• Geophysics and Geophysical Exploration
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• v.3 no.2
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• pp.39-47
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• 2000

A great deal of computing time and a large computer memory are needed to solve wave equation in a large complex subsurface layers using the finite difference method. The computing time and memory can be reduced by decreasing the number of grid points per minimum wave length. However, the decrease of grids may cause numerical dispersion and poor accuracy. In this study we performed the grid dispersion analysis for several rotated finite difference operators, which was commonly used to reduce grids per wavelength with accuracy in order to determine the solution for the acoustic wave equation in frequency domain. The rotated finite difference operators were to be extended to 81, 121 and 169 difference stars and studied whether the minimum grids could be reduced to 2 or not. To obtain accuracy (numerical errors less than $1\%$) the following was required: more than 13 grids for conventional 5 point difference stars, 9 grids for 9 difference stars, 3 grids for 25 difference stars, and 2.7 grids for 49 difference stars. After grid dispersion analysis for the new rotated finite difference operators, more than 2.5 grids for 81 difference stars, 2.3 grids for 121 difference stars and 2.1 grids for 169 difference stars were needed. However, in the 169 difference stars, there was no solution because of oscillation of the dispersion curves in the group velocity curves. This indicated that the grids couldn't be reduced to 2 in the frequency acoustic wave equation. According to grid dispersion analysis for the determination of grid points, the more rotated finite difference operators, the fewer grid points. However, the more rotated finite difference operators that are used, the more complex the difference equation terms.

• Bulletin of the Korean Mathematical Society
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• v.61 no.3
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• pp.745-762
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• 2024

In this paper, we study the existence of finite order meromorphic solutions of the following non-linear difference equation fⁿ(z) + P_d(z, f) = p₁e^α1z + p₂e^α2z + p₃e^α3z, where n ≥ 2 is an integer, P_d(z, f) is a difference polynomial in f of degree d ≤ n - 2 with small functions of f as its coefficients, p_j (j = 1, 2, 3) are small meromorphic functions of f and α_j (j = 1, 2, 3) are three distinct non-zero constants. We give the expressions of finite order meromorphic solutions of the above equation under some restrictions on α_j (j = 1, 2, 3). Some examples are given to illustrate the accuracy of the conditions.

• Tribology and Lubricants
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• v.16 no.4
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• pp.295-301
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• 2000

The dynamic simulation of slider in hard disk drive is performed using Factored Implicit Finite Difference method. The modified Reynolds equation with Fukui and Kaneko model is employed as a governing equation. Equations of motion for the slider of three degrees of freedom are solved simultaneously with the modified Reynolds equation. The transient responses of the slider for disk step bumps and slider impulse forces are shown for various cases and are compared for the iteration algorithm and new algorithm.

• Journal of applied mathematics & informatics
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• v.13 no.1_2
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• pp.11-27
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• 2003

A fourth order in time and second order in space scheme using a finite-difference method is developed for the non-linear Boussinesq equation. For the solution of the resulting non-linear system a predictor-corrector pair is proposed. The method is analyzed for local truncation error and stability. The results of a number of numerical experiments for both the single and the double-soliton waves are given.

• East Asian mathematical journal
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• v.24 no.4
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• pp.407-414
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• 2008

Finite difference schemes are considered for a $Ca^{2+}$ diffusion equations with damping and convection terms, which describe $Ca^{2+}$ buffering by using stationary and mobile buffers. Stability and $L^{\infty}$ error estimates of approximate solutions for the corresponding schemes are obtained using the extended Lax-Richtmyer equivalence theorem.

• Journal of applied mathematics & informatics
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• v.27 no.5_6
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• pp.1247-1256
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• 2009

Finite difference schemes are considered for a nonlinear $Ca^{2+}$ diffusion equations with stationary and mobile buffers. The scheme inherits mass conservation as for the classical solution. Stability and $L^{\infty}$ error estimates of approximate solutions for the corresponding schemes are obtained. using the extended Lax-Richtmyer equivalence theorem.

• Communications of the Korean Mathematical Society
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• v.30 no.4
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• pp.447-456
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• 2015

In this paper, we investigate the differential-difference equation $(f(z+c)-f(z))^2+P(z)^2(f^{(k)}(z))^2=Q(z)$, where P(z), Q(z) are nonzero polynomials. In addition, we also investigate Fermat type q-difference differential equations $f(qz)^2+(f^{(k)}(z))^2=1$ and $(f(qz)-f(z))^2+(f^{(k)}(z))^2=1$. If the above equations admit a transcendental entire solution of finite order, then we can obtain the precise expression of the solution.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.23 no.1
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• pp.19-30
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• 2019

In this paper, we develop an accurate explicit finite difference method for the two-dimensional Black-Scholes equation with a hybrid boundary condition. In general, the correlation term in multi-asset options is problematic in numerical treatments partially due to cross derivatives and numerical boundary conditions at the far field domain corners. In the proposed hybrid boundary condition, we use a linear boundary condition at the boundaries where at least one asset is zero. After updating the numerical solution by one time step, we reduce the computational domain so that we do not need boundary conditions. To demonstrate the accuracy and efficiency of the proposed algorithm, we calculate option prices and their Greeks for the two-asset European call and cash-or-nothing options. Computational results show that the proposed method is accurate and is very useful for nonlinear boundary conditions.

• Journal of applied mathematics & informatics
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• v.16 no.1_2
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• pp.53-68
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• 2004

Numerical solutions for the viscous Cahn-Hilliard equation are considered using the Crank-Nicolson type finite difference method which conserves the mass. The corresponding stability and error analysis of the scheme are shown. The decay speeds of the solution in $H^1-norm$ are shown. We also compare the evolution of the viscous Cahn-Hilliard equation with that of the Cahn-Hilliard equation numerically and computationally, which has been given as an open question in Novick-Cohen[13].

• 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.