• Communications for Statistical Applications and Methods
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• v.9 no.2
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• pp.555-563
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• 2002

This paper discusses cubic equations in general saddlepoint approximations. Exact roots are found for various cases by trigonometric identities, the root which is appropriate for the general saddlepoint approximations is selected and discussed, and the defective cases in which the general saddlepoint approximations cannot be used are found.

• Honam Mathematical Journal
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• v.30 no.2
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• pp.233-246
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• 2008

In this paper, we obtain the general solution of the following cubic type functional equation and establish the stability of this equation (0.1) $kf({{\sum}\limits^{n-1}_{j=1}}x_j+kx_n)+kf({{\sum}\limits^{n-1}_{j=1}}x_j-kx_n)+2{{\sum}\limits^{n-1}_{j=1}}f(kx_j)+(k^3-1)(n-1)[f(x_1)+f(-x_1)]=2kf({\sum\limits^{n-1}_{j=1}}x_j)=K^3{\sum\limits^{n-1}_{j=1}[f(x_j+x_n)+f(x_j-x_n)]$ for any integers k and n with k ${\geq}$ 2 and n ${\geq}$ 3.

• East Asian mathematical journal
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• v.34 no.4
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• pp.403-427
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• 2018

The purpose of this paper is to reinterpret and visualize the medieval Arab's studies on the geometric methods of solving the cubic equation by utilizing Apollonius' symptom of the parabola. In particular, we investigate the results of $Kam{\bar{a}}l$ $al-D{\bar{i}}n$ ibn $Y{\bar{u}}nus$, Alhazen, Umar al-$Khayy{\bar{a}}m$ and $Al-T{\bar{u}}s{\bar{i}}$ by 4 steps(analysis, construction, proof and examination) which are called the complete solution in the constructions. This paper is available in the current middle school curriculum through dynamic geometry program(Geogebra).

• East Asian mathematical journal
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• v.33 no.1
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• pp.53-65
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• 2017

Numerical solutions of the Rosenau-Burgers equation based on the cubic B-spline finite element method are introduced. The backward Euler method is used for discretization in time, and the obtained nonlinear algebraic system is changed to a linear system by the Newton's method. We show that those methods are unconditionally stable. Two test problems are studied to demonstrate the accuracy of the proposed method. The computational results indicate that numerical solutions are in good agreement with exact solutions.

• Kyungpook Mathematical Journal
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• v.50 no.2
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• pp.315-327
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• 2010

We give a xed point approach to the generalized Hyers-Ulam stability of the cubic equation f(2x + y) + f(2x - y) = 2f(x + y) + 2f(x - y) + 12f(x) in non-Archimedean normed spaces. We will give an example to show that some known results in the stability of cubic functional equations in real normed spaces fail in non-Archimedean normed spaces. Finally, some applications of our results in non-Archimedean normed spaces over p-adic numbers will be exhibited.

• Journal of Environmental Science International
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• v.11 no.9
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• pp.907-914
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• 2002

Various Eulerian-Lagrangian models for the one-dimensional longitudinal dispersion equation in nonuniform flow were studied comparatively. In the models studied, the transport equation was decoupled into two component parts by the operator-splitting approach; one part is governing advection and the other is governing dispersion. The advection equation has been solved by using the method of characteristics following fluid particles along the characteristic line and the results were interpolated onto an Eulerian grid on which the dispersion equation was solved by Crank-Nicholson type finite difference method. In the solution of the advection equation, Lagrange fifth, cubic spline, Hermite third and fifth interpolating polynomials were tested by numerical experiment and theoretical error analysis. Among these, Hermite interpolating polynomials are generally superior to Lagrange and cubic spline interpolating polynomials in reducing both dissipation and dispersion errors.

• Bulletin of the Korean Mathematical Society
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• v.55 no.1
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• pp.267-296
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• 2018

This paper studies the Cauchy problem and blow-up phenomena for a new generalized Camassa-Holm equation with cubic nonlinearity in the nonhomogeneous Besov spaces. First, by means of the Littlewood-Paley decomposition theory, we investigate the local well-posedness of the equation in $B^s_{p,r}$ with s > $max\{{\frac{1}{p}},\;{\frac{1}{2}},\;1-{\frac{1}{p}}\},\;p,\;r{\in}[0,{\infty}]$. Second, we prove that the equation is locally well-posed in $B^s_{2,r}$ with the critical index $s={\frac{1}{2}}$ by virtue of the logarithmic interpolation inequality and the Osgood's Lemma, and it is shown that the data-to-solution mapping is $H{\ddot{o}}lder$ continuous. Finally, we derive two kinds of blow-up criteria for the strong solution by using induction and the conservative property of m along the characteristics.

• Journal of applied mathematics & informatics
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• v.25 no.1_2
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• pp.109-118
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• 2007

A parametric boundary equation is established for the principal period-2 bulb in the cubic Mandelbrot set. Using its geometry, an efficient escape-time algorithm which reduces the construction time for the period-2 bulbs in the cubic Mandelbrot set is introduced and the implementation graphic results display the fascinating fractal beauty.

• Journal of the Chungcheong Mathematical Society
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• v.30 no.2
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• pp.239-249
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• 2017

In this paper, we approximate a fuzzy almost cubic function by a cubic function in a fuzzy sense. Indeed, we investigate solutions of the following cubic functional equation $$3f(kx+y)+3f(kx-y)-kf(x+2y)-2kf(x-y)-3k(2k^2-1)f(x)+6kf(y)=0$$. and prove the generalized Hyers-Ulam stability for it in fuzzy Banach spaces.

• Bulletin of the Korean Mathematical Society
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• v.44 no.2
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• pp.309-313
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• 2007

By applying ideas from [M. S. Moslehian, et al., Norms of operators in $X_{\lambda}$ spaces, Appl. Math. Lett. (2007), doi:10.1016/j.aml.2006. 11.009], we investigate the norm of the cubic operator on the function space $X_{\lambda}$.