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2,601
result(s) for
"Hyperbolic functions"
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Lump solution and its interaction to (3+1)-D potential-YTSF equation
This paper studies the
(
3
+
1
)
-dimensional potential-Yu–Toda–Sasa–Fukuyama (YTSF) equation by implementing the Hirota bilinear method. As a consequence, the Hirota bilinear method is successfully employed and acquired a type of the lump solution and five types of interaction solutions in terms of a new merge of positive quadratic functions, trigonometric functions and hyperbolic functions. All solutions have been verified back into its corresponding equation by Maple. We depicted the physical explanation of the extracted solutions with the free choice of the different parameters by plotting some 3D and 2D illustrations. Finally, we believe that the executed method is robust and efficient than other methods and the obtained solutions are trustworthy in the applied sciences.
Journal Article
New acoustic wave behaviors to the Davey–Stewartson equation with power-law nonlinearity arising in fluid dynamics
This manuscript investigates the new acoustic wave behaviors of the Davey–Stewartson equation with power-law nonlinearity with the help of sine-Gordon expansion method. This technique yields many new acoustic wave behaviors such as hyperbolic, exponential and complex function structures to the problem considered. Wolfram Mathematica 9 has been used throughout the paper for mathematical calculations along with obtaining two- and three-dimensional surfaces of results.
Journal Article
Dynamical behavior of analytical soliton solutions to the Kuralay equations via symbolic computation
The paper aims to establish diverse soliton solutions for the integrable Kuralay equations and to explore the integrable motion of space curves induced by these equations. The solitons arising from the integrable Kuralay equations are examined through qualitative studies. They are considered highly significant for understanding various phenomena in fields such as nonlinear optics, optical fibers, and ferromagnetic materials. This model is analyzed using the new generalized exponential rational function method and the new extended hyperbolic function method. With symbolic computations, the new extended hyperbolic function generates closed-form solutions to the integrable Kuralay equations, expressed in hyperbolic, trigonometric, and exponential forms. In contrast, the new extended hyperbolic function method provides hyperbolic, trigonometric, polynomial, and exponential solutions. The model is found to exhibit soliton solutions like periodic oscillating nonlinear waves, kink-wave profiles, multiple soliton profiles, singular solution, mixed singular solution, mixed hyperbolic solution, periodic pattern with anti-troughs and anti-peaked crests, mixed periodic, mixed complex solitary shock solution, mixed shock singular solution, mixed trigonometric solution, and periodic solution. These solutions are novel and have not been previously reported in the open literature. Using symbolic computation by
Mathematica
11.3 or
Maple
, these newly derived soliton solutions are verified by substituting them back into the associated system.
Journal Article
Multivariate Random Fields Evolving Temporally Over Hyperbolic Spaces
Gaussian random fields are completely characterised by their mean value and covariance function. Random fields on hyperbolic spaces have been studied to a limited extent only, namely for the case of scalar-valued fields that are not evolving over time. This paper challenges the problem of the second-order characteristics of multivariate (vector-valued) random fields that evolve temporally over hyperbolic spaces. Specifically, we characterise the continuous space–time covariance functions that are isotropic (radially symmetric) over space (the hyperbolic space) and stationary over time (the real line). Our finding is the analogue of recent findings that have been shown for the case where the space is either the
n
-dimensional sphere or more generally a two-point homogeneous space. Our main result can be read as a spectral representation theorem, and we also detail the main result for the subcase of covariance functions having a spectrum that is absolutely continuous with respect to the Lebesgue measure (technical details are reported below).
Journal Article
Some Geometrical Results Associated with Secant Hyperbolic Functions
In this paper, we examine the differential subordination implication related with the Janowski and secant hyperbolic functions. Furthermore, we explore a few results, for example, the necessary and sufficient condition in light of the convolution concept, growth and distortion bounds, radii of starlikeness and partial sums related to the class Ssech∗.
Journal Article
An electronic implementation for Morris–Lecar neuron model
In this paper, the threshold dynamics of Morris–Lecar neuron model is firstly analyzed by bifurcation diagram of interspike interval as a function of external forcing current, and then the discharge series, phase portraits and nullclines of the neuron under different conditions are investigated in a numerical way. The results show that the electrical activities, such as quiescent state, spiking and bursting, can be observed when the values of external forcing current beyond certain thresholds. Finally, based on the 2-D nonlinear differential equations of Morris–Lecar neuron model, a complete electronic implementation of this model is proposed and studied in detail. At the same time, a circuitry realization of the hyperbolic cosine function
τ
W
(
V
)
in the Morris–Lecar neuron model is put forward and described carefully. The outputs of designed circuits are consistent well with the theoretical predictions, which validate the design methods. Moreover, the circuit presented in this paper can be used as an experimental unit to investigate the dynamics of a single neuron or collective behaviors of a large-scale neural network.
Journal Article
Reciprocal Hyperbolic Series of Ramanujan Type
This paper presents an approach to summing a few families of infinite series involving hyperbolic functions, some of which were first studied by Ramanujan. The key idea is based on their contour integral representations and residue computations with the help of some well-known results of Eisenstein series given by Ramanujan, Berndt, et al. As our main results, several series involving hyperbolic functions are evaluated and expressed in terms of z=F12(1/2,1/2;1;x) and z′=dz/dx. When a certain parameter in these series is equal to π, the series are expressed in closed forms in terms of some special values of the Gamma function. Moreover, many new illustrative examples are presented.
Journal Article
Bridging the p-Special Functions between the Generalized Hyperbolic and Trigonometric Families
Here, we study the extension of p-trigonometric functions sinp and cosp family in complex domains and p-hyperbolic functions sinhp and the coshp family in hyperbolic complex domains. These functions satisfy analogous relations as their classical counterparts with some unknown properties. We show the relationship of these two classes of special functions viz. p-trigonometric and p-hyperbolic functions with imaginary arguments. We also show many properties and identities related to the analogy between these two groups of functions. Further, we extend the research bridging the concepts of hyperbolic and elliptical complex numbers to show the properties of logarithmic functions with complex arguments.
Journal Article
Sharp Bounds on Hankel Determinant of q-Starlike and q-Convex Functions Subordinate to Secant Hyperbolic Functions
In the present paper, using the q-difference operator, we introduce two classes of q-starlike functions and q-convex functions subordinate to secant hyperbolic functions. As functions in these classes have unique characteristic of missing coefficients on the second term in their analytic expansions, we define a new functional to unify the Hankel determinants with entries of the original coefficients, inverse coefficients, logarithmic coefficients, and inverse logarithmic coefficients for these functions. We obtain the sharp bounds on the new functional for functions in the two classes, and as a consequence, the best results on Hankel determinant for the starlike and convex functions subordinate to secant hyperbolic functions are given. The outcomes include some existing findings as corollaries and may help to deepen the understanding the properties of q-analogue analytic functions.
Journal Article