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ARTICLE

### Lemniscate of Leaf Function

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

ARTICLE

### Addition Formulas of Leaf Functions and Hyperbolic Leaf Functions

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

ARTICLE

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

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 >

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