Home / Advanced Search

  • Title/Keywords

  • Author/Affliations

  • Journal

  • Article Type

  • Start Year

  • End Year

Update SearchingClear
  • Articles
  • Online
Search Results (4)
  • Open Access


    Lemniscate of Leaf Function

    Kazunori Shinohara*

    CMES-Computer Modeling in Engineering & Sciences, Vol.126, No.1, pp. 275-292, 2021, DOI:10.32604/cmes.2021.012383

    Abstract A lemniscate is a curve defined by two foci, F1 and F2. If the distance between the focal points of F1F2 is 2a (a: constant), then any point P on the lemniscate curve satisfy the equation PF1 · PF2 = a2. Jacob Bernoulli first described the lemniscate in 1694. The Fagnano discovered the double angle formula of the lemniscate (1718). The Euler extended the Fagnano’s formula to a more general addition theorem (1751). The lemniscate function was subsequently proposed by Gauss around the year 1800. These insights were summarized by Jacobi as the theory of elliptic functions. A… More >

  • Open Access


    Addition Formulas of Leaf Functions and Hyperbolic Leaf Functions

    Kazunori Shinohara*

    CMES-Computer Modeling in Engineering & Sciences, Vol.123, No.2, pp. 441-473, 2020, DOI:10.32604/cmes.2020.08656

    Abstract Addition formulas exist in trigonometric functions. Double-angle and half-angle formulas can be derived from these formulas. Moreover, the relation equation between the trigonometric function and the hyperbolic function can be derived using an imaginary number. The inverse hyperbolic function is similar to the inverse trigonometric function , such as the second degree of a polynomial and the constant term 1, except for the sign − and +. Such an analogy holds not only when the degree of the polynomial is 2, but also for higher degrees. As such, a function exists with respect… More >

  • Open Access


    Damped and Divergence Exact Solutions for the Duffing Equation Using Leaf Functions and Hyperbolic Leaf Functions

    Kazunori Shinohara1, *

    CMES-Computer Modeling in Engineering & Sciences, Vol.118, No.3, pp. 599-647, 2019, DOI:10.31614/cmes.2019.04472

    Abstract According to the wave power rule, the second derivative of a function x(t) with respect to the variable t is equal to negative n times the function x(t) raised to the power of 2n-1. Solving the ordinary differential equations numerically results in waves appearing in the figures. The ordinary differential equation is very simple; however, waves, including the regular amplitude and period, are drawn in the figure. In this study, the function for obtaining the wave is called the leaf function. Based on the leaf function, the exact solutions for the undamped and unforced Duffing equations are presented. In the… More >

  • Open Access


    Exact Solutions of the Cubic Duffing Equation by Leaf Functions under Free Vibration

    Kazunori Shinohara1

    CMES-Computer Modeling in Engineering & Sciences, Vol.115, No.2, pp. 149-215, 2018, DOI: 10.3970/cmes.2018.02179

    Abstract Exact solutions of the cubic Duffing equation with the initial conditions are presented. These exact solutions are expressed in terms of leaf functions and trigonometric functions. The leaf function r=sleafn(t) or r=cleafn(t) satisfies the ordinary differential equation dx2/dt2=-nr2n-1. The second-order differential of the leaf function is equal to -n times the function raised to the (2n-1) power of the leaf function. By using the leaf functions, the exact solutions of the cubic Duffing equation can be derived under several conditions. These solutions are constructed using the integral functions of leaf functions sleaf2(t) and cleaf2(t) for the phase of a trigonometric… More >

Displaying 1-10 on page 1 of 4. Per Page