We will investigate the superstability of the hyperbolic trigonometric functional equation from the following functional equations: fx y ± gx − y λfxgy andfx y ± gx − y λgxfy, which can be considered the mixed functional equations of the sine function and cosine function, the hyperbolic sine function and hyperbolic cosine function, and the exponential functions, respectively.

We describe and characterize rigorously the chaotic behavior of the sine– Gordon equation. The existence of invariant manifolds and the persistence of homoclinic orbits for a perturbed sine–Gordon equation are established. We apply a geometric method based on Mel’nikov’s analysis to derive conditions for the transversal intersection of invariant manifolds of a hyperbolic point of the perturbed ...

We prove the existence of random attractors for nonlinear sto-chastic hyperbolic diierential equations. The nonlinearity of this equation has similar properties as the nonlinearity of the Sine Gordon equation. We consider a diiusion term depending on the state variable. To prove the existence of a random attractor we transform the stochastic hyperbolic equation into a random hyperbolic equation...

The double sine-Gordon equation is studied. Some complex hyperbolic functions are proposed to derive travelling wave solutions. Based on the symbolic computation program MAPLE, many new exact solutions are obtained. 2006 Elsevier Inc. All rights reserved.

In this paper, we explore more applications of the hyperbolic function approach, which was used to find new exact travelling wave solutions of nonlinear partial differential equations or coupled nonlinear partial differential equations (PDES), to special discrete nonlinear equations. Some exact travelling wave solution of the discrete sine-Gordon equation are obtained in terms of hyperbolic fun...

A comprehensive symmetry analysis of the N = 1 supersymmetric sine–Gordon equation is performed. Two different forms of the supersymmetric system are considered. We begin by studying a system of partial differential equations corresponding to the coefficients of the various powers of the anticommuting independent variables. Next, we consider the supersine-Gordon equation expressed in terms of a...

In this paper based on a system of Riccati equations with variable coefficients, we presented a new Riccati equation with variable coefficients expansion method and its algorithm, which are direct and more powerful than the tanh-function method, sine-cosine method, the generalized hyperbolic-function method and the generalized Riccat equation with constant coefficient expansion method to constr...

1. Error calculus 2 1.1. Ulp calculus 2 1.2. Relative error analysis 4 1.3. Generic error of addition/subtraction 4 1.4. Generic error of multiplication 5 1.5. Generic error of inverse 5 1.6. Generic error of division 6 1.7. Generic error of square root 7 1.8. Generic error of the exponential 7 1.9. Generic error of the logarithm 8 1.10. Ulp calculus vs relative error 9 2. Low level functions 9...