Teaching the Notion Conic Section with Computer

The article discusses the problem of introducing and constructing mathematical concepts using a computer. The Wolfram Mathematica 12 symbolic calculation system is used at each stage of the complex spiral process to form the notion of conic section and the related concepts of focus, directrix and eccentricity. The nature of these notions implies the use of appropriate animations, 3D graphics and symbolic calculations. Our vision of the process of formation of mathematical concepts is presented. The notions ellipse, parabola and hyperbola are defined as the intersection of a conical surface with a plane not containing the vertex of the conical surface. The conical section is represented as a geometric location of points on the plane for which the ratio of the distance to the focus to the distance to the directrix is a constant value. The lines of hyperbola and ellipse are determined by their foci. The equivalence of different definitions for conical sections is commented.

Weak Bi-Center and Critical Period Bifurcations of a Z2-Equivariant Quintic System

With the help of computer algebra system-Mathematica, this paper considers the weak center problem and local critical periods for bi-center of a [Formula: see text]-equivariant quintic system with eight parameters. The order of weak bi-center is identified and the exact maximum bifurcation number of critical periods generated from the bi-center is given via the combination of symbolic calculation and numerical analysis.

Symbolic Calculations

The chapter is devoted to symbolic calculations in which the variables and commands operate on mathematical expressions containing symbolic variables. The representation of a symbolic expression, its simplification, the solution of algebraic expressions, symbolic differentiation and integration, and conversion of the symbolic numbers to their decimal form are described. ODEs solutions are also presented. The final sections of the chapter give examples of the symbolic calculation implementation for some mechanical and tribological problems that were solved numerically in previous chapters, namely lengthening a two-spring scale, shear stress in a lubrication film, a centroid of a certain plate, and two-way solutions of the ODE describing the second order dynamical system – traditional and using the Laplace transform.

Study on the applications of two analytical methods for the construction of traveling wave solutions of the modified equal width equation

In this article, the Sinh–Gordon function method and sub-equation method are used to construct traveling wave solutions of modified equal width equation. Thanks to the proposed methods, trigonometric soliton, dark soliton, and complex hyperbolic solutions of the considered equation are obtained. Common aspects, differences, advantages, and disadvantages of both analytical methods are discussed. It has been shown that the traveling wave solutions produced by both analytical methods with different base equations have different properties. 2D, 3D, and contour graphics are offered for solutions obtained by choosing appropriate values of the parameters. To evaluate the feasibility and efficacy of these techniques, a nonlinear evolution equation was investigated, and with the help of symbolic calculation, these methods have been shown to be a powerful, reliable, and effective mathematical tool for the solution of nonlinear partial differential equations.

New Analytical Solutions of Fractional Symmetric Regularized-Long-Wave Equation

In this study, new extended direct algebraic method is successfully implemented to acquire new exact wave solution sets for symmetric regularized-long-wave (SRLW) equation which arise in long water flow models. By the help of Mathematica symbolic calculation package, the method produced a great number of analytical solutions. We also presented a few graphical illustrations for some surfaces. The fractional derivatives are considered in the conformable sense. All of the solutions were checked by substitution to ensure the reliability of the method. Obtained results confirm that the method is straightforward, powerful and effective method to attain exact solutions for nonlinear fractional differential equations. Therefore, the method is a good candidate to take part in the existing literature.

On the Reversibility of Discretization

“Discretization” usually denotes the operation of mapping continuous functions to infinite or finite sequences of discrete values. It may also mean to map the operation itself from one that operates on functions to one that operates on infinite or finite sequences. Advantageously, these two meanings coincide within the theory of generalized functions. Discretization moreover reduces to a simple multiplication. It is known, however, that multiplications may fail. In our previous studies, we determined conditions such that multiplications hold in the tempered distributions sense and, hence, corresponding discretizations exist. In this study, we determine, vice versa, conditions such that discretizations can be reversed, i.e., functions can be fully restored from their samples. The classical Whittaker-Kotel’nikov-Shannon (WKS) sampling theorem is just one particular case in one of four interwoven symbolic calculation rules deduced below.

Dynamic Modeling and Development of Symbolic Calculation Software for N-DOF Flexible-Link Manipulators Incorporating Lumped Mass

Considering the complex dynamic modeling of multi-DOF planar flexible manipulators, a general-purpose method for the rigid-flexible coupling dynamic modeling of N-DOF flexible manipulators is proposed in this paper, and symbolic calculation software is developed. The modeling method is based on the Lagrange equation and assumed mode method (AMM). First, the N-DOF flexible manipulator is divided into two parts, which are assumed to be rigid and flexible. On this basis, the rigid part and the flexible part are coupled, and the calculation process of the model is further simplified. Then, the simplest general symbolic expression of the dynamic model of the N-DOF flexible manipulator is obtained with the induction method. According to the modeling method, “symbolic expression computation software for dynamic equations of N-DOF flexible manipulators” is developed using the symbolic calculation software Mathematica. Finally, the dynamic modeling method and the symbolic calculation software are verified by a trajectory tracking experiment with a PD control applied to a 2-DOF flexible manipulator. Compared with the traditional modeling method, the calculation time can be reduced by more than 90% using the modeling method proposed in this paper, which significantly reduces the complexity of the modeling process.