• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.18 no.1
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• pp.27-41
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• 2014

In this paper, we present an efficient numerical method for multiphase image segmentation using a multiphase-field model. The method combines the vector-valued Allen-Cahn phase-field equation with initial data fitting terms containing prescribed interface width and fidelity constants. An efficient numerical solution is achieved using the recently developed hybrid operator splitting method for the vector-valued Allen-Cahn phase-field equation. We split the modified vector-valued Allen-Cahn equation into a nonlinear equation and a linear diffusion equation with a source term. The linear diffusion equation is discretized using an implicit scheme and the resulting implicit discrete system of equations is solved by a multigrid method. The nonlinear equation is solved semi-analytically using a closed-form solution. And by treating the source term of the linear diffusion equation explicitly, we solve the modified vector-valued Allen-Cahn equation in a decoupled way. By decoupling the governing equation, we can speed up the segmentation process with multiple phases. We perform some characteristic numerical experiments for multiphase image segmentation.

• Communications of the Korean Mathematical Society
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• v.36 no.1
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• pp.51-62
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• 2021

Variational method has played an important role in solving problems of uniqueness and existence of the nonlinear works as well as analysis. It will also be extremely useful for researchers in all branches of natural sciences and engineers working with non-linear equations economy, optimization, game theory and medicine. Recently, the existence of infinitely many weak solutions for some non-local problems of Kirchhoff type with Dirichlet boundary condition are studied [14]. Here, a suitable method is presented to treat the elliptic partial derivative equations, especially (p(x), q(x))-Laplacian-like systems. This kind of equations are used in the study of fluid flow, diffusive transport akin to diffusion, rheology, probability, electrical networks, etc. Here, the existence of infinitely many weak solutions for some boundary value problems involving the (p(x), q(x))-Laplacian-like operators is proved. The method is based on variational methods and critical point theory.

• Korea-Australia Rheology Journal
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• v.18 no.2
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• pp.51-65
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• 2006

Of concern in the present theoretical investigation is the study of blood flow and convection-dominated diffusion processes in a model bifurcated artery under stenotic conditions. The geometry of the bifurcated arterial segment having constrictions in both the parent and its daughter arterial lumen frequently appearing in the diseased arteries causing malfunction of the cardiovascular system, is constructed mathematically with the introduction of suitable curvatures at the lateral junction and the flow divider. The streaming blood contained in the bifurcated artery is treated to be Newtonian. The flow dynamical analysis applies the two-dimensional unsteady incompressible nonlinear Wavier-Stokes equations for Newtonian fluid while the mass transport phenomenon is governed by the convection diffusion equation. The motion of the arterial wall and its effect on local fluid mechanics is, however, not ruled out from the present model. The main objective of this study is to demonstrate the effects of constricted flow characteristics and the wall motion on the wall shear stress, the concentration profile and on the mass transfer. The ultimate numerical solutions of the coupled flow and diffusion processes following a radial coordinate transformation are based on an appropriate finite difference technique which attain appreciable stability in both the flow phenomena and the convection-dominated diffusion processes.

• International Journal of Aeronautical and Space Sciences
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• v.2 no.2
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• pp.65-72
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• 2001

This paper demonstrates an explicit-implicit, finite element analysis for linear as well as nonlinear hygrothermal stress problems. Additional features, such as moisture diffusion equation, crack element and virtual crack extension(VCE) method for evaluating J-integral are implemented in this program. The Linear Elastic Fracture Mechanics(LEFM) Theory is employed to estimate the crack driving force under the transient condition for an existing crack. Pores in materials are assumed to be saturated with moisture in the liquid form at the room temperature, which may vaporize as the temperature increases. The vaporization effects on the crack driving force are also studied. The ideal gas equation is employed to estimate the thermodynamic pressure due to vaporization at each time step after solving basic nodal values. A set of field equations governing the time dependent response of porous media are derived from balance laws based on the mixture theory. Darcy's law is assumed for the fluid flow through the porous media. Perzyna's viscoplastic model incorporating the Von-Mises yield criterion are implemented. The Green-Naghdi stress rate is used for the invariant of stress tensor under superposed rigid body motion. Isotropic elements are used for the spatial discretization and an iterative scheme based on the full Newton-Raphson method is used for solving the nonlinear governing equations.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.14 no.3
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• pp.189-200
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• 2010

A numerical method is proposed and analyzed to approximate a mathematical model of age-dependent population dynamics with spatial diffusion. The model takes a form of nonlinear and nonlocal system of integro-differential equations. A finite difference method along the characteristic age-time direction is considered and primal mixed finite elements are used in the spatial variable. A priori error estimates are derived for the relevant variables.

• Journal of Korean Society for Atmospheric Environment
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• v.8 no.3
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• pp.162-168
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• 1992

Numerical solutions to the advection equations used for long-range transport air pollution models are calculated using three numerical methods; Antidiffusion correction method(Smolarkiewicz, 1983), Positive definite advecton scheme obtained by nonlinear renormalization of the advective fluxes(Bott, 1989), and Positive definite pseudospectral method(Bartnicki, 1989). Accuracy, numerical diffusion and computational time requirement are compared for two-dimensional transport calculations in a uniform rotational flow field. The solutions from three methods are positive definite. Bartnicki(1989)'s method is most conservative but requires approximately 10 times as much computational time as Smolarkiewicz(1983)'s method of which numerical diffusion is the largest. All three methods are more conservative for a cone shape initial condition than for a rectangular block initial condition with a steep gradient.

• Journal of the Korean Mathematical Society
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• v.36 no.1
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• pp.229-241
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• 1999

The fractal space is often associated with natural phenomena with many length scales and the functions defined on this space are usually not differentiable. First we define a $\sigma$-multifractal from $\sigma$-iterated function systems with probability. We introduce the measure derivative through the invariant measure of the $\sigma$-multifractal. We show that the non-differentiable function on the $\sigma$-multifractal can be differentiable with respect to this measure derivative. We apply this result to some examples of ordinary differential equations and diffusion processes on $\sigma$-multifractal spaces.

• Steel and Composite Structures
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• v.36 no.1
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• pp.87-101
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• 2020

A multi-scale epoxy/CNT/fiberglass annular sector plate is studied in this paper in the view of determining nonlinear forced vibration characteristics. A 3D Mori-Tanaka model is employed for evaluating multi-scale material properties. Thus, all of glass fibers are assumed to have uni-direction alignment and CNTs have random diffusion. The geometry of annular sector plate can be described based on the open angle and the value of inner/outer radius. In order to solve governing equations and derive exact forced vibration curves for the multi-scale annular sector, Jacobi elliptic functions are used. Obtained results demonstrate the significance of CNT distribution, geometric nonlinearity, applied force, fiberglass volume, open angle and fiber directions on forced vibration characteristics of multi-scale annular sector plates.

• Communications of the Korean Mathematical Society
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• v.34 no.4
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• pp.1365-1388
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• 2019

In this paper we consider a class of nonlinear nonlocal parabolic equations involving p-Laplacian operator where the nonlocal quantity is present in the diffusion coefficient which depends on $L^p$-norm of the gradient and the nonlinear term is of polynomial type. We first prove the existence and uniqueness of weak solutions by combining the compactness method and the monotonicity method. Then we study the existence of global attractors in various spaces for the continuous semigroup generated by the problem. Finally, we investigate the existence and exponential stability of weak stationary solutions to the problem.

• East Asian mathematical journal
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• v.39 no.3
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• pp.271-278
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• 2023

We consider a 1-dimensional reaction diffusion equation with the following boundary conditions arising in a theory of the thermal explosion {-u"(t) = λf(u(t)), t ∈ (0, l), -u'(0) + C(0)u(0) = 0, u'(l) + C(l)u(l) = 0, where C : [0, ∞) → (0, ∞) is a continuous and nondecreasing function, λ > 0 is a parameter and f : [0, ∞) → (0, ∞) is a continuous function. We establish the extension of Quadrature method introduced in [8]. Using this extension, we provide numerical results for models with a typical function of f and C in a thermal explosion theory, which verify the existence, uniqueness and multiplicity results proved in [6].