Mustafa Bayram

Biruni University · Department of Computer Engineering

The aim of this paper is to establish -extension of the Grüss type integral inequality related to the integrable functions whose bounds are four integrable functions, involving Riemann-Liouville fractional -integral operators. The results given earlier by Zhu et al. (2012) and Tariboon et al. (2014) follow the special cases of our findings.

The aim of this study is to peruse the dispersive Schrödinger–Hirota equation (SH equation) with parabolic law linearity to get optical soliton solutions by utilizing the generalized Kudryashov method (GKM). Firstly, we introduced the solution algorithm of GKM. Then, we substituted the traveling wave transform to dispersive SH equation with parabol...

This article investigates the optical soliton solutions of the Kundu–Mukherjee–Naskar (KMN) equation with the help of the enhanced modified extended tanh method (eMETEM). The equation is crucial for simulating the bending of light beam, rogue waves in the seas, and fiber pulses in optics. Some different kinds of solitons such as singular-periodic,...

This paper tackles the recently introduced (4+1)-dimensional Boiti–Leon–Manna–Pempinelli equation (4D-BLMPE) utilized to model wave phenomena in incompressible fluid and fluid mechanics. Modified extended tanh expansion method (METEM) and the new Kudryashov scheme are implemented to produce analytical soliton solutions for the presented equation. T...

Purpose: In this paper, optical solitons of higher-order nonlinear Schrödinger equation with Kudryashov's sextic power-law of nonlinear refractive index are investigated via the direct mapping method. The considered model identifies the optical soliton pulse propagation in the optical fibers. Deriving the optical solutions of investigated model suc...

Purpose: In this article, two main subjects are discussed. First, the nonlinear Schrödinger equation (NLSE) with an anti-cubic (AC) nonlinearity equation is examined, which has a great working area, importance and popularity among the study areas of soliton behavior in optical fibers, by using the enhanced modified extended tanh expansion method an...

We have discussed the perturbed Gerdjikov-Ivanov (pGI) equation describing optical pulse propagation (PP) with perturbation effects, which has various applications in optical fibers, especially in photonic crystal fibers. According to our literature review, we have discovered new and original soliton types using the Sardar sub-equation and the modi...

We have extracted some soliton solutions of the fractional longitudinal wave equation with the M-truncated derivative (M-LWE), which emerges in a magneto electro-elastic circular rod. To obtain new results of this model, the unified Riccati equation expansion and new Kudryashov methods have been utilized for the first time. The presented methods ha...

We have discussed the perturbed Gerdjikov-Ivanov (pGI) equation describing optical pulse propagation (PP) with perturbation effects, which has various applications in optical fibers, especially in photonic crystal fibers. According to our literature review, we have discovered new and original soliton types using the Sardar sub-equation and the modi...

Purpose: This study includes the examination of the cases where the [Formula: see text]-dimensional chiral nonlinear Schrödinger equation also has Bohm potential. This review is not to obtain different soliton solutions for both cases but to obtain a certain type of soliton and to observe the effect of the problem parameters. By using the modified...

This study investigates the nonlinear Klein–Gordon equation (KGE). We successfully construct some new topological kink-type, non-topological, singular solitons, periodic waves and singular periodic wave solutions to this nonlinear model by using the extended ShGEEM, rational sine-cosine extended (ERSC), and sinh-cosh (ERSCh) methods. In addition, a...

In this research paper, we take into account the ([Formula: see text])-dimensional Kadomtsev–Petviashvili equation which is important in the soliton theory of nonlinear physics. To get the desired soliton solutions, the modified F-expansion method using the Riccati equation which has many solution functions, as well as the modified generalized Kudr...

In this research paper, the generalized projective Riccati equations method (GPREM) is applied successfully to procure the soliton solutions of the local M-fractional longitudinal wave equation (LWE) arising in mathematical physics with dispersion caused by the transverse Poisson's effect in a magneto-electro-elastic circular rod (MEECR). Applying...

Because the importance of the optical wave propagation in fibers, we seek for the optical travelling wave solutions and effect of the third-order dispersion parameter for the extended nonlinear Schrödinger equation (NLSE) which is used to describe the femtosecond(fs) pulse propagation in optical fiber. We have used the modified F-expansion method b...

We examined the (3+1)-dimensional Kadomtsev–Petviashvili–Boussinesq (KP-B) equation, which arises not only in fluid dynamics, superfluids, physics, and plasma physics but also in the construction of connections between the hydrodynamic and optical model fields. Moreover, unlike the Kadomtsev–Petviashvili equation (KPE), the KP-B equation allows the...

This article investigates the optical soliton solutions of the time-fractional Biswas–Arshed (BA) equation, which is a new model for soliton transmission through optical fibers where the eigen-phase modulation is negligible and thus removed. We have studied both models including M-truncated or beta derivative operators. Firstly, the time-fractional...

In this paper, some analytical solutions for a model of dual-core optical fibers governed by a system of coupled non-linear Schrödinger equations (NLSEs) and the effect of the coefficient of the group velocity dispersion term on the considered model are investigated. The group velocity dispersion (GVD) has a important role in the optical wave propa...

This article is devoted to the even entire functions, which are the exact solution for the Laplace and diffusion equations. These functions are considered in the algebraic number field. We guess that the functions have purely real zeros in the entire complex plane. These are proposed as new connections with algebraic number theory and mathematical...

The unstable nonlinear Schrödinger equations (UNLSEs) are universal equations of the class of nonlinear integrable systems, which reveal the temporal changing of disruption in slightly stable and unstable media. In current paper, an improved auxiliary equation technique is proposed to obtain the wave results of UNLSE and modified UNLSE. Numerous va...

Numerical methods play an important role in modern mathematical research, especially studying the symmetry analysis and obtaining the numerical solutions of fractional differential equation. In the current work, we use two numerical schemes to deal with fractional differential equations. In the first case, a combination of the group preserving sche...

In this scientific research article, we consider the (2+1)-Date–Jimbo– Kashiwara–Miwa equation with conformable derivative (C-DJKME), a water wave model with low surface tension and long wavelengths with weakly nonlinear restoring forces and frequency dispersion. Since the solutions of C-DJKME constitute the basis and model of many physical phenome...

In this paper, we consider the Boussinesq equation which is an important equation and it is widely used in coastal engineering, harbors, shallow seas and water wave to model weakly nonlinear and long wave approximation. Exact traveling wave solutions in such equations are extremely valuable in analytical and numerical theories. To compute the solit...

Purpose In this study, we investigated the time-fractional coupled nonlinear Schrödinger (TF-CNLS) system in conformable fractional sense. TF-CNLS system models the circularly-polarized waves in the optic branch of physics frequently. Methodology To obtain optical solitons of the model via new Kudryashov method and Kudryashov auxiliary equation me...

Purpose: When it comes to third and higher-order dispersion, the Schrödinger–Hirota equation is one of the main models developed outside the classical NLSE management models for optical soliton transmission. The cubic–quartic Fokas–Lenells equation is also one of the recently developed equations, which has importance in the field of telecommunicati...

This article deals with the study of ultrasound propagation, which propagates the mechanical vibration of the molecules or of the particles of a material. It measures the speed of sound in air. For this reason, the third-order non-linear model of the Westervelt equation was chosen to be studied, as the solutions to such problems have much importanc...

In this paper, we aim to derive new soliton solutions of [Formula: see text]- and [Formula: see text]-dimensional generalized Sasa–Satsuma equations via the new Kudryashov method. In optical fiber transmission systems, the Sasa–Satsuma equation describes the effects of third-order dispersion, self-steepening and stimulated Raman scattering in the p...

In this scientific research article, the new Kudryashov method and the tanh-coth method, which have not been applied before, are employed to construct analytical and soliton solutions of the (2+1)-dimensional Hirota–Maccari system. The (2+1)-dimensional Hirota–Maccari system is a special kind of nonlinear Schrödinger equation (NLSEs) that models th...

In this paper, we studied the (3 + 1)-dimensional nonlinear Kadomtsev-Petviasvili equation (3D-KPE) that is utilized in order to describe 3D solitons in weakly dispersive media, long wavelength water waves with weak nonlinear restoring forces, waves in ferromagnetic media, nonlinear wave propagation in supefluids, plasma physics and fluid dynamics...

In this work, we aim to derive various soliton solutions to the Wazwaz–Benjamin–Bona–Mahony equation with conformable and M-truncated derivatives. The considered equation models long waves in the ocean engineering field. Unified Riccati equation expansion and Kudryashov auxiliary equation methods are used to the model, and so, kink, singular, and p...

Purpose This study aims to examine the optical soliton solutions of the nonlinear Schrödinger form of the (2+1)-Biswas-Milovic equation with Kerr, power and parabolic law nonlinearity. Methodology Obtaining the nonlinear ordinary differential equation (NODE) and constraint relations for each Kerr, power and parabolic nonlinearity form by using wav...

In this paper, we have investigated the perturbed Chen–Lee–Liu equation which describes the pulse propagation in the optical fibers, under the impact of the inter-modal dispersion, self-steepening and nonlinear dispersion terms. By using the enhanced modified extended tanh expansion method, bright, singular, periodic singular and periodic bright so...

In this study, we have focused on finding soliton solutions of the cubic–quartic Fokas–Lenells equation, which models the nonlinear pulse transmission through optical fiber, is a pretty new and updated model. The main motivation of this study is to produce new solutions with previously unused methods for data transmission models in fiber optic cabl...

In this paper, we have successfully extracted many analytic solutions for the (1+2)-dimensional Chiral non-linear Schrödinger equation (NLSE) by the enhanced modified extended tanh expansion method (eMETEM). The considered method is a recently enhanced version of the classical modified extended tanh expansion method. So, we have successfully extrac...

This study investigates various analytic soliton solutions of the generalized (2+1)-dimensional Calogero-Bogoyavlenskii-Schiff (CBS) equation in fluid dynamics and plasma physics using a recently introduced technique which is the New Kudryashov method. Moreover, it is examined how the wave propagation in both directions represented by the CBS equat...

Purpose The main motivation of the study is to present the (3+1) version of the Biswas-Milovic equation (BME) for the first time and to obtain effective bullet (soliton) solutions. In the literature, it is observed that this equation is generally solved in (1+1) form. Methodology The part of the (3+1) dimensional Biswas-Milovic equation containing...

In this scientific work, we explore the solutions of an important mathematical model appearing in the nonlinear elastic inhomogeneous Murnaghan’s rod (NEIMR) or called doubly dispersive equation (DDE) with a useful and powerful analytical technique; which is the extended Kudryashov method with the Bernoulli–Riccati approach. In our research, we hav...

In this paper, we present the higher-order nonlinear Schrödinger equation (NLSE) with third order dispersion (3OD), fourth-order dispersion (4OD), and cubic-quintic nonlinearity (CQNL) terms that define the propagation of ultrashort pulses. Two analytical methods, which are the new Kudryashov’s method and the unified Riccati equation expansion meth...

In the present work, an efficient Lie group integrator called group preserving scheme (GPS) is utilized for the numerical treatment of the Sine-Gordon problems. With the help of a suitable transformation, we turn the main problem into a new one and consequently solve the converted problem using GPS. Indeed, preserving the Lie group structure under...

Purpose This manuscript addresses to investigate the optical soliton solution of the Manakov model which governs soliton transmission technology by utilizing the auxiliary equation technique. Methodology After obtaining the nonlinear ordinary differential equation form (NODE) of investigated nonlinear differential partial problem (NLPDE) with the...

In this study, we have focused on finding soliton solutions of the cubic-quartic Fokas-Lenells equation, which models the nonlinear pulse transmission through optical fiber, is a pretty new and updated model. We have used three different efficient analytical methods, namely, the Sinh-Gordon expansion method, enhanced modified extended tanh expansio...

This paper presents an investigation of soliton solutions for the perturbed Fokas-Lenells (pFL) equation, which has a vital role in optics, using Sardar sub-equation method. The equation models the propagation of ultrashort light pulses in optical fibers. Using appropriate wave transformation, the pFL equation is reduced to a nonlinear ordinary dif...

Purpose In this article, we establish novel solutions for the perturbed nonlinear Schrödinger–Fokas–Lenells equation, which is one of the model to investigate soliton dynamics through a polarization–preserving optical fiber, by using an enhanced modified extended tanh expansion method. Our aim is, not only gain various analytical optical soliton so...

Purpose This study aimed to examine the effects of the parameters, the group velocity dispersion (GVD), the third-order dispersion (3OD), spatio-temporal dispersion (STD), the third-order spatio-temporal dispersion (TO-STD), self-steepening effect and nonlinear dispersions terms, included in the equations modeling optical fiber phenomena on the opt...

Some new optical solitons find for the Kudryashov equation (KE) in this study. These solutions are in the form of dark, bright, singular, singular-dark solitons and other solutions with certain conditions. These new solutions may be applied in the demonstration of Kudryashov equation in some better way. Modified integration method, the extended Sin...

This paper carries out the analytical optical soliton solutions of perturbed Radhakrishnan–Kundu–Lakshmanan (pRKL) equation with Kerr law nonlinearity using an efficient modified extended tanh expansion method, enhanced with the new Riccati solutions (eMETEM). In this study, we have established robust solutions for the pRKL by using the eMETEM meth...

Purpose In the last decades, many researchers have performed their best in order to find the solution of the nonlinear evolution equations, nonlinear partial differential equations (NLPDEs), fractional nonlinear partial differential equations and nonlinear optical models by using Kudryashov methods, which have various forms. However, most researche...

Purpose The importance of efficient analytical methods in solving nonlinear partial differential equations and nonlinear optical differential equations cannot be underestimated. Because of this importance, to solve efficiently above mentioned partial differential equations, a method has been suggested to use the improved Kudryashov and a sub equati...

Objective The principal purpose of this paper is to examine the perturbed Schrödinger-Hirota equation with the effect of spatio-temporal dispersion and Kerr Law nonlinearity which governs the propagation of dispersive pulses in optical fibers by proposing and using a direct algebraic form of the enhanced modified extended tanh expansion method for...

This paper extracts some analytical solutions of simplified modified Camassa-Holm (SMCH) equations with various derivative operators, namely conformable and M-truncated derivatives that have been recently introduced. The SMCH equation is used to model the unidirectional propagation of shallow-water waves. The extended rational sine−cosine and sinh−...

We introduce optical soliton solutions of the Kundu–Mukherjee–Naskar (KMN) equation by using the Sardar subequation (SSM) and the new Kudryashov methods (nKM). The KMN equation plays an important role in modeling the fiber pulse in optics, the rogue waves in the oceans and the bending of the light beam. In order to motivate and contribute to resear...

Objective – The main objective of this paper is to investigate analytic soliton solutions of a nonlinear Schrödinger equation (NLSE), including Kudryashov’s sextic power-law nonlinearity by introducing new approaches of two efficient analytical methods. The considered equation has been recently introduced by N. A. Kudryashov to describe pulse propa...

In this research article, the Sardar subequation method is used to retrieve new analytical solutions to the space-time local derivative Sasa–Satsuma equation with Atangana’s conformable derivative, which defines short pulse propagation in an optical fiber area. This equation is the integrable extension of the nonlinear Schrödinger equation. First,...

In this research paper, we scrutinize the novel traveling wave solutions and other solutions with conformable, M-truncated and beta fractional derivatives for the nonlinear fractional Hirota–Maccari system. In order to acquire the analytical solutions, the Riccati–Bernoulli sub-ODE technique is implemented. Presented method is the very powerful tec...

In this paper, the time-fractional Jaulent–Miodek system associated with energy-dependent Schrödinger potential is solved by the modified Laplace decomposition method. The Caputo fractional derivative is considered throughout the paper. The attained solutions using the method are analyzed and compared with the solutions of the existing studies in t...

In this paper, we investigate resonanat nonlinear Schrödinger equation (RNLSE) with self steeping phenomena to obtain some chirped periodic (CP) and soliton waves. A chirp is a signal in which the frequency increases (up chirp) or decreases (down-chirp) with time. It is commonly used in sonar, radar and laser systems and in other applications, such...

This study aims to investigate the complicated dynamical HCO3-/CO2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$HCO_3^-/CO_2$$\end{document} buffering system using fr...

We have searched for optical solutions of Kaup–Newell equation (KNE) used in nonlinear optical and plasma physics, which have a very important role in the class of derivative nonlinear Schrödinger equations (dNLSEs). To obtain exact wave solutions of the proposed model, two analytical techniques, recently presented as new Kudryashov’s method and th...

A soliton is a packet of self-reinforcing waves that maintains its structure when moving at a constant speed. Solitons are caused by the cancellation of the medium’s nonlinear and dispersive effects. In plasmas, the bilinear form of Hirota will be utilized to investigate the (2+1)-dimensional Korteweg-de Vries equation with electrostatic wave poten...

This paper presents investigation of soliton solutions for the perturbed Fokas-Lenells (pFL) equation, has a vital role in optics, using Sardar sub-equation method. The equation models the propagation in ultrashort light pulses in optical fibers. Using appropriate wave transformation, the pFL equation is reduced to a nonlinear ordinary differential...

In this paper, we study the cubic-quintic nonlinear Schrödinger equation (CQ-NLSE) to describe the propagation properties of nonlinear periodic waves (PW) in an optical fiber. We find chirped periodic waves (CPW) with some Jacobi elliptic functions (JEF). We also obtain some solitary waves (SW) like dark, bright, hyperbolic and singular solitons. T...

4th International Conference on Pure and Applied Mathematics (ICPAM-VAN 2022) VAN, TURKEY Dear Colleague, Due to the global spread of COVID-19, we organize a virtual conference entitled 4th International Conference on Pure and Applied Mathematics (ICPAM-VAN 2022), which will be held on June 22-23, 2022. For more information, you can http://icpam.yy...

This paper considers deriving new exact solutions of a nonlinear complex generalized Zakharov dynamical system for two different definitions of derivative operators called conformable and $ M- $ truncated. The system models the spread of the Langmuir waves in ionized plasma. The extended rational $ sine-cosine $ and $ sinh-cosh $ methods are used t...

Using an efficient modified extended tanh expansion method, we establish robust solutions for the Radhakrishnan-Kundu-Lakshmanan equation in this work. We want to utilize the effective scheme to solve the problem for the governing model. Bright and mixed dark–bright optical solitons are successfully exposed using the Maple symbolic package. On the...

In this paper, we have successfully extracted novel analytic solutions for the (1+2)-dimensional Chiral non-linear Schrödinger (NLS) equation by modified extended tanh expansion method combined with new Riccati solutions (METEM-cNRCS) as far as we know. When a wave transformation is applied to the considered Chiral NLS equation, a nonlinear ODE is...

In this research, Genocchi wavelets method, a quite new type of wavelet-like basis, is adopted to obtain a numerical solution for the classical and time-fractional Rosenau-Hyman or K(n,n) equation arising in the formation of patterns in liquid drops. The considered partial differential equation can be transformed into a system of non-linear algebra...

This study presents a large family of the traveling wave solutions to the two fourth-order nonlinear partial differential equations utilizing the Riccati-Bernoulli sub-ODE method. In this method, utilizing a traveling wave transformation with the aid of the Riccati-Bernoulli equation, the fourth-order equation can be transformed into a set of algeb...

This article develops a method based on the generalized Gegenbauer–Humbert wavelets in concert with their operational matrices of fractional integration to deal with the fractional partial differential equations and find the approximate solutions of it. The goal is to show that the proposed method is appropriate for boundary and initial-boundary pr...

The modified extended tanh expansion method is applied to the perturbed Chen–Lee–Liu equation where the perturbation terms are with full nonlinearity. This method contributes a variety of optical soliton solutions including dark, singular, dark-singular soliton, singular periodic waves and rational function solutions to the perturbed Chen–Lee–Liu e...

In this research paper, a numerical method, named the three-step Ultraspherical wavelet collocation method, is presented for solving some nonlinear multi-dimensional parabolic partial differential equations. The method is third-order accurate in time. In this method, the three-step Taylor method is used to get the time derivative, while the Ultrasp...

To establish the effects of the Higher-Order Dispersion (HOD) and Raman Scattering (RS) on Modulation Instability (MI) in the presence of the Saturable Function (SF), it is used the Coupled Nonlinear Schrödinger Equation (CNLSE) which describes pulses propagating in a higher optical beam in oppositely guided coupler. It has been shown through the M...

The current study employs the novel Hirota bilinear scheme to investigate the nonlinear model. Thus, we acquire some two-wave and br