of truncated series with visual representation.

• Honam Mathematical Journal
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• v.45 no.4
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• pp.619-628
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• 2023

In this paper, the fractional order time-varying linear dynamical systems are investigated by using a residual power series method. A residual power series method (RPSM) is constructed for this problem. The exact solution is obtained by the Laplace transform method and the analytical solution is calculated via the residual power series method (RPSM). As an application, some examples are tested to show the accuracy and efficacy of the proposed methods. The obtained result showed that the proposed methods are effective and accurate for this type of problem.

• Journal of Advanced Research in Ocean Engineering
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• v.1 no.3
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• pp.157-167
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• 2015

In order to provide simple and accurate wave theory in design of offshore structure, an analytical approximation is introduced in this paper. The solution is limited to flat bottom having a constant water depth. Water is considered as inviscid, incompressible and irrotational. The solution satisfies the continuity equation, bottom boundary condition and non-linear kinematic free surface boundary condition exactly. Error for dynamic condition is quite small. The solution is suitable in description of breaking waves. The solution is presented with closed form and dispersion relation is also presented with closed form. In the last century, there have been two main approaches to the nonlinear problems. One of these is perturbation method. Stokes wave and Cnoidal wave are based on the method. The other is numerical method. Dean's stream function theory is based on the method. In this paper, power series method was considered. The power series method can be applied to certain nonlinear differential equations (initial value problems). The series coefficients are specified by a nonlinear recurrence inherited from the differential equation. Because the non-linear wave problem is a boundary value problem, the power series method cannot be applied to the problem in general. But finite number of coefficients is necessary to describe the wave profile, truncated power series is enough. Therefore the power series method can be applied to the problem. In this case, the series coefficients are specified by a set of equations instead of recurrence. By using the set of equations, the nonlinear wave problem has been solved in this paper.

• Journal of Advanced Marine Engineering and Technology
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• v.8 no.1
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• pp.100-111
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• 1984

The methods solving the 2nd order linear ordinary differential equations of the form y"+H(x)y'+G(x)y=P(x) using Fourier series are presented in this paper. These methods are applied to the differential equations of which the exact solutions are known, and the solutions by Fourier series are compared with the exact solutions. The main results obtained in these studies are summarized as follows; 1) The product and the quotient of two functions expressed in Fourier series can be expressed also in Fourier series and the relations between the Fourier coefficients of the series are obtained by multiplying term by term. 2) If the solution of the 2nd order lindar ordinary differential equation exists in a certain interval, the solution can be obtained using Fourier series and can be expressed in Fourier series. 3) The absolute errors of Fourier series solutions are generally less in the center of the interval than in the end of the interval. 4) The more terms are considered in Fourier series solutions, the less the absolute errors.rors.

• Geomechanics and Engineering
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• v.6 no.1
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• pp.47-63
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• 2014

In this paper, asymmetric drainage boundaries modeled by exponential functions which can simulate intermediate drainage from pervious to impervious boundary is proposed for the one-dimensional consolidation problem, and the solution for the new boundary conditions was derived. The new boundary conditions satisfy the initial and the steady state conditions, and the solution for the new boundary conditions can be degraded to the conventional solution by Terzaghi. Convergence study on the infinite series solution showed that only one term in the series is needed to meet the precision requirement for larger degree of consolidation, and that more terms in the series for smaller degree of consolidation. Comparisons between the present solution with those by Terzaghi and Gray are also provided.

• Structural Engineering and Mechanics
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• v.45 no.5
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• pp.631-641
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• 2013

In this paper, the problem of axisymmetric deformation of the circular membrane fixed at its perimeter under the action of uniformly-distributed loads was resolved by exactly using power series method, and the solution of the problem was presented. An investigation into the so-called Hencky transformation was carried out, based on the solution presented here. The results obtained here indicate that the well-known Hencky solution is, without doubt, correct, but in the published papers the statement about its derivation is incorrect, and the so-called Hencky transformation is invalid and hence may not be extended to use as a general mathematical method.

• Bulletin of the Korean Chemical Society
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• v.10 no.6
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• pp.529-535
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• 1989

A general perturbation series solution of the Smoluchowski equation is applied to investigate the rate of recombination and the remaining probability of a pair of particles in liquids. The radiative boundary condition is employed and the convergence of the perturbation series is analyzed in terms of a convergene factor in time domain. The upper bound to the error introduced by the n-th order perturbation scheme is also evaluated. The long time behaviors of the rate of recombination and the remaining probability are found to be expressed in closed forms if the perturbation series is convergent. A new and efficient method of purely numerical integration of the Smoluchowski equation is proposed and its results are compared with those obtained by the perturbation method. For the two cases where the interaction between the particles is given by (i) the Coulomb potential and (ii) the shielded Coulomb potential, the agreement between the two results is found to be excellent.

• Korean Journal of Materials Research
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• v.28 no.12
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• pp.738-747
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• 2018

For the aerospace structural application of high-strength 2xxx series aluminum alloys, stress corrosion cracking(SCC) behavior in aggressive environments needs to be well understood. In this study, the SCC sensitivities of 2024-T62, 2124-T851 and 2050-T84 alloys in a 3.5 % NaCl solution are measured using a constant load testing method without polarization and a slow strain rate test(SSRT) method at a strain rate of 10-6 /sec under a cathodic applied potential. When the specimens are exposed to a 3.5 % NaCl solution under a constant load for 10 days, the decrease in tensile ductility is negligible for 2124-T851 and 2050-T84 specimens, proving that T8 heat treatment is beneficial in improving the SCC resistance of 2xxx series aluminum alloys. The specimens are also susceptible to SCC in a hydrogen-generating environment at a slow strain rate of $10^{-6}/sec$ in a 3.5 % NaCl solution under a cathodic applied potential. Regardless of the test method, low impurity 2124-T851 and high Cu/Mg ratio 2050-T84 alloys are found to have relatively lower SCC sensitivity than 2024-T62. The SCC behavior of 2xxx series aluminum alloys in the 3.5 % NaCl solution is discussed based on fractographic and micrographic observations.

• Proceedings of the SAREK Conference
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• 2007.11a
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• pp.534-539
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• 2007

The objectives of the present work is to investigate the influence of the solution cooled absorber(SCA), refrigerant drain heat exchanger(RSX), exhaust gas/solution heat exchanger(ESX) and high efficiency solution heat exchanger on COP for a double-effect series-flow absorption chiller. A simulation program has been prepared for the cycle analysis of absorption chillers. As a result, Solution heat exchangers(LSX, HSX) are a most effective element for the COP than the others. In spite of the poor contribution to COP, SCA make a rule to reduce the crystallization phenomena of LiBr solution at solution heat exchanger. And the optimum solution split ratio are varied with the relative size of RSX and LSX.

• Korean Journal of Air-Conditioning and Refrigeration Engineering
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• v.20 no.10
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• pp.662-668
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• 2008

The objectives of the present work are to investigate the influence of the solution cooled absorber(SCA), refrigerant drain heat exchanger(RSX), exhaust gas/solution heat exchanger(ESX) and high efficiency solution heat exchanger on COP for a double-effect series-flow absorption chiller. A simulation program has been prepared for the cycle analysis of absorption chillers. As a result, solution heat exchangers(LSX, HSX) are the most effective element for the COP than the others. In spite of the poor contribution to COP, SCA plays an important role to reduce the crystallization phenomena of LiBr solution at solution heat exchanger. And the optimum solution split ratio varies with the relative size of RSX and LSX.