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3,489
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
Some mathematical properties of flexible hyperbolic tangent activation function with application to deep neural networks
This study rigorously delves into some analytic properties of an improved activation function (referred as$${ flx}\\tanh $$flx tanh ). The determined results appear as a generalization of some results known in the literature. The$${ flx}\\tanh $$flx tanh is created by taking into account symmetry property, parameterizability, and deformability of the classical$$\\tanh $$tanh function. Under the regime of certain parameters, we examine the behaviours of$${ flx}\\tanh $$flx tanh . Moreover, these dynamic properties of the function$${ flx}\\tanh $$flx tanh yields promising results as an activation function in deep neural networks. We utilize the PyTorch library running on Python 3.9 to evaluate the performance of our activation function. Additionally, we aim to encourage the readers to improve their computer programming language skills by making the Python 3.9 codes available on GitHub.
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
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
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
On Asymptotic Series for Generalized Airy, Circular, and Hyperbolic Functions
The paper concerns the solution of the ordinary differential equation y″±xmy=0, which may be designated the generalized Airy equation, since the original Airy equation corresponds to the particular case m=1 with the + sign. The solutions may be designated generalized circular (hyperbolic) sines and cosines for the + (−) sign, since the particular case m=0 corresponds to the elementary circular (hyperbolic) sines and cosines. There are 3 cases of solution of the generalized Airy equation, depending on the parameter m: (I) for m a non-negative integer, the coefficient xm is an analytic function, and the solutions are also analytic series; (II) for m complex other than an integer, the coefficient xm has a branch point at the origin, and the solutions also have a branch point multiplied by an analytic series; (III) for m a negative integer, the coefficient xm has a pole of order m, and the generalized Airy equation is singular. Case III has four subcases: (III-A) for m=−1, the coefficient x−1 is a simple pole, and the solutions are Frobenius–Fuchs series of two kinds; (III-B) for m=−2, the coefficient is a double pole, and the solutions are a combination of elementary functions, namely exponential, logarithmic, and circular (hyperbolic) sine and cosine for the + (−) sign; (III-C,D) for m=−3,−4,…, the coefficient is a pole of multiplicity m, and the generalized Airy differential equation has an irregular singularity of degree m−2 at the origin. In the sub-cases (III-C,D), the solutions can be obtained by inversion as asymptotic series of descending powers specified by (III-C) Frobenius–Fuchs series of two kinds for a triple pole m=−3; (III-D) for higher-order poles m=−4,−5,… by generalized circular (hyperbolic) sines and cosines of 1/x. It is shown that in all cases the ascending and descending series are absolutely and uniformly convergent with the n-th term decaying like On2. This enables the use of a few terms of the series to obtain tables and plot graphs of the solutions of the generalized Airy differential equation as generalized circular and hyperbolic sines and cosines for several values of the parameter m. As a physical application, it is shown that the generalized circular (hyperbolic) cosines and sines specify the motion of a linear oscillator with natural frequency a power of time in the oscillatory (monotonic) case when the origin is an attractor (repeller).
Journal Article