• Communications of the Korean Mathematical Society
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• v.34 no.3
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• pp.1015-1027
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• 2019

A class of systems of singularly perturbed convection diffusion type equations with integral boundary conditions is considered. A numerical method based on a finite difference scheme on a Shishkin mesh is presented. The suggested method is of almost first order convergence. An error estimate is derived in the discrete maximum norm. Numerical examples are presented to validate the theoretical estimates.

• Journal of the Korean Mathematical Society
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• v.58 no.3
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• pp.553-569
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• 2021

In this paper, we present and analyze a fully discrete numerical method for solving the time-fractional diffusion wave equation: ∂^β_tu - div(a∇u) = f, 1 < β < 2. We first construct a difference formula to approximate ∂^β_tu by using an interpolation of derivative type. The truncation error of this formula is of O(△t^2+δ-β)-order if function u(t) ∈ C^2,δ[0, T] where 0 ≤ δ ≤ 1 is the Hölder continuity index. This error order can come up to O(△t^3-β) if u(t) ∈ C³ [0, T]. Then, in combinination with the linear finite volume discretization on spatial domain, we give a fully discrete scheme for the fractional wave equation. We prove that the fully discrete scheme is unconditionally stable and the discrete solution admits the optimal error estimates in the H¹-norm and L₂-norm, respectively. Numerical examples are provided to verify the effectiveness of the proposed numerical method.

• Journal of Biosystems Engineering
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• v.12 no.2
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• pp.38-43
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• 1987

An experimental study was conducted to develop a thin-layer rewetting equation of short grain rough rice of Akihikari variety. Four thin-layer rewetting equations were experimentally determined from $25^{\circ}C$ to $45^{\circ}C$ and 70%RH to 85%RH conditions. Diffusion, Henderson, Page, and Thompson equations widely used as thin-layer drying equations were selected. Experimental data were fitted to these equations using linear regression analysis except diffusion equation. The diffusivity in the diffusion equation was determined by optimization method. Four equations were highly significant. In order to compare the goodness of fit of each equation, the error mean square of each equawas calculated. The diffusion model was not a very good model because the error mean square was very large. The other three models showed the same level or error mean square and could predict satisfactorily the rewetting rate or short grain rough rice.

• Journal of the Institute of Electronics Engineers of Korea SP
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• v.43 no.1 s.307
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• pp.9-16
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• 2006

This Paper proposes an adaptive error diffusioThis paper proposes an adaptive error diffusion algorithm for text enhancement followed by an efficient text segmentation that uses the maximum gradient difference (MGD). The gradients are calculated along with scan lines, and the MGD values are filled within a local window to merge the potential text segments. Isolated segments are then eliminated in the non-text region filtering process. After the left segmentation, a conventional error diffusion method is applied to the background, while the edge enhancement error diffusion is used for the text. Since it is inevitable that visually objectionable artifacts are generated when using two different halftoning algorithms, the gradual dilation is proposed to minimize the boundary artifacts in the segmented text blocks before halftoning. Sharpening based on the gradually dilated text region (GDTR) prevents the printing of successive dots around the text region boundaries. The error diffusion algorithm with edge enhancement is extended to halftone color images to sharpen the tort regions. The proposed adaptive error diffusion algorithm involves color halftoning that controls the amount of edge enhancement using a general error filter. The multiplicative edge enhancement parameters are selected based on the amount of edge sharpening and color difference. Plus, the additional error factor is introduced to reduce the dot elimination artifact generated by the edge enhancement error diffusion. By using the proposed algorithm, the text of a scanned image is sharper than that with a conventional error diffusion without changing background.

• Journal of the Semiconductor & Display Technology
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• v.3 no.3
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• pp.19-21
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• 2004

This paper presents a 3-dimensional diffusion simulation with adaptive solution strategy. The developed diffusion simulator VLSIDIF-3 was designed to re-refine areas. Refine scheme was calculated by the difference of doping concentration between any of two nodes. Each element is greater than tolerance and redo diffusion process until error is tolerable. Numerical experiment in low doping diffusion problem showed that this adaptive solution strategy is very efficient in both memory and time, and expected this scheme would be more powerful in complex diffusion model.

• The Transactions of the Korea Information Processing Society
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• v.4 no.7
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• pp.1893-1899
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• 1997

Spatial dithering techniques are the method of rendering the illusion of continuous-tone pictures on displays that are capable of producing only binary picture elements. In this paper, we propose a new dithering algorithm which diffuses error into nearby pixels symmetrically. This method complements the artifacts of the error diffusion dither for the graphic images and the short-comings of the ordered dither that can't display some intensity level. We applied this method to graphic images and obtained results that complement the short-comings of conventional method.

• The Journal of the Korea Contents Association
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• v.5 no.2
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• pp.221-228
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• 2005

The paper proposes the improved error diffusion halftoning system to enhance the edges using the spatial perceptual characteristics of the human visual system. The proposed method computes a spatial variation(SV), which is the difference between a pixel luminance and the average of its $3{\times}3$ neighborhood pixels' luminances weighted according to the spatial positioning. Information of edge enhancement(IEE) Is computed using the SV and the local average luminance. The IEE is added to the quantizer's input pixel and feeds into the halftoning quantizer. The quantizer produces the halftone image having the enhanced edge. The performance of the proposed method is compared with conventional methods by measuring the edge correlation. The halftone images by using the proposed method show better quality due to the enhanced edge. And the detailed edge is preserved in the halftone images by using the proposed method.

• Nuclear Engineering and Technology
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• v.55 no.6
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• pp.2172-2194
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• 2023

A variational nodal method (VNM) with unstructured-mesh is presented for solving steady-state and dynamic neutron diffusion equations. Orthogonal polynomials are employed for spatial discretization, and the stiffness confinement method (SCM) is implemented for temporal discretization. Coordinate transformation relations are derived to map unstructured triangular nodes to a standard node. Methods for constructing triangular prism space trial functions and identifying unique nodes are elaborated. Additionally, the partitioned matrix (PM) and generalized partitioned matrix (GPM) methods are proposed to accelerate the within-group and power iterations. Neutron diffusion problems with different fuel assembly geometries validate the method. With less than 5 pcm eigenvalue (k_eff) error and 1% relative power error, the accuracy is comparable to reference methods. In addition, a test case based on the kilowatt heat pipe reactor, KRUSTY, is created, simulated, and evaluated to illustrate the method's precision and geometrical flexibility. The Dodds problem with a step transient perturbation proves that the SCM allows for sufficiently accurate power predictions even with a large time-step of approximately 0.1 s. In addition, combining the PM and GPM results in a speedup ratio of 2-3.

• Journal of applied mathematics & informatics
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• v.15 no.1_2
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• pp.1-28
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• 2004

We first apply a first order splitting to a semilinear reaction-diffusion equation and then discretize the resulting system by an $H^1$-Galerkin mixed finite element method in space. This semidiscrete method yields a system of differential algebraic equations (DAEs) of index one. A priori error estimates for semidiscrete scheme are derived for both differ-ential as well as algebraic components. For fully discretization, an implicit Runge-Kutta (IRK) methods is applied to the temporal direction and the error estimates are discussed for both components. Finally, we conclude the paper with a numerical example.

• Proceedings of the Korean Powder Metallurgy Institute Conference
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• 2006.09a
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• pp.72-73
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• 2006

The error function can be calculated based on the Simpson method through a subroutine program. An integration program by FORTRAN language was made for diffusion equations of extended source with infinite extent and limited extent. The results on some alloying elements such as C, Co, Cr, Mn, Mo, Ni and V's diffusion in iron, showed the diffusion distance for Ni and Mo can only be $1{\sim}3\;{\mu}m$ and more distance for Co at common sintering temperature of $1120^{\circ}C$. To refine the particle size of the added elements down to a scale of micrometers is an effective way to get homogeneous distribution.