Hyperbolic Function Method Research Articles

Modelling and Simulation of Microbial Activity in Stratified and Homogeneous Biofilms: Hyperbolic Function Method

Modelling and Simulation of Microbial Activity in Stratified and Homogeneous Biofilms: Hyperbolic Function Method

Solitons solutions to the high-order dispersive cubic–quintic Schrödinger equation in optical fibers

In this paper, solitons solutions of higher-order dispersive cubic–quintic Schrödinger equationincluding third-order as well as fourth-order derivatives with respect to time, that describes the dynamics of ultrashort pulses in optical fibers are investigated in detail. In this respect,a solution procedure in the locality of applied mathematics called the hyperbolic function method is appliedusing multi-linear variable separation approach (MLVSA). As an outcome, a bunch of soliton solutions isderived in conjunction with plotting dark and periodic wave solutions. The credibility of the results is examined by setting each solution back into its governing equation. Through portraits, different forms of wave solutions are depicted. Moreover, the restrictions on the parameters are also given for the existence of the obtained solutions.

Cryptology Based on Laplace Transform of Hyperbolic Function and Matrix Decomposition Method

Cryptography is the art of coding and decoding the communication. Cryptography ensures the security of delicate information over some confidentiality breaching resources. In the recent age Cryptography has turned into a battleground of some of the world’s best mathematicians and computer scientists. A number of transforms like Sumudu transform, Laplace transform, Fourier transform, Kamal transform, Mellin transform, Jafari transform, Aboodh transform, N-transform, ELzaki transform, MAHGOUB transform are frequently used in cryptography. Cryptography is of great importance in every field of life including data transmission, data storage, E-Commerce, confidential communication such as credit card transactions and Emails. Cryptography has been a hot topic among the researchers from 4000 years ago until now. In this work, a new cryptographic technique will be introduced. . In this work mathematical transformation known as Laplace transform on hyperbolic function is used for advancement of security purpose in the field of computers and telecommunications, this concept is known as Cryptography. This work includes a mathematical method known as “Laplace Transform” along with “Matrix Decomposition or Triangularization method” in cryptographic process (Encryption and Decryption). Both these mathematical concepts are used in this work to create a new cryptographic technique for the security of data and information in the world of computer. In this technique, Laplace transform on hyperbolic functions along with Matrix decomposition method will be incorporated. The Laplace Transform of hyperbolic function and Lower Triangular Matrix will be used as encryption and inverse Laplace Transformand upper triangular matrix will be used for decryption. Furthermore, the technique will also be explained with its applications.

Extended hyperbolic function method for the (2 +1)-dimensional nonlinear soliton equation

By employing the extended hyperbolic function method (EHFM), we extract the exact solutions of the (2+1)-dimensional nonlinear soliton equation (SE). A soliton equation is used for investigation of the dynamics of nonlinear waves in plasma physics and fluid dynamics. A various new techniques for finding exact solutions of the (2+1)-dimensional nonlinear SE are satisfactorily acquired with the help of EHFM. The EHFM presents various types of new solutions in the form of dark, singular, periodic, bright solitons and some rational function solutions. In addition, for the physical characterization of the acquired solutions of (2+1)-dimensional SE, some 2-dim and 3-dim plots are drawn. The attained results are novel for the considered equation, and results reveal that the method is concise, direct and competent which can be assembled in other complex phenomena.

Dynamics behavior of solitons solutions of Chen-Lee-Liu equation using analytical techniques

The main concern of the this article is to study the non-linear Chen-Lee-Liu equation which describes themotion of waves in shallow water. For this purpose two analytical techniques namely, the Sardar-Subequationmethod and the new extended hyperbolic function method are utilized. Also, we established the idea ofthe construction of solitons solutions of non-linear evolution equations which are rising in fluid dynamics,nonlinear optics, mathematical biological models, mechanics, waves theory, quantum mechanics. Acquiredsolutions are demonstrated graphically to reveal the dynamics behavior of solitons solutions. It hoped thatthe established solutions can be used to enrich the dynamic behaviors of Chen-Lee-Liu equation. Further,these solutions disclose that our techniques are up-to-date, suitable and straightforward.

Reaction-diffusion in a packed-bed reactors: Enzymatic isomerization with Michaelis-Menten Kinetics

A theoretical model for the enzyme catalysed isomerisation reaction in a packed-bed reactors made up of microporous particles is discussed. The system is described in terms of finite-range Fickian diffusion and non linear Michaelis-Menten reaction kinetics to produce a non linear reaction/diffusion equation which is solved both numerically by MATLAB and analytically by hyperbolic function methods (HFM). Approximate analytical expressions of the substrate concentration within the microparticle, the related flux and the effectiveness factor for a wide range of values of pore-level Thiele modulus and kinetic parameters are reported using these simple, and robust methods. A satisfactory agreement is obtained between the estimated analytical solution and the numerical simulation.

Construction of optical solitons of Radhakrishnan–Kundu–Lakshmanan equation in birefringent fibers

Abstract In this article, we are attracted to discover the multiple-optical soiltons in birefringent fibers for Radhakrishnan–Kundu–Lakshmanan equation (RKLE) by applying the Sardar-subequation method (SSM) and the new extended hyperbolic function method (EHFM). We construct the solutions in the form of exponential, trigonometric, and hyperbolic functions solitons solutions like mixed complex solitons and multiple-optical solitons solutions. In addition, singular periodic wave solutions are constructed, and the restraint conditions for the presence of soliton solutions are also defined. Moreover, the physical interpretation of the obtained solutions is disclosed in forms of 3D and 2D plots for different suitable parameters. The attained results indicate that the implemented computational scheme is straight, proficient, and brief and can be applied in more complex phenomena with the associate of representative computations. We have obtained several sorts of new solutions.

Optical solitons for conformable space-time fractional nonlinear model

In search of the exact solutions of nonlinear partial differential equations in solitons form has become most popular to understand the internal features of physical phenomena. In this paper, we discovered various type of solitons solutions for the conformable space-time nonlinear Schrodinger equation (CSTNLSE) with Kerr law nonlinearity. To seek such solutions, we applied two proposed methods which are Sardar-subequation method and new extended hyperbolic function method. In this way several types of solitons obtained for example bright, dark, periodic singular, combined dark-bright, singular, and combined singular solitons. Some of the acquired solutions are interpreted graphically. These solutions are specific, novel, correct and may be beneficial for edifying precise nonlinear physical phenomena in nonlinear dynamical schemes. It is revealed that the proposed methods offer a straightforward and mathematical tool for solving nonlinear conformable space-time nonlinear Schrodinger equation. These results support in attaining nonlinear optical fibers in the future.

Study of power law non-linearity in solitonic solutions using extended hyperbolic function method

<abstract><p>This paper retrieves the optical solitons to the Biswas-Arshed equation (BAE), which is examined with the lack of self-phase modulation by applying the extended hyperbolic function (EHF) method. Novel constructed solutions have the shape of bright, singular, periodic singular, and dark solitons. The achieved solutions have key applications in engineering and physics. These solutions define the wave performance of the governing models. The outcomes show that our scheme is very active and reliable. The acquired results are illustrated by 3-D and 2-D graphs to understand the real phenomena for such sort of non-linear models.</p></abstract>

On soliton solutions of fractional-order nonlinear model appears in physical sciences

<abstract><p>In wave theory, the higher dimensional non-linear models are very important to define the physical phenomena of waves. Herein study we have built the various solitons solutions of (4+1)-dimensional fractional-order Fokas equation by using two analytical techniques that is, the Sardar-subequation method and new extended hyperbolic function method. Different types of novel solitons are attained such as, singular soliton, bright soliton, dark soliton, and periodic soliton. To understand the physical behavior, we have plotted 2D and 3D graphs of some selected solutions. From results we concluded that the proposed methods are straightforward, simple, and efficient. Moreover, this paper offers a hint, how we can convert the fractional-order PDE into an ODE to acquire the exact solutions. Also, the proposed methods and results can be help to examine the advance fractional-order models which seem in optics, hydrodynamics, plasma and wave theory etc.</p></abstract>

NEW FAMILIES OF SOLUTIONS FOR THE SPACE-TIME FRACTIONAL BURGERS’ EQUATION

In this paper, the hyperbolic tangent function method is applied for constructing exact solutions for space-time conformal fractional Burgers’ equation. Furthermore, the space-time conformal fractional Burgers’ equation is tested for the Painlevé property, and consequently, new numerous exact solutions are generated via Bäcklund transform.

On Soliton Solutions of Perturbed Boussinesq and KdV-Caudery-Dodd-Gibbon Equations

Nonlinear science is a fundamental science frontier that includes research in the common properties of nonlinear phenomena. This article is devoted for the study of new extended hyperbolic function method (EHFM) to attain the exact soliton solutions of the perturbed Boussinesq equation (PBE) and KdV–Caudery–Dodd–Gibbon (KdV-CDG) equation. We can claim that these solutions are new and are not previously presented in the literature. In addition, 2d and 3d graphics are drawn to exhibit the physical behavior of obtained new exact solutions.

Family of optical solitons for perturbed Fokas–Lenells equation

The current study investigates optical solitons with the perturbed Fokas–Lenells equation under the influences of analytical procedures, namely, the modified extended tangent hyperbolic function method, extended rational sinh–cosh method, and extended rational sin–cos method. Dark soliton, hole-like dark soliton, chirped-like dark soliton, and kink/antikink-like cuspon are revealed.

Dynamics of optical solitons in higher-order Sasa–Satsuma equation

Higher order and multidimensional generalizations of the nonlinear Schrödinger equation are useful in a variety of applications. A generalized (3+1)-dimensional nonlinear Sasa–Satsuma equation is studied via the improved generalized Riccati equation mapping method and its Bäcklund transformation and the extended hyperbolic function method. This nonlinear model can be use to describe the physical processes of wave propagation in optical fibers. We explicitly retrieved varieties of optical solitons like bright, dark, singular, combined bright–dark and combined singular-bright to this multidimensional generalized nonlinear model. Other solutions, such as periodic, rational, and exponential solutions of different structures, are also produced as a result of the evolution of this improved technique. Furthermore, the graphs in two and three dimensions are provided to visualize the structure of the solutions acquired by choosing certain parameter values.

On the exact solutions for a type of nonlinear Schrödinger equations with a harmonic potential

The nonlinear Schrödinger equation with harmonic potential (NLSE) plays an important role in quantum mechanics, so the exact solutions of this equation is studied in this paper. The NLSE is transformed into the classical nonlinear Schrödinger equation by a new class of traveling wave transformation. Next, the problem of exact solutions is changed into the solutions of ordinary differential equation (ODE) by the method of undetermined function. Then, through low-order sub-ODE method and hyperbolic function method, we get two class of solutions of the ODEs. Finally, a series of new exact solutions of the NLSE are obtained. Meanwhile the related numerical simulations are presented.

Two component model of microtubules and continuum approximation

In the present work, we study the nonlinear dynamics of microtubules, the basic components of the eukaryotic cytoskeleton. We introduce a two-component model describing tangential oscillations of dimers. A crucial nonlinear differential equation is solved using continuum approximation. We show that the dynamics of microtubules can be explained in terms of kink and antikink solitary waves. We used two mathematical procedures, that is the tangent hyperbolic function method and, more general, the simplest equation method. It is shown that both procedures bring about equal solutions.

New optical soliton solutions via two distinctive schemes for the DNA Peyrard–Bishop equation in fractal order

The deoxyribonucleic acid (DNA) dynamical equation, which emerges from the oscillator chain known as the Peyrard–Bishop (PB) model for abundant optical soliton solutions, is presented, along with a novel fractional derivative operator. The Kudryashov expansion method and the extended hyperbolic function (HF) method are used to construct novel abundant exact soliton solutions, including light, dark, and other special solutions that can be directly evaluated. These newly formed soliton solutions acquired here lead one to ask whether the analytical approach could be extended to deal with other nonlinear evolution equations with fractional space–time derivatives arising in engineering physics and nonlinear sciences. It is noted that the newly proposed methods’ performance is most reliable and efficient, and they will be used to construct new generalized expressions of exact closed-form solutions for any other NPDEs of fractional order.

Solitary wave solutions of (2+1)-dimensional Maccari system

In this article, three mathematical techniques have been operationalized to discover novel solitary wave solutions of (2+1)-dimensional Maccari system, which also known as soliton equation. This model equation is usually of applicative relevance in hydrodynamics, nonlinear optics and plasma physics. The [Formula: see text] function, the hyperbolic function and the [Formula: see text]-expansion techniques are used to obtain the novel exact solutions of the (2+1)-dimensional Maccari system (arising in nonlinear optics and in plasma physics). Many novel solutions such as periodic wave solutions by [Formula: see text] function method, singular, combined-singular and periodic solutions by hyperbolic function method, hyperbolic, rational and trigonometric solutions by [Formula: see text]-expansion method are obtained. The exact solutions are shown through 3D graphics which present the movement of the obtained solutions.

Peakon and cuspon excitations in optical fibers for eighth order nonlinear Schrödi̇nger’s model

This paper investigates, the ultrashort or ultrafast pulses propagates through the optical fiber for eighth order nonlinear Schrödinger (NLS) equation. We employ the modified extended tangent hyperbolic function method to obtain cuspon and peakon like structures.

The exact solutions for the [formula omitted]-dimensional Boiti-Leon-Manna-Pempinelli equation

In the paper, we study the Boiti-Leon-Manna-Pempinelli equation with (3 + 1) dimension. By using the modified hyperbolic tangent function method, we obtain more new exact solutions for the (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation, which expand the solutions in previous literature. Those solutions have three forms determined according to the choice of parameters: trigonometric function form, rational function form, hyperbolic function form. Both trigonometric function form solutions and hyperbolic function form solutions include logarithmic function. And we give some examples and draw the some figures of the exact solutions.