• Journal of the Korean Society of Visualization
• /
• v.20 no.3
• /
• pp.74-85
• /
• 2022

We propose a Cubic-Interpolated Pseudo-Particle Lattice Boltzmann method (CIP-LBM) for the convection-diffusion equation (CDE) based on the Bhatnagar-Gross-Krook (BGK) scheme equation. The CIP-LBM relies on an accurate numerical lattice equilibrium particle distribution function on the advection term and the use of a splitting technique to solve the Lattice Boltzmann equation. Different schemes of lattice spaces such as D1Q3, D2Q5, and D2Q9 have been used for simulating a variety of problems described by the CDE. All simulations were carried out using the BGK model, although another LB scheme based on a collision term like two-relation time or multi-relaxation time can be easily applied. To show quantitative agreement, the results of the proposed model are compared with an analytical solution.

• Korean Chemical Engineering Research
• /
• v.43 no.5
• /
• pp.637-640
• /
• 2005

The density of polar-nonpolar liquid mixtures composed of methyl tert-butyl ether(MTBE) and 2,2,4-trimethylpentane, and methyl ethyl ketone (MEK) and 2,2,4-trimethylpentane, and the density of polar-polar liquid mixture of MTBE and MEK were measured by densitometer at 278.15 K, 288.15 K and 298.15 K, respectively. The excess molar volume of the binary systems calculated from the measured density was shown good agreement with the calculated one by the cubic Peng-Robinson- Stryjek-Vera (PRSV) equation of state together with Huron-Vidal mixing rule and it confirmed that the cubic PRSV equation of state could be used in the molar volume calculation of polar mixture.

• The Pure and Applied Mathematics
• /
• v.19 no.4
• /
• pp.383-396
• /
• 2012

In this paper, we investigate a fuzzy version of stability for the functional equation $$f(x+2y)-3f(x+y)+3f(x)-f(x-y)-3f(y)+3f(-y)=0$$ in the sense of M. Mirmostafaee and M. S. Moslehian.

• Journal of the Chungcheong Mathematical Society
• /
• v.28 no.3
• /
• pp.353-363
• /
• 2015

In this paper, we investigate the functional equation f(3x+y)+f(3x-y) = f(x+2y)+2f(x-y)+6f(2x)+3f(x)-6f(y) and prove the generalized Hyers-Ulam stability for it in non-Archimedean normed spaces.

• Honam Mathematical Journal
• /
• v.32 no.1
• /
• pp.29-43
• /
• 2010

In this note, by using the fixed point method, we prove the generalized stability for a mixed type functional equation in random normed spaces of which the general solution is either cubic or quadratic.

• Communications of Mathematical Education
• /
• v.22 no.4
• /
• pp.505-514
• /
• 2008

In this paper we study on derivation of formulas for roots of quadratic equation, cubic equation, and quartic equation through method analogy. Our argument is based on the norm form of polynomial. We also present some mathematical content knowledge related with main discussion of this article.

• School Mathematics
• /
• v.18 no.3
• /
• pp.589-609
• /
• 2016

In this study, researchers have modernly reinterpreted geometric solving of cubic equations presented by an arabic mathematician, Omar Khayyam in medieval age, and have considered the pedagogical significance of geometric solving of the cubic equations using two conic sections in terms of analytic geometry. These efforts allow to analyze educational application of mathematics instruction and provide useful pedagogical implications in school mathematics such as 'connecting algebra-geometry', 'induction-generalization' and 'connecting analogous problems via analogy' for the geometric approaches of cubic equations: $x^3+4x=32$, $x^3+ax=b$, $x^3=4x+32$ and $x^3=ax+b$. It could be possible to reciprocally convert between algebraic representations of cubic equations and geometric representations of conic sections, while geometrically approaching the cubic equations from a perspective of connecting algebra and geometry. Also, it could be treated how to generalize solution of cubic equation containing variables from geometric solution in which coefficients and constant terms are given under a perspective of induction-generalization. Finally, it could enable to provide students with some opportunities to adapt similar solving procedures or methods into the newly-given cubic equation with a perspective of connecting analogous problems via analogy.

• International Journal of Fuzzy Logic and Intelligent Systems
• /
• v.5 no.4
• /
• pp.286-290
• /
• 2005

We consider the poisson equation where the functions involved are periodic including the solution function. Let $R=[0,1]{\times}[0,l]{\times}[0,1]$ be the region of interest and let $\phi$(x,y,z) be an arbitrary periodic function defined in the region R such that $\phi$(x,y,z) satisfies $\phi$(x+1, y, z)=$\phi$(x, y+1, z)=$\phi$(x, y, z+1)=$\phi$(x,y,z) for all x,y,z. We describe a very simple method for solving the equation ${\nabla}^2u(x, y, z)$ = $\phi$(x, y, z) based on the cubic spline interpolation of u(x, y, z); using the requirement that each interval [0,1] is a multiple of the period in the corresponding coordinates, the Laplacian operator applied to the cubic spline interpolation of u(x, y, z) can be replaced by a square matrix. The solution can then be computed simply by multiplying $\phi$(x, y, z) by the inverse of this matrix. A description on how the storage of nearly a Giga byte for $20{\times}20{\times}20$ nodes, equivalent to a $8000{\times}8000$ matrix is handled by using the fuzzy rule table method and a description on how the shape preserving property of the Laplacian operator will be affected by this approximation are included.

• Journal of Educational Research in Mathematics
• /
• v.27 no.1
• /
• pp.89-112
• /
• 2017

In this study we have analyzed processes of generalization in which students have geometrically solved cubic equation $x^3+ax=b$, regarding geometrical solution of cubic equation $x^3+4x=32$ as examples. The result of this research indicate that students could especially re-interpret the geometric solution of the given cubic equation via dynamically understanding the variables in dynamic geometry environment. Furthermore, participants could simultaneously re-interpret the given geometric solution and then present a different geometric solutions of $x^3+ax=b$, so that the level of generalization could be improved. In conclusion, the study could provide useful pedagogical implications in school mathematics that the dynamic geometry environment performs significant function as a means of students-centered exploration when understanding variables and enhancing the level of generalization in problem solving.

• East Asian mathematical journal
• /
• v.37 no.4
• /
• pp.401-426
• /
• 2021

In order to propose ways to implement mathematical connection between algebra and geometry, this study reinterpreted and visualized the Khayyam's geometric method solving the cubic equations using two conic sections and the Al-Kāshi's method of constructing of angle trisection using a cubic equation. Khayyam's method is an example of a geometric solution to an algebraic problem, while Al-Kāshi's method is an example of an algebraic a solution to a geometric problem. The construction and property of conics were presented deductively by the theorem of "Stoicheia" and the Apollonius' symptoms contained in "Conics". In addition, I consider connections that emerged in the alternating process of algebra and geometry and present meaningful Implications for instruction method on mathematical connection.