• Journal of the Chungcheong Mathematical Society
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• v.25 no.2
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• pp.149-157
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• 2012

By means of fractional calculus techniques, we find explicit solutions of confluent hypergeometric equations. We use the N-fractional calculus operator $N^{\mu}$ method to derive the solutions of these equations.

• Journal of the Korean Mathematical Society
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• v.61 no.4
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• pp.621-657
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• 2024

In this study, the numerical solution of a space-fractional Burgers' equation with initial and boundary conditions is considered. This equation is the simplest nonlinear model for diffusive waves in fluid dynamics. It occurs in a variety of physical phenomena, including viscous sound waves, waves in fluid-filled viscous elastic pipes, magneto-hydrodynamic waves in a medium with finite electrical conductivity, and one-dimensional turbulence. The proposed QBS/CNG technique consists of the Galerkin method with a function basis of quadratic B-splines for the spatial discretization of the space-fractional Burgers' equation. This is then followed by the Crank-Nicolson approach for time-stepping. A linearized scheme is fully constructed to reduce computational costs. Stability analysis, error estimates, and convergence rates are studied. Finally, some test problems are used to confirm the theoretical results and the proposed method's effectiveness, with the results displayed in tables, 2D, and 3D graphs.

• Journal of the Korean Society for Precision Engineering
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• v.12 no.4
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• pp.127-134
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• 1995

The dynamic characteristics of a system can be critically influenced by system uncertainty, so the dynamic system must be analyzed stochastically in consideration of system uncertainty. This study presents a generalized stochastic model of dynamic system subjected to bot external and parametric nonstationary stochastic input. And this stochastic system is analyzed by a new stochastic process closure method and moment equation method. The first moment equation is numerically evaluated by Runge-Kutta method. But the second moment equation is founded to constitute an infinite coupled set of differential equations, so this equations are numerically evaluated by cumulant neglect closure method and Runge-Kutta method. Finally the accuracy of the present method is verified by Monte Carlo simulation.

• Journal of the Korean Statistical Society
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• v.34 no.2
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• pp.85-98
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• 2005

A Bayesian analysis for the product of different powers of k independent Poisson rates, written ${\theta}$, is developed. This is done by considering a prior for ${\theta}$ that satisfies the differential equation due to Tibshirani and induces a proper posterior distribution. The Gibbs sampling procedure utilizing the rejection method is suggested for the posterior inference of ${\theta}$. The procedure is straightforward to specify distributionally and to implement computationally, with output readily adapted for required inference summaries. A salient feature of the procedure is that it provides a unified method for inferencing ${\theta}$ with any type of powers, and hence it solves all the existing problems (in inferencing ${\theta}$) simultaneously in a completely satisfactory way, at least within the Bayesian framework. In two examples, practical applications of the procedure is described.

• Transactions of the Korean Society of Machine Tool Engineers
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• v.10 no.1
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• pp.112-119
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• 2001

Design sensitivity is an important is an important device in improving a mechanical system design. A continuum design consists of the shape and orientation design. This research develops the shape and orientation design sensitivity method. The configura-tion design variables of multibody systems define the shape and orientation changes. The equations of motion are directly differentiated to obtain the governing equations for the design sensitivity. The governing equation of the design sensitivity is formulated as an over determined differential algebraic equation and treated as ordinary differential equations on mani-folds. The material derivative of a domain functional is performed to obtain the sensitivity due to shape and orientation changes. The configuration design sensitivities of a fly-ball governor system and a spatial four bar mechanism are obtained using the proposed method and are validated against those obtained from the finite difference method.

• Journal of applied mathematics & informatics
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• v.26 no.3_4
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• pp.689-706
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• 2008

In this paper, a fifth order numerical method is presented for solving singularly perturbed two point boundary value problems with a boundary layer at one end point. The two point boundary value problem is transformed into general first order ordinary differential equation system. A discrete approximation of a fifth order compact difference scheme is presented for the first order system. An asymptotically equivalent first order equation of the original singularly perturbed two point boundary value problem is obtained from the theory of singular perturbations. It is used in the fifth order compact difference scheme to get a two term recurrence relation and is solved. Several linear and non-linear singular perturbation problems have been solved and the numerical results are presented to support the theory. It is observed that the present method approximates the exact solution very well.

• Journal of applied mathematics & informatics
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• v.33 no.5_6
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• pp.485-502
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• 2015

A computational method for solving singularly perturbed boundary value problem of differential equation with shift arguments of mixed type is presented. When shift arguments are sufficiently small (o(ε)), most of the existing method in the literature used Taylor's expansion to approximate the shift term. This procedure may lead to a bad approximation when the delay argument is of O(ε). The main idea for this work is to deal with constant shift arguments, which are independent of ε. In the present method, we construct the formally asymptotic solution of the problem using the method of composite expansion. The reduced problem is solved numerically by using operator compact implicit method, and the second problem is solved analytically. Error estimate is derived by using the maximum norm. Numerical examples are provided to support the theoretical results and to show the efficiency of the proposed method.

• Journal of applied mathematics & informatics
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• v.8 no.3
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• pp.743-753
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• 2001

We provide local convergence results in affine form for inexact Newton-like as well as quasi-Newton iterative methods in a Banach space setting. We use hypotheses on the second or on the first and mth Frechet-derivative (m≥2 an integer) of the operator involved. Our results allow a wider choice of starting points since our radius of convergence can be larger than the corresponding one given in earlier results using hypotheses on the first-Frechet-derivative only. A numerical example is provided to illustrate this fact. Our results apply when the method is, for example, a difference Newton-like or update-Newton method. Furthermore, our results have direct applications to the solution of autonomous differential equations.

• Advances in nano research
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• v.15 no.1
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• pp.75-89
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• 2023

In this paper, Bishop theory performs longitudinal vibration analysis of Nano-beams. Its governing equation, due to integrated displacement field and more considered primarily effects compared with other theories, enjoys fully completed status, and more reliable results as well. This article aims to find how Bishop theory and Two-phase elasticity work together. In other words, whether Bishop theory will be compatible with Two-phase local/nonlocal elasticity. Hamilton's principle is employed to derive governing equation of motion, and then the 6th order of Generalized Differential Quadrature Method (GDQM) as a constructive numerical method is utilized to attain the discretized two-phase formulation. To acquire a proper verification procedure, exact solution is prepared to be compared with current results. Furthermore, the effects of key parameters on the objective are investigated.

• Proceedings of the KIEE Conference
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• 1996.07b
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• pp.677-679
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• 1996

This paper describes a method to improve the speed of a distance relay based on a differential equation of transmission lines using feedforward artificial neural networks (ANN) on an EHV system. For the impedance calculation an integration approximation to the differential equation is used and then an ANN is trained with the impedance convergence characteristic. The ANN predicts the fault distance with some calculated resistances and reactances before they reach trip zone. Thus, the proposed method can improve the speed of distance relays, significantly if a high sampling rate such as 48 samples per cycle is employed.