• Journal of applied mathematics & informatics
• /
• v.22 no.1_2
• /
• pp.49-65
• /
• 2006

A robust numerical method for a singularly perturbed second-order ordinary differential equation having two parameters with a discontinuous source term is presented in this article. Theoretical bounds are derived for the derivatives of the solution and its smooth and singular components. An appropriate piecewise uniform mesh is constructed, and classical upwind finite difference schemes are used on this mesh to obtain the discrete system of equations. Parameter-uniform error bounds for the numerical approximations are established. Numerical results are provided to illustrate the convergence of the numerical approximations.

• Journal of applied mathematics & informatics
• /
• v.23 no.1_2
• /
• pp.141-152
• /
• 2007

In this paper a numerical method is presented to solve singularly perturbed two points boundary value problems for second order ordinary differential equations consisting a discontinuous source term. First, in this method, an asymptotic expansion approximation of the solution of the boundary value problem is constructed using the basic ideas of a well known perturbation method WKB. Then some initial value problems and terminal value problems are constructed such that their solutions are the terms of this asymptotic expansion. These initial value problems are happened to be singularly perturbed problems and therefore fitted mesh method (Shishkin mesh) are used to solve these problems. Necessary error estimates are derived and examples provided to illustrate the method.

• Journal of applied mathematics & informatics
• /
• v.28 no.5_6
• /
• pp.1035-1054
• /
• 2010

We consider a mixed type singularly perturbed one dimensional elliptic problem with discontinuous source term. The domain under consideration is partitioned into two subdomains. A convection-diffusion and a reaction-diffusion type equations are posed on the first and second subdomains respectively. Two hybrid difference schemes on Shishkin mesh are constructed and we prove that the schemes are almost second order convergence in the maximum norm independent of the diffusion parameter. Error bounds for the numerical solution and its numerical derivative are established. Numerical results are presented which support the theoretical results.

• Communications of the Korean Mathematical Society
• /
• v.37 no.3
• /
• pp.939-955
• /
• 2022

In this paper, we study a first-order non-linear singularly perturbed Volterra integro-differential equation (SPVIDE). We discretize the problem by a uniform difference scheme on a Bakhvalov-Shishkin mesh. The scheme is constructed by the method of integral identities with exponential basis functions and integral terms are handled with interpolating quadrature rules with remainder terms. An effective quasi-linearization technique is employed for the algorithm. We establish the error estimates and demonstrate that the scheme on Bakhvalov-Shishkin mesh is O(N^-1) uniformly convergent, where N is the mesh parameter. The numerical results on a couple of examples are also provided to confirm the theoretical analysis.

• Proceedings of the KSME Conference
• /
• 2004.04a
• /
• pp.1869-1874
• /
• 2004

Acoustic pressure field around the cross-flow fan is predicted by a hydrodynamic-acoustic splitting method. Unsteady flow field is obtained by solving the incompressible Navier-Stokes equations using an unstructured finite-volume method on the triangular meshes, while the acoustic waves generated inside the cross-flow fan are predicted by solving the perturbed compressible equations(PCE) with a 6th-order compact finite difference method. Computational results show that the acoustic waves of BPF tone are generated by interactions of the blades wakes with the stabilizer, which then are reflected from the rear-guider and mainly propagate towards the fan inlet. Also, a directivity of BPF noise predicted by the PCE is noticeably different from that of the FW-H equations, in which a fan casing effect cannot be included.

• Journal of The Korean Astronomical Society
• /
• v.42 no.5
• /
• pp.107-123
• /
• 2009

We investigate the wave properties for isothermal plasma state around to the de Sitter black hole's horizon using 3+1 split of spacetime. The corresponding Fourier analyzed perturbed perfect GRMHD equations are used to obtain the complex dispersion relations. We obtain the real values of the wave number k, from these relations, which are used to evaluate the quantities like phase and group velocities etc. These have been analyzed graphically in the neighborhood of the horizon.

• Communications of the Korean Mathematical Society
• /
• v.21 no.3
• /
• pp.543-558
• /
• 2006

We consider a type of large deviation Principle(LDP) using Freidlin-Wentzell exponential estimates for the solutions to perturbed stochastic differential equations(SDEs) driven by Martingale measure(Gaussian noise). We are using exponential tail estimates and exit probability of a diffusion process. Referring to Freidlin-Wentzell inequality, we want to show another approach to get LDP for the solutions to SDEs.

• Journal of the Korean Mathematical Society
• /
• v.60 no.2
• /
• pp.465-477
• /
• 2023

In this study, we consider a heat equation with a variable-coefficient linear forcing term and a time-periodic boundary condition. Under some decay and smoothness assumptions on the coefficient, we establish the existence and uniqueness of a time-periodic solution satisfying the boundary condition. Furthermore, possible connections to the closed boundary layer equations were discussed. The difficulty with a perturbed leading order coefficient is demonstrated by a simple example.

• Proceedings of the KSME Conference
• /
• 2007.05b
• /
• pp.3209-3214
• /
• 2007

This study numerically investigates the unsteady flow and acoustic characteristics of a flapping wing using a hydrodynamic/acoustic splitting method. The Reynolds number based on the maximum translation velocity of the wing is Re=8800 and Mach number is M=0.0485. The flow around the flapping wing is predicted by solving the two-dimensional incompressible Navier-Stokes equations (INS) and the acoustic field is calculated by the linearized perturbed compressible equations (LPCE), both solved in moving coordinates. Numerical results show that the hovering sound is largely generated by wing translation (transverse and tangential), which have different dipole sources with different mechanisms. As a distinctive feature of the flapping sound, it is also shown that the dominant frequency varies around the wing.

• Bulletin of the Korean Mathematical Society
• /
• v.53 no.3
• /
• pp.765-777
• /
• 2016

In this paper, we prove the existence of nontopological solutions to the self-dual equations arising from the Chern-Simons gauged O(3) sigma models. The property of solutions depends on a parameter ${\tau}{\in}[-1,1]$ appearing in the nonlinear term. The case ${\tau}=1$ lies on the borderline for the existence of solutions in the previous results [4, 5, 7]. We prove the existence of solutions in this case when there are only vortex points. Moreover, if $-1{\leq}{\tau}$<1, we establish solutions which are perturbed from the solutions of singular Liouville equations.