• 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.

• 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.

• 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.

• 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.

• 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 $X_{\lambda}$.

• Journal of Embryo Transfer
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• v.18 no.3
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• pp.215-225
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• 2003

This study was carried out to investigate the change of blood compositions by age in Hanwoo, and a total of 866 of Hanwoo, which consisted with 638 of steer and 228 of bulls, were used to measure serum concentrations. A multiple regression equation was estimated with collection age and blood composition as independent and dependent variables, respectively. Complicated regression equations for blood compositions in steer and bulls were IGF-I(cubic), calcium (linear), and IP(linear). Linear and cubic equations were fitted to testosterone in steer and creatinine in bulls, respectively. A cubic equation in steer and linear equation in bulls were fitted to HDLC. Equations of quadratic in steer and cubic in bulls were fitted to concentration of triglyceride, globulin, and A/G ratio. BUN was fitted by equations of cubic in steer and quadratic in bulls. TP and albumin were fitted by equations of quadratic in steer and linear in bulls. A cubic regression equation did not explain the change of cortisol by age in steer and bulls. A cubic regression equation did explain the change of glucose by age in steer, but not in bulls. Higher R-square values (R-SQUARE>0.1) were estimated to IGF-1, albumin, creatinine, Inorganic phosphorous(IP) and HDLC in steer, and testosterone, IGF-I, TP, albumin, glucose, creatinine, IP, and HDLC in bulls for the fitted regression equations of blood compositions. Therefore, IGF-I, albumin, creatinine, IP, and HDLC were regarded as comparatively large variation by age in steer and bulls.