• Journal of Mechanical Science and Technology
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• v.17 no.4
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• pp.538-544
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• 2003

Although many methodologies exist for determining the constrained equations of motion, most of these methods depend on numerical approaches such as the Lagrange multiplier's method expressed in differential/algebraic systems. In 1992, Udwadia and Kalaba proposed explicit equations of motion for constrained systems based on Gauss's principle and elementary linear algebra without any multipliers or complicated intermediate processes. The generalized inverse method was the first work to present explicit equations of motion for constrained systems. However, numerical integration results of the equation of motion gradually veer away from the constraint equations with time. Thus, an objective of this study is to provide a numerical integration scheme, which modifies the generalized inverse method to reduce the errors. The modified equations of motion for constrained systems include the position constraints of index 3 systems and their first derivatives with respect to time in addition to their second derivatives with respect to time. The effectiveness of the proposed method is illustrated by numerical examples.

• Journal of Korean Society of Coastal and Ocean Engineers
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• v.11 no.4
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• pp.197-200
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• 1999

An extension of the modified leap-frog scheme is made to solve the nonlinear shallow-water equations. In the extended model. the physical dispersion of the Boussinesq equations is replaced by the numerical dispersion resulted from the leap-frog finite difference scheme. The model is used to simulate propagations of a solitary wave over a constant water depth and a linearly varying water depth. Obtained numerical results are compared with available analytical and other numerical solutions. A reasonable agreement is observed.

• Journal of computational fluids engineering
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• v.7 no.4
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• pp.19-27
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• 2002

An efficient implicit multigrid method is presented for the Navier-Stokes and k-ω turbulence equations. Freezing and limiting strategies are applied to improve the robustness and convergence of the multigrid method. The eddy viscosity and strongly nonlinear production terms of turbulence are frozen in the coarser grids by passing down the values without update of them. The turbulence equations together with the Navier-Stokes equations, however, are consecutively solved on the coarser grids in a loosely coupled fashion. A simple limit for k is also introduced to circumvent slow-down of convergence. Numerical results for the unseparated and separated transonic airfoil flows show that all computations converge well without any robustness problem and the computing time is reduced to a factor of about 3 by the present multigrid method.

• Journal of the Korean Society of Safety
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• v.17 no.4
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• pp.126-132
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• 2002

By using the reference data, the empirical equations which describe the interrelationships of explosion properties and physical properties of n-chlorinated hydrocarbons have been derived. The properties which have been correlated are the lower and upper explosive limits, the stoichiometric coefficients, the heats of combustion, the carbon numbers. Also, the new equations using the mathematical and statistical methods for predicting the temperature dependence of lower explosive limits(LEL) of chlorinated hydrocarbons on the basis of the literature data are proposed. The fire and explosion properties calculated by the proposed equations in this research were a good agrement with literature data within a few A.A.P.E.(Average Absolute Percent Error) and A.A.D.(Average Absolute Deviation.) From a given explosive properties, by using the proposed equations, it is possible to predict to the fire and explosion characteristics for the other chlorinated hydrocarbons.

• Bulletin of the Korean Mathematical Society
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• v.30 no.1
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• pp.35-51
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• 1993

This paper concerns collocation methods for integro-differential equations in which memory kernels have a singularity at t = 0. There has been extensive research in recent years on Volterra integral and integro-differential equations for physical systems with memory effects in which the stabilty and asymtotic stability of solutionsl have been the main interest. We will study a class of hereditary equations with singular kernels which interpolate between well known model equations as the order of singularity varies. We are also concerned with the smoothing effect of singular kernels, but we use energy methods and our results involve fractional time in fixed spatial norms. Galerkin methods for our models was studied and existence, uniqueness and stability results was obtained in [4]. Our major goal is to study collocation methods.

• Journal of the Korean Mathematical Society
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• v.58 no.4
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• pp.1001-1017
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• 2021

In this paper, we first apply parabolic inequalities and a maximum principle to give a new proof for symmetry and monotonicity of solutions to fractional elliptic equations with gradient term by the method of moving planes. Under the condition of suitable initial value, by maximum principles for the fractional parabolic equations, we obtain symmetry and monotonicity of positive solutions for each finite time to nonlinear fractional parabolic equations in a bounded domain and the whole space. More generally, if bounded domain is a ball, then we show that the solution is radially symmetric and monotone decreasing about the origin for each finite time. We firmly believe that parabolic inequalities and a maximum principle introduced here can be conveniently applied to study a variety of nonlocal elliptic and parabolic problems with more general operators and more general nonlinearities.

• Journal of applied mathematics & informatics
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• v.42 no.4
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• pp.899-913
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• 2024

In this article, the Series Solution Method (SSM) is employed to solve the linear or non-linear Volterra integro-differential equations. Numerous examples have been presented to explain the numerical results, which is the comparison between the exact solution and the numerical solution, and it is found through the tables. The amount of error between the exact solution and the numerical solution is very small and almost nonexistent, and it is also illustrated through the graph how the exact solution completely applies to the numerical solution. This proves the accuracy of the method, which is the Series Solution Method (SSM) for solving the linear or non-linear Volterra integro-differential equations using Mathematica. Furthermore, this approach yields numerical results with remarkable accuracy, speed, and ease of use.

• The Pure and Applied Mathematics
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• v.23 no.3
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• pp.223-236
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• 2016

We show how to evaluate the cubic continued fraction $G(e^{-{\pi}\sqrt{n}})$ and $G(-e^{-{\pi}\sqrt{n}})$ for n = 4^m, 4^−m, 2 · 4^m, and 2⁻¹ · 4^−m for some nonnegative integer m by using modular equations of degree 9. We then find some explicit values of them.

• Water Engineering Research
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• v.5 no.4
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• pp.177-183
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• 2004

This paper introduces an eigenvalue method of transforming the hyperbolic partial differential equations of a particular unsteady friction water hammer model into characteristic form. This method is based on the solution of the corresponding one-dimensional Riemann problem that transforms hyperbolic quasi-linear equations into ordinary differential equations along the characteristic directions, which in this case arises as the eigenvalues of the system. A mathematical justification and generalization of the eigenvalues method is provided and this approach is compared to the traditional characteristic method.

• Journal of the Chungcheong Mathematical Society
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• v.20 no.4
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• pp.465-476
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• 2007

The aim of this paper is to investigate the stability problem bounded by function for the generalized sine functional equations as follow: $f(x)g(y)=f(\frac{x+y}{2})^2-f(\frac{x+{\sigma}y}{2})^2\\g(x)g(y)=f(\frac{x+y}{2})^2-f(\frac{x+{\sigma}y}{2})^2$. As a consequence, we have generalized the superstability of the sine type functional equations.