English العربية کوردی
01007018

Dr. Karmina Kamal Ali

Assistant Professor
College of Science / Department of Mathematics
9647504717745
karmina.ali@uoz.edu.krd
Number of Publications: 75
0
Awards
75
Published Journal Articles
1
Thesis
2
Training Course

1
Book Chapter
0
Conference
0
Presentation
0
Workshop
2022
Doctor of Philosophy

Mathematics
Firat University

2013
Master

Zakho
Zakho

2008
Bachelor

Department of Mathematics
University of Salahaddin

Teaching undergraduate courses in Applied Mathematics, Linear Algebra, Foundation of Mathematics, Geometry and Advanced Calculus.

Worked as a member in quality assurance of undergraduate students in Mathematics Department.

Supervising undergraduate student projects.

Google scholar
Researchgate
Scopus
Assistant Professor

Date: 2022-12-11

Lecturer

Date: 2017-07-23

Assistant Lecturer

Date: 2013-11-04

Assistant Researcher

Date: 2009-03-04

2026

Dynamical behavior and exact solutions of the third-order (1+1)-dimensional nonlinear Schrödinger equation with general power-law non linearity
Save article Optical and Quantum Electronics (Issue : 400) (Volume : 58)
This study examines bifurcation structures, optical soliton solutions, chaotic dynamics, and sensitivity analysis for a third-order (1+1)-dimensional nonlinear Schrödinger equation with power-law self-phase modulation, spatio-temporal dispersion, and inter-modal dispersion, neglecting chromatic dispersion. A traveling-wave transformation reduces the model to a singular dynamical system, which is regularized through a suitable change of the independent variable; both systems have the same first integrals. Bifurcation theory for planar systems is applied to analyze the corresponding phase portraits. The modified extended hyperbolic function method yields new exact traveling-wave solutions, including exponential, bright, singular, and single-periodic forms. Chaotic behavior is investigated via Poincaré sections, bifurcation diagrams and Lyapunov exponents, together with a detailed analysis of sensitivity to initial conditions. The results broaden the understanding of nonlinear wave propagation in optical media, with applications in quantum optics, nonlinear optics, and plasma physics. Unlike Hamad and Ali (2025b), who considered only n=1, this work addresses a general nonlinearity index.
2026-07
Bifurcation analysis and exploration of new optical soliton solutions in parabolic law medium with weak non-local nonlinearity
Communications in Theoretical Physics
In the present study, we examine how optical solitons behave in a synthetic nonlocal nonlinear medium. An analytical investigation is conducted into the onlineardynamical model that represents the propagation of optical solitons in weakly nonlocal nonlinear media with parabolic law nonlinearity.Duetoitscapacityto explain intricate physical phenomena and display dynamic structures of localized wave solutions, this equation has attracted a lot of attention. The Khater method and the (1/G′-expansion approacharethetwoanalytical techniquesusedtoobtainawiderangeofsolitonsolutions, including mixed, kink, and anti-kink profiles.The obtained solitonsolutionsindicatethatthemodelcandescribestablepulsetransmission,opticalswitchingfronts, and nonlinear wave modulation.These results are potentially useful for applications in optical communication systems, pulse shaping, and nonlinear signal processing.Wealsoofferathoroughbifurcationanalysisthatgoes beyondsolutionformationtoidentifydistinctdynamicalregimesthatmarktheboundariesofstability, instability, and periodicity transitions. To investigate the dynamical behavior of the solutions, graphical representations and numerical simulations are offered.Thisstudyprovidesafoundationforfurtherinvestigationandillustration of other dynamical features present in the physical processes.To the best of our knowledge, the results reported in this work are completely novel and have not been examined in previous literature.
2026-04
LIE GROUP STRUCTURES AND SOLITON SOLUTIONS TO THE NONLINEAR SCHRO¨ DINGER EQUATION
Romanian Reports in Physics (Issue : 105) (Volume : 78)
This paper investigates the Lie symmetry structure, conservation laws, and novel soliton solutions of a nonlinear Schr¨odinger equation. We determine the admitted Lie point symmetries and establish the associated Lie algebra generated by 􀀀1 = @x, 􀀀2 = @t, and 􀀀3 = x@x +2t@t. The algebraic structure is explored through the commutator table, adjoint representation, and classification of the optimal system of subalgebras. Corresponding similarity reductions transform the PDE system into reduced ordinary differential equations, for which approximate solutions are constructed using power series methods. In addition, conservation laws are systematically derived via conservation theorem, linking the symmetries to physically meaningful invariants. Novel soliton solutions have been established using the modified generalized Riccati equation mapping method (MGREMM). The results provide a comprehensive symmetry-based framework for the system under study, contributing to the broader understanding of nonlinear coupled PDEs and their analytical properties
2026-04
Dynamics of wave propagation for (1+ 2)-dimensional Chiral nonlinear Schrodinger in quantum theory
Computational Methods for Differential Equations
We investigate and extracted many solutions for the (1+2)-dimensional Chiral nonlinear Schrödinger equation by new modified unified auxiliary equation method in this paper. The considered method is a enhanced version of the unified auxiliary equation method. The mentioned model is used to discuss the wave in quantum field theory. Consequently, some different and more general solutions are obtained such as singular, periodic, hyperbolic, dark-bright, trigonometric, exponential, and Jacobi elliptic functions. These solutions are more general in nature and offer different approaches to address the problem at hand. To elucidate the dynamics of these solutions, their figures are presented. These figures provide a visual representation of the behavior of the solutions, enabling a better understanding of their characteristics.
2026-04
Dynamical system, chaotic behavior, and sensitivity of the resonant nonlinear Schrödinger equation with the parabolic law
Journal of Optics
In this study, we investigate the cubic-quartic resonant nonlinear Schrödinger equation (CQRNSE) under a parabolic law to obtain various categories of optical solutions, including bright, dark, singular soliton, rational wave and singular periodic wave solutions. The CQRNSE describes wave propagation in fiber optics. The modified F-expansion method is a powerful technique employed to construct these solutions. Additionally, using bifurcation theory, we derive the associated Hamiltonian function and analyze the corresponding phase portraits. To graphically illustrate and exhibit the obtained solutions, we present them in both 3D and 2D formats. Furthermore, Poincaré sections and bifurcation diagram are used to analyze periodic, quasi-periodic, and chaotic behaviors of the related dynamic system, along with the system’s sensitivity to initial conditions. The findings of this research are fascinating and make a significant contribution to the domain of solitons in particular, and to the broader field of mathematical physics. The approach used in this work yields a variety of new solitary and soliton wave solutions, which will be valuable for researchers applying these models to real-world problems.
2026-04
Bifurcation analysis and exploration of new optical soliton solutions in parabolic law medium with weaknon-local nonlinearity
Scientific Reports
In the present study, we examine how optical solitons behave in a synthetic nonlocal nonlinear medium. An analytical investigation is conducted into the nonlinear dynamical model that represents the propagation of optical solitons in weakly nonlocal nonlinear media with parabolic law nonlinearity. Due to its capacity to explain intricate physical phenomena and display dynamic structures of localized wave solutions, this equation has attracted a lot of attention. The Khater method and the $\left( {\frac{1}{{G'}}} \right)$-expansion approach are the two analytical techniques used to obtain a wide range of soliton solutions, including mixed, kink and anti-kink profiles. The obtained soliton solutions indicate that the model can describe stable pulse transmission, optical switching fronts, and nonlinear wave modulation. These results are potentially useful for applications in optical communication systems, pulse shaping, and nonlinear signal processing. We also offer a thorough bifurcation analysis that goes beyond solution formation to identify distinct dynamical regimes that mark the boundaries of stability, instability, and periodicity transitions. To investigate the dynamical behavior of the solutions, graphical representations and numerical simulations are offered. This study provides a foundation for further investigation and illustration of other dynamical features present in the physical processes. To the best of our knowledge, the results reported in this work are completely novel and have not been examined in previous literature.
2026-03

2025

The formation of optical soliton solutions of the Kudryashov equation using the unified auxiliary equation method
Ain Shams Engineering Journal (Issue : 12) (Volume : 16)
In this work, we examine the dynamic properties of the well-known Kudryashov equation in the context of pulse propagation within optical fibers. For this purpose, we utilize the unified auxiliary equation method (UAEM), which is a powerful technique for deriving abundant exact solutions for nonlinear partial differential equations. Using a suitable wave transformation changes the given equation is converted into an ordinary differential equation, which makes it possible to treat it systematically. Through an extensive list of Jacobi elliptic functions in the UAEM, a large number of exact solutions are created, comprising singular, kink, dark, bright, rational, and combined solutions. Soliton theory is especially significant because solitons are stable, localized nonlinear waves that preserve their shape and energy during propagation and interaction. Due to this property, they are essential for comprehending nonlinear wave dynamics as well as for real-world uses like fluid dynamics, plasma waves, and long-distance optical communication.
2025-12
Chaotic behavior and traveling wave solutions of the paraxial nonlinear Schrödinger equation
Modern Physics Letters A (Issue : 33) (Volume : 40)
The paraxial nonlinear Schrödinger equation is used in many domains such as mathematical physics, optical communication systems, nonlinear optics, and plasma physics. In this work, we aim to investigate the dynamic properties of the model employing bifurcation theory and determining the corresponding Hamiltonian function. By applying the bifurcation and planar dynamical system theories, we obtain the traveling wave solutions with the corresponding phase portraits. The purpose of the carefully selected parameter values and the resulting plot presentations is to guarantee the physical validity of our results. These results provide answers that are thorough and reliable by illuminating the applicability, effectiveness, and computing speed of the employed scheme. In addition, chaotic behavior and sensitive analysis of the model under consideration are studied. This work has directly opened up new directions for investigation and possible applications in other nonlinear science fields.33
2025-10
Bifurcation analysis, chaotic pattern, and optical solutions for the pure-cubic optical Schrödinger equation with Kerr law nonlinearity
Ain Shams Engineering Journal (Issue : 19) (Volume : 16)
In this article, we are devoted to extracting optical solutions to the pure-cubic optical non-linear Schrödinger equation with Kerr law non-linearity. Applying bifurcation theory, we investigate the dynamic properties and the determination of the corresponding Hamiltonian function. Through the complete discriminant system technique and symbolic computation, several traveling wave solutions are obtained, such as the singular periodic, bright, dark, singular solitons, combo snoidal-cnoidal, exponential solutions, and cnoidal wave solutions. Furthermore, using planar dynamical system theories and bifurcation theories, some other new traveling wave solutions are derived, like the cnoidal, snoidal, dark, and singular periodic solutions. Additionally, the chaotic behavior of dynamic systems is examined using 3D phase portraits, time series, and Poincaré sections. Furthermore, the sensitivity to initial conditions is also studied. The purpose of the carefully selected parameter values and the resulting plot presentations is to guarantee the physical validity of our results.
2025-08
Chaotic phenomena, quasi-periodic, and optical solutions to the third-order (1+ 1)-nonlinear Schrödinger equation with a power law of self-phase modulation equation
Ain Shams Engineering Journal (Issue : 10) (Volume : 16)
The main objective of this paper is to study optical soliton solutions, bifurcations, chaotic behavior, sensitive analysis, and Lyapunov exponents for the third-order (1+1)-nonlinear Schrödinger equation with a power law of self-phase modulation, spatiotemporal dispersion, and inter-modal dispersion in the absence of chromatic dispersion. Firstly, bifurcation theory is applied to investigate the phase portraits and dynamical behavior of this model and then to determine the associated Hamiltonian function. Secondly, by applying a complete discriminant system method with symbolic computation and the modified F-expansion method, distinct solutions are constructed, which are categorized as periodic singular, bright, dark, and singular solutions. Additionally, the chaotic property of dynamic systems is studied by time series, Poincaré sections, 3D phase portraits, and Lyapunov exponents. These results provide new viewpoints on how nonlinear optical systems respond to random perturbations. Finally, numerical simulations are provided to show the physical characteristics of the derived optical soliton solutions.
2025-07
Bifurcation Analysis, Chaotic Pattern, and Optical Solutions for the Pure-Cubic Optical Schrödinger Equation with Kerr Law Nonlinearity
Physics Letters A
In this article, we are devoted to extracting optical solutions to the pure-cubic optical non-linear Schrödinger equation with Kerr law non-linearity. Applying bifurcation theory, we investigate the dynamic properties and the determination of the corresponding Hamiltonian function. Through the complete discriminant system technique and symbolic computation, several traveling wave solutions are obtained, such as the singular periodic, bright, dark, singular solitons, combo snoidal-cnoidal, exponential solutions, and cnoidal wave solutions. Furthermore, using planar dynamical system theories and bifurcation theories, some other new traveling wave solutions are derived, like the cnoidal, snoidal, dark, and singular periodic solutions. Additionally, the chaotic behavior of dynamic systems is examined using 3D phase portraits, time series, and Poincaré sections. Furthermore, the sensitivity to initial conditions is also studied. The purpose of the carefully selected parameter values and the resulting plot presentations is to guarantee the physical validity of our results.
2025-07
Soliton-M-lump wave formation and other interacting propagation for the nonlinear Geng equation
Springer Berlin Heidelberg (Issue : 6) (Volume : 140)
In this work, we obtain various kind of lump interaction solutions for the (3 + 1)-dimensional nonlinear Geng equation, which was derived in relation to Hamiltonian flows on nonlinear subvarieties of hyperelliptic Jacobian. The well-known technique namely Hirota bilinear is used to accomplish the task. A lump solution is a real analytical rational function solution that decays in all directions of the space variables. The governing equation describes wave dynamical behaviors in more complex applications related to shallow water wave or other similar fluids. Numerical simulations using the three-dimensional and contour profiles are carried out with careful consideration for the values of the involved parameters in order to shed more light on the properties of the acquired solutions.
2025-06
The modulation instability gain spectrum assessment and generalized propagating wave structures of nonlinear cubic-quartic Schrödinger equation
Modern Physics Letters B (Issue : 11) (Volume : 40)
This study investigates a cubic-quartic nonlinear Schrödinger model with third- and fourth-order dispersions without the disturbance in parabolic law media and group velocity dispersion (GVD). The analytical method is used to produce traveling wave solutions because the inverse scattering transform is unable to solve the Cauchy problem for this equation. The utilized analytical approach is novel and providing generalized families of solutions as compared to other, thus, prior to this study, there was not existing any work in which suck kinds of solutions were exist. The propagating solitons are driven using the unified auxiliary equation approach. As a result, the aforementioned technique is learned in the following areas: singular soliton, shock solution, mixed-singular, mixed trigonometric, mixed shock single solution, plane solutions, exponential, trigonometry, mixed-hyperbolic solution, periodic and mixed periodic solutions, and trigonometry. The stability of the topic under consideration is shown by the modulational instability (MI) study. The appropriate parametric values are shown alongside the graphical representation.
2025-03

2024

Kudryashov-Sinelshchikov equation: phase portraits, bifurcation analysis and solitary waves
Optical and Quantum Electronics (Volume : 56)
This work investigates the Kudryashov-Sinelshchikov equation, which explains the influence of viscosity and heat transfer on pressure wave propagation. This equation is transformed into a singular dynamical system by applying the proper wave transformation. Then, by applying the proper point transformation for the independent variable, it may be transformed again into a regular dynamical system. We show that the first integral of both systems is the same. We provide a brief overview of the phase planes of both systems, taking into account the topological equivalence between their phase orbits. Furthermore, the new Kudryashov method and the sine-Gordon expansion method are considered to construct some novel exact solutions to the Kudryashov-Sinelshchikov equation. Using the wave transform, the studied equation is converted to the ODE. Then, the proposed methods are successfully applied to the equation to reveal some new solutions. The obtained solutions are stated as hyperbolic and exponential functions, which are classified as singular, bright, dark, kink, and anti-kink solutions. To better understand the physical properties of the solutions, the two-dimensional and three-dimensional graphs for the solutions are presented by selecting appropriatore values for the including physical parameters. Exploring these behaviors may play an important role in understanding the propagation of nonlinear waves in a combination of liquid with gas bubbles.
2024-09
Bifurcations of phase portraits and exact solutions of the ( 2 + 1)‑dimensional integro‑differential Jaulent–Miodek equation
Optical and Quantum Electronics (Issue : 2024) (Volume : 56)
This paper is dedicated to exterminate the ( 2 + 1)-dimensional integro-differential Jaulent–Miodek equation, a prominent model linked to energy-dependent Schrödinger potential. This equation is employed in a wide array of disciplines, including fluid dynamics, condensed matter physics, optics, and various engineering systems. First, we are given to derive exact wave solutions for the ( 2 + 1)-dimensional integro-differential Jaulent–Miodek equation using an innovative approach known as the new modified unified auxiliary equation method. We offer a comprehensive visual representation and some exact solutions propagate of these solutions in 2D and 3D plots, using various parametric values for a comprehensive analysis. In addition, we employed the planar dynamical system method to study phase portraits and chaotic behavior of the governing equation. Through the analysis of phase portraits, the sensitivity of dynamic system is examined. The chaotic behavior of dynamic system is examined by time series, Poincaré section, and 2D, 3D phase portraits.
2024-07
Exact solutions of the (2+1)-dimensional Konopelchenko–Dubrovsky system using Sardar sub-equation method
Modern Physics Letters B (Issue : 31) (Volume : 38)
In this paper, the new modification of the Sardar sub-equation method is used to generate a wide variety of exact solutions for the (2+1)-dimensional Konopelchenko–Dubrovsky system. We focus on investigating the Konopelchenko–Dubrovsky equation, which serves as a mathematical model for studying nonlinear waves in the field of mathematical physics. This equation specifically captures the behavior of waves with weak dispersion, allowing us to explore the intricate dynamics and characteristics associated with such wave phenomena. By delving into the properties and solutions of this system, we aim to deepen our understanding of nonlinear wave propagation and its implications in the broader field of mathematical physics. The exact solutions generated through this modified method provide valuable insights into the propagation and interaction of waves with weak dispersion in the system. The obtained novel solutions are expressed as hyperbolic, and trigonometric functions. The proposed model successfully constructs various types of solutions, including singular, dark, bright, trigonometric, periodic, dark–bright, exponential, and hyperbolic. These solutions are presented with appropriate parameter values in both 3D and 2D graphics.
2024-07
Soliton waves with optical solutions to the three-component coupled nonlinear Schrödinger equation
Modern Physics Letters A
This study uses the modified Sardar sub-equation method to find novel soliton solutions to the nonlinear three-component coupled nonlinear Schrödinger equation (NLSE), which is used for pulse propagation in nonlinear optical fibers. Multi-component NLSE equations are widely used because they can represent a wide range of complex observable systems and more dynamic patterns of localized wave solutions. The optical solutions proposed in this study are novel and can be described using hyperbolic, trigonometric, and exponential functions. These solutions are categorized as bright, dark, singular, combo bright-singular, and periodic solutions. Some solutions’ dynamic behaviors are demonstrated by selecting appropriate physical parameter values. The results and computational analysis indicate that the techniques provided are simple, effective, and adaptable. They can be applied to a variety of nonlinear evolution equations, whether stable or unstable, and can be used in fields such as mathematics, mathematical physics, and applied sciences.
2024-06
Dynamics of pulse propagation with solitary waves in monomode optical fibers with nonlinear Fokas system
Modern Physics Letters B (Issue : 31) (Volume : 38)
In this study, a unified auxiliary equation method, which is one of the powerful methods for exploring nonlinear model solutions, is used in the Fokas system, with complex functions representing nonlinear pulse propagation in monomode optical fibers. As a result, we get some solutions, including dark–bright, singular, periodic, bright–dark, Jacobi elliptic functions, trigonometric, hyperbolic and exponential ones. In addition, we use a computer program to generate 3D, 2D and counterplot graphics from the obtained solutions by assigning specific values to the involved parameters. While discussing, the graphs for various values of an arbitrary constant are examined. These findings constitute an important step in understanding how solitary waves are generated in nonlinear media. Since the studied model is used in many domains, including Bose–Einstein condensates and plasma physics, these results improve our theoretical knowledge and open up new avenues for potential real-world applications and the development of cutting-edge technologies.
2024-06
Optical soliton solutions of generalized Pochammer Chree equation
Optical and Quantum Electronics (Issue : 2024) (Volume : 56)
This research investigates the utilization of a modified version of the Sardar sub-equation method to discover novel exact solutions for the generalized Pochammer Chree equation. The equation itself represents the propagation of longitudinal deformation waves in an elastic rod. By employing this modified method, we aim to identify previously unknown solutions for the equation under consideration, which can contribute to a deeper understanding of the behavior of deformation waves in elastic rods. The solutions obtained are represented by hyperbolic, trigonometric, exponential functions, dark, dark-bright, periodic, singular, and bright solutions. By selecting suitable values for the physical parameters, the dynamic behaviors of these solutions can be demonstrated. This allows for a comprehensive understanding of how the solutions evolve and behave over time. The effectiveness of these methods in capturing the dynamics of the solutions contributes to our understanding of complex physical phenomena. The study’s findings show how effective the selected approaches are in explaining nonlinear dynamic processes. The findings reveal that the chosen techniques are not only effective but also easily implementable, making them applicable to nonlinear model across various fields, particularly in studying the propagation of longitudinal deformation waves in an elastic rod. Furthermore, the results demonstrate that the given model possesses solutions with potentially diverse structures.
2024-04
Optical solutions to the stochastic Fokas–Lenells equation with multiplicative white noise in Itô sense using Jacobi elliptic expansion function method
Optical and Quantum Electronics (Volume : 56)
In this paper, we study the stochastic Fokas–Lenells equation with multiplicative white noise in the Itô sense. For this purpose, we will use the Jacobi elliptic expansion function method. The nonlinear partial differential equation is transformed into ordinary differential equations through the symmetry reduction technique. New optical solutions are constructed as dark, combined hyperbolic, periodic, bright, and rational solutions. Depending on the parameters, these optical solutions exhibit varying periods of qualitative characteristics. Furthermore, the effects of time on the solutions of the nonlinear stochastic Fokas–Lenells equation are examined. In addition, graphical representations of a few solutions are included to supplement our analysis.
2024-04
Investigation of Brownian motion in stochastic Schrödinger wave equation using the modified generalized Riccati equation mapping method
Optical and Quantum Electronics (Volume : 56)
This work investigates the stochastic nonlinear Schrödinger equation in a more extended form, influenced by multiplicative noise which represents the temporal change of fluctuations. To obtain novel optical soliton solutions, the nonlinear partial differential equation is transformed into a nonlinear ordinary differential equation via symmetry reduction (nonclassical symmetry). This stochastic nonlinear problem is proposed to be solved by using the modified generalized Riccati equation mapping method. The resulting stochastic solitons show how waves dispersion throughout transmissions via optical fibers. With the use of this method, we can investigate an expansive range of solutions from important physical standpoints, such as dark, bright, singular, and periodic solitons, as well as their noise term effects related to Brownian motion based on the Itô sense. Both 2D and 3D graphs are used to display the impact of the noise term on the solitons. The comes about and calculations show the importance, exactness, and effectiveness of the strategy. The model under consideration is also examined through the concept of modulation instability analysis. Different stable and unstable nonlinear stochastic differential equations that are found in mathematics, physics, and other connected ranges can be solved by utilizing this technique. Furthermore, the bifurcation of phase portraits are studied.
2024-04
Phase trajectories and Chaos theory for dynamical demonstration and explicit propagating wave formation
Chaos Soliton and Fractals (Issue : 2024) (Volume : 182)
This paper is subjected to study the nonlinear integrable model which is the (3+1)-dimensional Boussinesq equation which has a lot of applications in engineering and modern sciences. To find and examine the analytical exact solitary wave solutions of (3+1)-dimensional Boussinesq equation, a modified generalized exponential rational functional method is exerted. As a result, waves, singular periodic, hyperbolic, and trigonometric type solutions are obtained. These acquired solutions are more innovative and encouraging to researchers in their endeavor to study physical marvels. To illustrate how some selected exact solutions propagate, the graphical representation in 2D, Contour, and 3D of those solutions is provided with various parametric values. The considered equation is additionally transformed into the planar dynamical structure by applying the Galilean transformation. All potential phase portraits of the dynamical system are investigated using the theory of bifurcation. The Hamiltonian function of the dynamical system of differential equations is established to see that, the system is conservative over time. The presentation of energy levels through graphics provides valuable insights, and it demonstrates that the model has solutions that can be expressed in closed form. The periodic, quasi-periodic, and chaotic behaviors of the 2D, 3D, and time series are also observable once the dynamical system is subjected to an external force. Meanwhile, the sensitivity of the derived solutions is carefully examined for a range of initial conditions.
2024-03
Bifurcation analysis, chaotic structures and wave propagation for nonlinear system arising in oceanography
Results in Physics (Issue : 2024) (Volume : 57)
This study focuses on the variant Boussinesq equation, which is used to model waves in shallow water and electrical signals in telegraph lines based on tunnel diodes. The aim of this study is to find closed-form wave solutions using the extended direct algebraic method. By employing this method, a range of wave solutions with distinct shapes, including shock, mixed-complex solitary-shock, singular,mixed-singular, mixed trigonometric, periodic, mixed-shock singular, mixed-periodic, and mixed-hyperbolic solutions, are attained. To illustrate the propagation of selected exact solutions, graphical representations in 2D, contour, and 3D are provided with various parametric values. The equation is transformed into a planar dynamical structure through the Galilean transformation. By utilizing bifurcation theory, the potential phase portraits of nonlinear and super-nonlinear traveling wave solutions are investigated. The Hamiltonian function of the dynamical system of differential equations is established, revealing the system’s conservative nature over time. The graphical representation of energy levels offers valuable insights and demonstrates that the model has closed-form solutions.
2024-01

2023

Abundant optical soliton solutions to the Kudryashov equation and its modulation instability analysis
Optical and Quantum Electronics (Issue : 129) (Volume : 56)
This study investigate the Kudryashov equation, which is used to describe the propagation pulses in optical fibers. The aim of this study is to find closed-form wave solutions using the new modified generalized Riccati equation mapping method. By employing the proposed method, a range of new optical solutions with distinct shapes including dark, bright, combined dark-bright, singular periodic solutions are attained. These solutions are integrated with physical parameters and exhibit special waveform characteristics under specific constraints. The modulation instability analysis of the governing model is also presented. To illustrate the propagation of selected exact solutions, graphical representations in 2D, and 3D are provided with various parametric values. Additionally, the impact of the time parameter and the non-linearity degree n are also presented. It can be convincingly stated that the employed method is remarkably efficient and achieves significant success in discovering exact solutions for nonlinear partial differential equations.
2023-12
Optical waves solutions for the perturbed Fokas–Lenells equation through two different methods
Results in Physics (Issue : 2023) (Volume : 53)
This study examines the perturbed Fokas–Lenells equation using two methods: the Bernoulli sub-equation function method and the (1∕𝐺′)-expansion method. A traveling wave transformation is used to convert the governing equation into a nonlinear ordinary differential equation. The perturbed Fokas–Lenells equation arises in optical fibers, and the article constructs various optical soliton solutions, including periodic dark-bright, singular, and periodic singular soliton solutions. The article also presents the necessary constraint conditions for the existence of valid solitons and graphs 2D, 3D, and contour surfaces based on different parameter values.
2023-09
Exploring new optical solutions for nonlinear Hamiltonian amplitude equation via two integration schemes
Physica Scripta (Issue : 2023) (Volume : 98)
This research explores the Jacobi elliptic expansion function method and a modified version of the Sardar sub-equation method to discover new exact solutions for the nonlinear Hamiltonian amplitude equation. By applying these techniques, the study seeks to uncover previously unknown solutions for this equation, contributing to the understanding of its behavior and opening up new possibilities for its applications. The solutions obtained using these methods are represented by hyperbolic, trigonometric, and exponential functions, and they include optical dark-bright, periodic, singular, and bright solutions. The dynamic behaviors of these solutions are demonstrated by selecting appropriate values for physical parameters. By assigning values to these parameters, the study aims to showcase how the solutions of the nonlinear Hamiltonian amplitude equation behave under different conditions. This analysis provides insights into the system’s response and enables a deeper comprehension of its complex dynamics in various scenarios, contributing to the overall understanding of the equation’s behavior and potential real-world implications. Overall, these methods are effective in analyzing and obtaining analytic solutions for nonlinear partial differential equations.
2023-08
New physical structures and patterns to the optical solutions of the nonlinear Schrödinger equation with a higher dimension
Communications in Theoretical Physics (Issue : 2023) (Volume : 75)
It is commonly recognized that, despite current analytical approaches, many physical aspects of nonlinear models remain unknown. It is critical to build more efficient integration methods to design and construct numerous other unknown solutions and physical attributes for the nonlinear models, as well as for the benefit of the largest audience feasible. To achieve this goal, we propose a new extended unified auxiliary equation technique, a brand-new analytical method for solving nonlinear partial differential equations. The proposed method is applied to the nonlinear Schrödinger equation with a higher dimension in the anomalous dispersion. Many interesting solutions have been obtained. Moreover, to shed more light on the features of the obtained solutions, the figures for some obtained solutions are graphed. The propagation characteristics of the generated solutions are shown. The results show that the proper physical quantities and nonlinear wave qualities are connected to the parameter values. It is worth noting that the new method is very effective and efficient, and it may be applied in the realisation of novel solutions.
2023-07
Physical wave propagation and dynamics of the Ivancevic option pricing model
Results in Physics (Issue : 2023) (Volume : 51)
In this study, the Ivancevic option pricing model (IOPM), a nonlinear wave alternative to the traditional Black-Scholes option pricing model, is investigated. The Ivancevic option pricing model is formalized by adaptive nonlinear Schrodinger equations that characterize the option-pricing wave function in terms of stock price and time. This model exhibits the characteristic regulated Brownian behavior observed in financial markets. We use the unified auxiliary equation method, which is one of the most powerful methods to explore analytical solutions of the Ivancevic option pricing model. Via this method, we obtain several new exact solutions such as singular, periodic, bright, dark, hyperbolic, trigonometric, exponential and Jacobi elliptic functions solutions. In addition, we simulate 3D surface 2D graphs and counter plots graphics for some obtained solutions by giving specific values for the involved parameters. To further analyze the obtained solutions, we conduct simulations and generate visual representations in the form of 3D surface plots, 2D graphs, and counter plots. By assigning specific values to the parameters involved in the model, we are able to visualize and examine the dynamics of the solutions in a more tangible and intuitive manner. These graphical representations provide valuable insights into the behavior and patterns exhibited by the solutions, enhancing our understanding of the model’s implications and potential applications Furthermore, we present the option price wave functions of the dependent variable in 2D graphs under the effect of the time variable t.
2023-07
Quantum‑mechanical properties of long‑lived optical pulses in the fourth‑order KdV‑type hierarchy nonlinear model
Optical and Quantum Electronics (Volume : 2023)
This study aims to investigate the (3+1)-dimensional fourth-order KdV-type hierarchy nonlinear euation. For this aim, we use Hirota’s simple method to handle one, two, three, and four soliton solutions. Furthermore, using the ansatz method in the form of the tanhp function, the symbolic computational method is used to construct kink solitary wave solutions, multi-wave solutions, X-type solitons, and Y-type solitons, as well as shock wave solutions. Furthermore, the physical structure and propagation characteristics of the obtained solutions are simulated.
2023-05
Applications of the generalized nonlinear evolution equation with symbolic computation approach
Modern Physics Letters B (Issue : 2350073) (Volume : 2023)
In this work, we will try to find lump solutions, interaction between lump wave and solitary wave solutions, kink-solitary wave solutions and shock wave-type solutions to ð3 þ 1Þ-dimensional generalized nonlinear evolution equation arising in the shallow water waves. The lump solutions, the interaction between lump wave and solitary wave solutions and kink-solitary wave solutions are derived with symbolic computation based on a logarithmic derivative transform which is derived by the help of Hirota's simple method. The shallow water waves in this equation are associated with some natural problems such as tides, storms, atmospheric currents and tsunamis. For the physical presentation of the solutions, we draw 3D and counter graphics by giving suitable values to include the free parameters. We believe that disciplines such as mathematical physics, nonlinear dynamics, °uid mechanics and engineering sciences can benefit from this study.
2023-05
Dynamical rational solutions and their interaction phenomena for an extended nonlinear equation
Communications in Theoretical Physics (Volume : 75)
In this paper, we analyze the extended Bogoyavlenskii-Kadomtsev-Petviashvili (eBKP) equation utilizing the condensed Hirota’s approach. In accordance with a logarithmic derivative transform, we produce solutions for single, double, and triple M-lump waves. Additionally, we investigate the interaction solutions of a single M-lump with a single soliton, a single M-lump with a double soliton, and a double M-lump with a single soliton. Furthermore, we create sophisticated single, double, and triple complex soliton wave solutions. The extended Bogoyavlenskii-Kadomtsev-Petviashvili equation describes nonlinear wave phenomena in fluid mechanics, plasma, and shallow water theory. By selecting appropriate values for the related free parameterswe also create three-dimensional surfaces and associated counter plots to simulate the dynamical characteristics of the solutions offered.
2023-03
Consistent solitons in the plasma and optical fiber for complex Hirota-dynamical model
Results in Physics (Issue : 2023) (Volume : 47)
In this paper, the nonlinear complex Hirota-dynamical model is explored using the unified auxiliary equation method, which is the most powerful method for studying exact solutions to nonlinear models. This model is one of the important standards of the nonlinear Schrödinger equation in which the third derivative term represents the self-interaction in the high-frequency subsystem. Typically, in plasma, this term is isomorphic to the so known self-focusing effect. The aforementioned equation is crucial to plasma physics due to the agreements between the self-interaction at high frequency and the well-known self-focusing effect in plasma. Different structures of solutions are successfully investigated for this model. Consequently, we obtain some solutions like dark–bright, bright, dark, singular, periodic, trigonometric function, Jacobi elliptic function, and exponential solutions. We simulate the 2𝐷, 3𝐷, and counter plots of the constructed solutions by choosing the suitable values of the parameters involved.
2023-03
Modulation instability analysis and optical solutions of an extended (2+1)-dimensional perturbed nonlinear Schrödinger equation
Results in Physics (Issue : 2023) (Volume : 45)
We examine an extended (2+1)-dimensional perturbed nonlinear Schrödinger equation with Kerr law nonlinearity in a nano-optical fiber with fourth-order spatial derivatives in this paper. We develop various optical solutions to the equation under study using the Jacobi elliptic expansion method. We investigate the consequences of nonlinearity and spatial dispersion in spatial directions 𝑥 and 𝑦. Furthermore, by investigating the stability condition of this nonlinear equation, we derive the linear stability analysis. The innovative evolutionary behaviors in the 3-dimensional profile are provided under constraint circumstances to indicate the wave propagation direction.
2023-02
Optical solitons to the Perturbed Gerdjikov‑Ivanov equation with quantic nonlinearity
Optical and Quantum Electronics (Issue : 55) (Volume : 2023)
In this article, we examine the Perturbed Gerdjikov-Ivanov equation by using the Jacobi elliptic function expansion method. This results in obtaining distinct solutions including dark, bright, singular solitons, periodic waves, singular periodic waves, and Jacobi elliptic function solutions. The 2- and 3-dimensional graphs of the reported solutions are presented. The reported results may be useful in explaining the physical features of the studied equation.
2023-01

2022

Dynamic behavior of optical solitons to the Coupled-Higgs equation through an efficient method
International Journal of Modern Physics B (Issue : 16) (Volume : 36)
In this study, through the -expansion method, we extract soliton solutions to the coupled-Higgs equation. The studied nonlinear model is known to describe Higgs mechanism. The Higgs mechanism is essential to explain the generation mechanism of the property “mass” for gauge bosons. The proposed method is one of the most powerful methods for constructing soliton solutions for nonlinear partial differential equations. The obtained wave solutions include exponential, hyperbolic, and distinct structures of complex function solutions. The presented results may be helpful in explaining the physical features of various nonlinear physical phenomena. In order to analyze the dynamic behavior of all obtained solutions, we plot three-dimensional and two-dimensional graphs for the obtained solutions.
2022-12
NEW EXPLICIT WAVE PROFILES OF KUNDU-MUKHERJEE-NASKAR EQUATION THROUGH JACOBI ELLIPTIC FUNCTION EXPANSION METHOD
Romanian Journal in Physics (Issue : 118) (Volume : 74)
In this paper, the Jacobi elliptic function expansion approach is used to explore the Kundu-Mukherjee-Naskar equation, which describes the fiber pulse in optics, rogue waves in the seas, and light beam bending. As a consequence, many structures of solutions based on the parameter module m, such as optical dark-bright, dark, bright, singular, Jacobi elliptic function, and exponential solutions, have been effectively developed. In the solutions produced to analyze wave propagation, specific values for free parameters are presented. We also used computer tools to display 3D surfaces, 2D images, and contour plots of some of the created solutions. To our knowledge, all the obtained solutions verify the partial differential equation.
2022-11
Newly modified unified auxiliary equation method and its applications
Optik (Issue : 169880) (Volume : 296)
This study aims to propose a new modified unified auxiliary equation method for the first time. To illustrate the validity of the new modified method, we consider the dimensionless time-dependent paraxial equation which is investigated previously in Tarla and Yilmazer (2022) using the unified auxiliary equation method. As a result, some different and more general solutions are obtained such as singular, periodic, hyperbolic, dark-bright, trigonometric, exponential, and Jacobi elliptic functions. The obtained results are compared with the solutions obtained by the unified auxiliary equation method in Tarla and Yilmazer (2022) in the result and discussion section. We conclude that our findings are novel and distinctive, and we give the comparison in the section on results and discussion. It is demonstrated that the modified unified auxiliary equation method provides a more efficient mathematical technique for solving nonlinear partial differential equations. In addition, we draw 3D profiles and counter plots for some obtained solutions with the assistance of a computer program by giving a specific value for the involved parameters.
2022-09
Optical solitons to the Fokas system equation in monomode optical fibers
Optical and Quantum Electronics (Issue : 707) (Volume : 54)
In this work, we investigate the Fokas system which expresses the nonlinear pulse propagation in mono-mode optical fibers. We use the (m+1/G')-expansion scheme and the novel modified generalized exponential rational function scheme to generate soliton solutions to the proposed model. As a result, some exact solutions are constructed including hyperbolic, exponential, dark, singular, and periodic singular solutions. In order to analyze the physical behavior of the absolute, and real parts of the obtained solutions we present the 3-D surfaces, 2-D graphs, and their contour graphs.
2022-09
Invariant optical soliton solutions to the Coupled‑Higgs equation
Optical and Quantum Electronics (Issue : 11) (Volume : 54)
In the present study, through the generalized exponential rational function method (GERFM), we construct optical soliton solution to the coupled-Higgs equation (CHE). The proposed method is one of the most powerful method to construct abundant exact solutions for nonlinear partial differential equations. The obtained wave solutions include hyperbolic, trigonometry, and exponential functions solutions. Furthermore, we draw three-dimensional surfaces and counter plots of the obtained solutions using the appropriate value for involved parameters.
2022-09
Closed form wave profiles of the coupled-Higgs equation via the Phi^6-model expansion method
International Journal of Modern Physics B (Volume : 37)
The coupled-Higgs equation, which depicts a system of preserved scalar nucleons cooperating with unbiased scalar mesons, is investigated using the Phi^6-model expansion approach in this paper. We want to look at the various particles that are created when scalar nucleons collide with unbiased scalar mesons. The modulus of the elliptic function m=0 and m=1 is used to obtain the trigonometric and hyperbolic functions. Under the stated limited constraints, all found solutions are used to validate the partial differential equation. We displayed 3D surfaces and 2D visuals of selected solutions produced with the assistance of a computer program to shed more light on the dynamic behavior of the acquired solutions.
2022-09
Closed form wave pro les of the coupled-Higgs equation via the 6-model expansion method
International Journal of Modern Physics B (Issue : 16) (Volume : 36)
The coupled-Higgs equation, which depicts a system of preserved scalar nucleons coop- erating with unbiased scalar mesons, is investigated using the 6-model expansion approach in this paper. We want to look at the various particles that are created when scalar nucleons collide with unbiased scalar mesons. The modulus of the elliptic function m ! 0 and m ! 1 is used to obtain the trigonometric and hyperbolic functions. Under the stated limited constraints, all found solutions are used to validate the partial di er- ential equation. We displayed 3D surfaces and 2D visuals of selected solutions produced with the assistance of a computer program to shed more light on the dynamic behavior of the acquired solutions.
2022-09
Investigation of the dynamical behavior of the Hirota-Maccari system in single-mode fibers
Optical and Quantum Electronics (Issue : 14) (Volume : 54)
In this paper, we use the Jacobi elliptic function expansion approach to investigate the Hirota-Maccari system which describes the dynamical behaviors of the femto-second soliton pulse in single-mode fibers. As a result, solutions with various structures are developed including dark-bright, dark, single, brilliant, exponential function, and Jacobi elliptic function solutions. In addition, a computer application is used to examine the solutions found on three-dimensional surfaces and two-dimensional visuals.
2022-08
New behavior of tsunami and tidal oscillations for Long-and short-wave interaction systems
Modern Physics Letters B (Issue : 23) (Volume : 36)
The Jacobi elliptic function expansion method is one of the most powerful tools for exploring exact solutions of nonlinear partial differential models, which is used in this work to characterize the interaction between one long longitudinal wave and one short transverse wave propagation in a generalized elastic medium. First, we applied a wave transform to the proposed system of equations and obtained an ordinary differential equation. We obtain the values of involved free parameters after performing necessary operations, then substitute the obtained values to the ordinary differential equation by considering the constructed solutions to the ordinary and then to the partial differential equation. As a result, different structures of solutions are constructed such as solitary dark–bright, dark, bright, singular, trigonometric function, Jacobi elliptic function, and hyperbolic function solutions. In order to illustrate the tsunami and tidal oscillations, we draw 3D surfaces and 2D graphics for the obtained solutions by giving a specific value for the involved parameters under the given conditions.
2022-08
The ion sound and Langmuir waves dynamical system via computational modified generalized exponential rational function
Chaos, Solitons & Fractals (Volume : 161)
In this study, we investigate a system of equations based on ion sound and Langmuir waves equations using a generalized exponential rational function, and a new technique named as a modified generalized exponential rational function method. The concept of the new approach is to modify the generalized exponential rational function method. To evaluate the efficiency of the new scheme, we applied the two methods to the given system of equations, and we chose the same values of the parameters in all families, as well as we presented the difference between the solutions in the result and the discussion section. One might easily conclude that the new approach is quite effective and successful in seeking the exact solutions of the nonlinear differential equations. As a result, we have identified a variety of new families of exact travel wave solutions. In addition, we plotted 2D, 3D and contour graphs for some reported solutions by choosing the suitable parameters values.
2022-07
M-lump solutions and interactions phenomena for the (2+ 1)-dimensional KdV equation with constant and time-dependent coefficients
Communications in Theoretical Physics
In this paper, Hirota’s simple method and long-wave method are used to study the (2+1)-dimensional KdV equation including constant and time-dependent coefficients. One-, two-, and three-M-lump solutions are constructed for both cases. Also, the interaction phenomena of M-lump solution with one-soliton, and two-soliton waves are derived for these equations. Moreover, to investigate the physical characteristics of the obtained results, the 3-dimensional figures and contour plots are graphed. All gained solutions verify the proposed equations.
2022-06
On dynamical behavior for optical solitons sustained by the perturbed Chen-Lee-Liu model
Communications in Theoretical Physics (Issue : 7) (Volume : 74)
This study investigates the perturbed Chen-Lee-Liu (CLL) model that represents the propagation of an optical pulse in plasma and optical fiber. The generalized exponential rational function method is used for this purpose. As a result, we obtain some non-trivial solutions such as the optical singular, periodic, hyperbolic, exponential, trigonometric soliton solutions. We aim to express the pulse propagation of the generated solutions, by taking specific values for the free parameters existed in the obtained solutions. The obtained results show that the generalized exponential rational function technique is applicable, simple and effective to get the solutions of non-linear engineering and physical problems. Moreover, the acquired solutions display rich dynamical evolutions that are important in practical applications
2022-06
Propagation of solitons for the Hamiltonian amplitude equation via an analytical technique
Modern Physics Letters B (Volume : 33)
This study investigates the Hamiltonian amplitude equation by using the generalized exponential rational function method. As a result, a variety of soliton solutions have been established with different wave structures such as optical exponential, dark, singular, periodic, periodic-singular, and hyperbolic solutions. 2D and 3D graphics of certain acquired solutions are drawn with the help of computer program. By picking appropriate values for the free parameters in the obtained solutions, the physical phenomena for these obtained solutions are graphically studied and depicted.
2022-06
Nonlinear pulse propagation for novel optical solitons modeled by Fokas system in monomode optical fibers
Results in Physics (Volume : 36)
In this study, the Fokas system which represents the spread of irregular pulse in monomode optical fibers is investigated via the Jacobi elliptic function expansion (JEFE) method. This method is the most powerful technique to explore solutions for a wide range of various models. As a result, different solutions such as dark–bright, singular, bright, Jacobi elliptic function, and exponential solutions are obtained. In addition, 3D and 2D graphics for the obtained solutions are investigated with the assist of a computer program by assigning particular values to the parameters involved.
2022-05
The dynamic behaviors of the Radhakrishnan–Kundu–Lakshmanan equation by Jacobi elliptic function expansion technique
Optical and Quantum Electronics (Issue : 5) (Volume : 54)
In this study, we express the Radhakrishnan–Kundu–Lakshmanan equation with an arbitrary index of. We investigated the solitary wave solutions of the Radhakrishnan–Kundu–Lakshmanan equation by mean of the Jacobi elliptic function expansion technique. As a result, we constructed several distinct solutions include dark, bright, singular, periodic, hyperbolic, trigonometric, and Jacobi elliptic function types solutions. To highlight the dynamic behavior of the generated solutions, specific values for the parameters are also assigned. The above techniques could also be employed to get a variety of exact solutions for other nonlinear models in physics, applied mathematics, and engineering.
2022-05
New Wave Behaviours of the Generalized Kadomtsev-Petviashvili Modified Equal Width-Burgers Equation
Appl. Math (Issue : 2) (Volume : 16)
In this article, we applied two different methods namely as the (1/G′)-expansion method and the Bernoulli sub-equation method to investigate the generalized Kadomtsev-Petviashvili modified equal width-Burgers equation, which is designated the propagation of long-wave with dissipation and dispersion in nonlinear media. To transform the given equation into a nonlinear ordinary differential equation, a traveling wave transformation has been carried out. As a result, we constructed distinct exact solutions like complex solutions, singular solutions, and complex singular solutions. Besides, 2D, 3D, and contour surfaces are illustrated to demonstrate the physical properties of the obtained solutions.
2022-03
Dynamic behavior of the (3+ 1)-dimensional KdV–Calogero–Bogoyavlenskii–Schiff equation
Optical and Quantum Electronics (Issue : 3) (Volume : 54)
In this article, the (3+1)-dimensional KdV–Calogero–Bogoyavlenskii–Schiff equation, which describes the interactions of long wave propagations and has diverse applications in physics, mathematics and engineering, is investigated. Lump and multi-lump wave solutions, one-, and two-soliton solutions, exploding and periodic wave solutions, localized and breather wave solutions, and interaction of lump waves with solitary waves are constructed through the symbolic computational method. In addition, multi-soliton solutions in complex form via using Hirota’s simple method and long-wave method are obtained. The dynamic behaviors of all obtained solutions are analyzed and illustrated in figures by choosing appropriate values of the parameters all of the obtained solutions are verified by a direct substitution in the original equation.
2022-03
New optical solitons based on the perturbed Chen-Lee-Liu model through Jacobi elliptic function method
Optical and Quantum Electronics (Issue : 2) (Volume : 54)
In this study, we investigate the perturbed Chen-Lee-Liu equation that represents the propagation of an optical pulse in plasma and optical fiber. The Jacobi elliptic function technique is used for this purpose. As a result, we obtain some new solitary wave solutions such as the Jacobi elliptic function, dark-bright, trigonometric, exponential, hyperbolic, periodic, and singular soliton solutions. To express the pulse propagation of the generated solutions, specific values for the free parameters under conditions are also given.
2022-02
Solving fractal differential equations via fractal Laplace transforms
Journal of Applied Analysis
The intention of this study is to investigate the fractal version of both one-term and three-term fractal differential equations. The fractal Laplace transform of the local derivative and the non-local fractal Caputo derivative is applied to investigate the given models. The analogues of both the Wright function with its related definitions in fractal calculus and the convolution theorem in fractal calculus are proposed. All results in this paper have been obtained by applying certain tools such as the general Wright and Mittag-Leffler functions of three parameters and the convolution theorem in the sense of the fractal calculus. Moreover, a comparative analysis is conducted by solving the governing equation in the senses of the standard version and fractal calculus. It is obvious that when , we obtain the same results as in the standard version.
2022-01

2021

Extended Calogero-Bogoyavlenskii-Schiff equation and its dynamical behaviors
Physica Scripta (Issue : 125249) (Volume : 96)
In this paper, we consider an extended Calogero-Bogoyavlenskii-Schiff (eCBS) equation. Based on a logarithmic derivative transform and with the aid of symbolic computation, we construct complex multiple solitons for this nonlinear model. Also, by using a symbolic computational method, onelump solution, two-soliton solution, localized and breather wave solution, as well as a periodic wave solution and multiple wave solutions are obtained. The constraint conditions which ensure the validity of the wave structures are also reported. Besides, the graphs of the solution attained are recorded in 3D graphs by fixing parameters to discuss their dynamical properties. The achieved outcomes show that the applied computational strategy is direct, efficient, concise and can be implemented in more complex phenomena with the assistant of symbolic computations.
2021-11
Fractal Kronig-Penney model involving fractal comb potential
Journal of Mathematical Modeling (Issue : 3) (Volume : 9)
In this article, we suggest a fractal Kronig-Penny model which includes a fractal lattice, a fractal potential energy comb, and a fractal Bloch's theorem on thin Cantor sets. We solve the fractal Schrodinger equation for a given potential, using an exact analytical method. We observe that the allowed band energies and forbidden bands in the fractal lattice are bigger than in the standard lattice. These results show the effect of fractal space-time or their fractal geometry on energy levels.
2021-09
Economic Models Involving Time Fractal
Journal of Mathematics and Modeling in Finance (Issue : 1) (Volume : 1)
In this article, the price adjustment equation has been proposed and studied in the frame of fractal calculus which plays an important role in market equilibrium. Fractal time has been recently suggested by researchers in physics due to the self-similar properties and fractional dimension. We investigate the economic models from the viewpoint of local and non-local fractal Caputo derivatives. We derive some novel analytical solutions via the fractal Laplace transform. In fractal calculus, a useful local fractal derivative is a generalized local derivative in the standard computational sense, and the non-local fractal Caputo fractal derivative is a generalization of the non-local fractional Caputo derivative. The economic models involving fractal time provide a new framework that depends on the dimension of fractal time. The suggested fractal models are considered as a generalization of standard models that present new models to economists for fitting the economic data. In addition, we carry out a comparative analysis to understand the advantages of the fractal calculus operator on the basis of the additional fractal dimension of time parameter, denoted by $alpha$, which is related to the local derivative, and we also indicate that when this dimension is equal to $1$, we obtain the same results in the standard fractional calculus as well as when $alpha$ and the nonlocal memory effect parameter, denoted by $gamma$, of the nonlocal fractal derivative are both equal to $1$, we obtain the same results in the standard calculus.
2021-09
Abundant exact solutions to the strain wave equation in micro-structured solids
Modern Physics Letters B (Issue : 26) (Volume : 35)
In this study, the strain wave equation in micro-structured solids which take an important place in solid physics is presented for consideration. The generalized exponential rational function method is used for this purpose which is one of the most powerful methods of constructing abundantly distinct, exact solutions of nonlinear partial differential equations. In micro-structured solids, wave propagation is based on the structure of the strain wave equation. As a consequence, we successfully received many different exact solutions, including non-topological solutions, periodic singular solutions, topological solutions, singular solutions, like periodic lump solutions. Furthermore, in order to better understand their physical interpretation, 2D, 3D, and counter plots are graphed for each of the solutions acquired.
2021-08
Battery discharging model on fractal time sets
International Journal of Nonlinear Sciences and Numerical Simulation
This article is devoted to propose and investigate the fractal battery discharging model, which is one of the well-known models with a memory effect. It is presented as to how non-locality affects the behavior of solutions and how the current state of the system is affected by its past. Firstly, we present a local fractal solution. Then we solve the non-local fractal differential equation and examine the memory effect that includes the Mittag-Leffler function with one parameter. For that aim, the local fractal and non-local fractal Laplace transforms are used to achieve fractional solutions. In addition, the simulation analysis is performed by comparing the underlying fractal derivatives to the classical ones in order to understand the significance of the results. The effects of the fractal parameter and the fractional parameter are discussed in the conclusion section.
2021-06
Electrical circuits involving fractal time
Chaos: An Interdisciplinary Journal of Nonlinear Science (Issue : 3) (Volume : 31)
In this paper, we develop fractal calculus by defining improper fractal integrals and their convergence and divergence conditions with related tests and by providing examples. Using fractal calculus that provides a new mathematical model, we investigate the effect of fractal time on the evolution of the physical system, for example, electrical circuits. In these physical models, we change the dimension of the fractal time; as a result, the order of the fractal derivative changes; therefore, the corresponding solutions also change. We obtain several analytical solutions that are non-differentiable in the sense of ordinary calculus by means of the local fractal Laplace transformation. In addition, we perform a comparative analysis by solving the governing fractal equations in the electrical circuits and using their smooth solutions, and we also show that when α=1, we get the same results as in the standard version. Fractal calculus is a new branch of calculus that includes ordinary calculus. In the future, it will find many applications in various branches of science. Fractal calculus development is in progress.
2021-03

2020

Propagation of dispersive wave solutions for -dimensional nonlinear modified Zakharov–Kuznetsov equation in plasma physics
International Journal of Modern Physics B (Issue : 25) (Volume : 34)
In the current study, we instigate the four-dimensional nonlinear modified Zakharov–Kuznetsov (NLmZK) equation. The NLmZK equation guides the attitude of weakly nonlinear ion-acoustic waves in a plasma comprising cold ions and hot isothermal electrons in the presence of a uniform magnetic field. Two different methods are used, namely the sine-Gordon expansion method (SGEM) and the (1/G') -expansion method to the proposed model. We have successfully constructed some topological, non-topological, and wave solutions. In addition, the 2D, 3D, and contour graphs of the solutions are also plotted under the choice of appropriate values of the parameters.
2020-10
Semi-analytical Method to Solve the Non-linear System of Equations to Model of Evolution for Smoking Habit in Spain
Int. J. Industrial Mathematics (Issue : 4) (Volume : 12)
An epidemiological model of smoking habit is studied by using one of flexible and accurate semi-analytical methods. For this reason, the homotopy analysis transform method (HATM) is applied. Convergence theorem is studied and several h-curves are demonstrated to show the convergence regions. Also, the optimal convergence regions are obtained by demonstrating the residual error functions versus h. The numerical tables are presented to show the precision of method.
2020-10
New wave behaviors and stability analysis of the Gilson–Pickering equation in plasma physics
Indian Journal of Physics (Volume : 94)
This study investigates the Gilson–Pickering equation by using the sine-Gordon expansion method. Sine-Gordon expansion method is one of the most powerful methods for solving the nonlinear partial differential equations. We successfully construct various exact solitary wave solutions to the governing equation, such as shock wave, topological, non-topological, compound topological, and non-topological soliton wave solutions. In addition, the stability of the studied nonlinear equation is analyzed via the linear stability analysis. The 2D, 3D, and contour surfaces are also plotted for all obtained solutions.
2020-07
Modulation instability analysis and analytical solutions to the system of equations for the ion sound and Langmuir waves
Physica Scripta (Issue : 6) (Volume : 95)
The core of this research is to investigate the system of equations for the ion sound and Langmuir waves through the Bernoulli sub-equation method. It is one of the most effective methods for solving nonlinear ordinary differential equations. To transform a given system to a nonlinear ordinary differential equation, a traveling wave transformation has been implemented. Novel wave behaviors of the guiding system are acquired, such as kink-type singular solutions, wave solutions, periodic wave solutions, and periodic singular solutions. Besides, the modulation instability analysis of the given system is investigated based on the standard linear stability analysis and the modulation instability gain spectrum analysis. Furthermore, 2D, 3D and contour graphs of all acquired solutions are also plotted under the selection of appropriate parameter values.
2020-05
Discrete fractional solutions to the k‐hypergeometric differential equation
Mathematical Methods in the Applied Sciences
In this study, the discrete fractional nabla calculus operator is used to investigate the k‐hypergeometric differential equation for both homogeneous and nonhomogeneous states. To solve the guiding equation, we implement certain classical transformations and also constrain the parameters needed to determine them valued. In order to achieve these results, some equipment like the Leibniz rule, the index law, the shift operator, and the power rule are set out in the frame of the discreet fractional calculus. We use all of these tools to the governing equation for homogeneous and nonhomogeneous situations. The major benefit of the fractional nabla operator is that it implements singular differential equations and converts them into fractional order equations. As a result, several new exact fractional solutions of the given equation are constructed.
2020-04
On the new wave behaviors of the Gilson-Pickering equation
Front. Phys. (Volume : 8)
In this article, we study the fully non-linear third-order partial differential equation, namely the Gilson-Pickering equation. The (1/G′)-expansion method, and the generalized exponential rational function method are used to construct various exact solitary wave solutions for a given equation. These methods are based on a homogeneous balance technique that provides an order for the estimation of a polynomial-type solution. In order to convert the governing equation into a nonlinear ordinary differential equation, a traveling wave transformation has been implemented. As a result, we have constructed a variety of solitary wave solutions, such as singular solutions, compound singular solutions, complex solutions, and topological and non-topological solutions. Besides, the 2D, 3D, and contour surfaces are plotted for all obtained solutions by choosing appropriate parameter values.
2020-03
Analytical solutions for the (3+ 1)-dimensional nonlinear extended quantum Zakharov–Kuznetsov equation in plasma physics
Physica A: Statistical Mechanics and its Applications (Volume : 548)
In the current study, we investigate the (3+ 1)-dimensional extended quantum Zakharov–Kuznetsov equation through two different methods namely the sine-Gordon expansion method and the (1∕ G′)-expansion method. The given model represent the nonlinear propagation of the quantum ion-acoustic wave. Traveling wave transformation has been used, in order to convert the governing equation into a non-linear ordinary differential equation. As a result, we successfully construct various exact wave solutions, such as shock, topological, non-topological solutions, periodic wave solutions, and singular solutions. In addition, 2D, 3D and contour surfaces are plotted according to the appropriate parameters values.
2020-03
Exact soliton solutions to the cubic-quartic nonlinear Schrödinger equation with conformable derivative
Front. Phys. (Volume : 8)
The research paper aims to investigate the space-time fractional cubic-quartic nonlinear Schrödinger equation in the appearance of the third, and fourth-order dispersion impacts without both group velocity dispersion, and disturbance with parabolic law media by utilizing the extended sinh-Gordon expansion method. This method is one of the strongest methods to find the exact solutions to the nonlinear partial differential equations. In order to confirm the existing solutions, the constraint conditions are used. We successfully construct various exact solitary wave solutions to the governing equation, for example, singular, and dark-bright solutions. Moreover, the 2D, 3D, and contour surfaces of all obtained solutions are also plotted. The finding solutions have justified the efficiency of the proposed method.
2020-03
Cross-Kink Wave Solutions and Semi-Inverse Variational Method for (3+ 1)-Dimensional Potential-YTSF Equation
East Asian Journal on Applied Mathematics (Issue : 3) (Volume : 10)
Periodic wave solutions of (3 + 1)-dimensional potential-Yu-Toda-Sasa-Fukuyama (YTSF) equation are constructed. Using the bilinear form of this equation, we chose ansatz as a combination of rational, trigonometric and hyperbolic functions. Density graphs of certain solutions in 3D and 2D situations show different cross-kink waveforms and new multi wave and cross-kink wave solutions. Moreover, we employ the semi-inverse variational principle (SIVP) in order to study the solitary, bright and dark soliton wave solutions of the YTSF equation.
2020-02
Discrete fractional solutions to the effective mass Schrödinger equation by mean of nabla operator
AIMS Mathematics (Issue : 2) (Volume : 5)
In the current article, we investigate the second order singular differential equation namely the effective mass Schrödinger equation by means of the fractional nabla operator. We apply some classical transformations in order to reduce the governing equation, and also restrict the difference parameters involved in order to find them values. In order to achieve these important results, certain tools such as the Leibniz rule, the index law, the shift operator, and the power rule are provided in view of the discrete fractional calculus. We use all these mentioned data for two representations of the given model for homogeneous and non-homogeneous instances. The main advantage of the fractional nabla operator is to apply the singular differential equations and transform them into a fractional order model. As a result, we produce some new exact fractional solutions to the present model for a given potential.
2020-01

2019

ON DISCRETE FRACTIONAL SOLUTIONS OF THE HYDROGEN ATOM TYPE EQUATIONS
Thermal Science (Issue : 6) (Volume : 23)
Discrete fractional calculus deals with sums and differences of arbitrary orders. In this study, we acquire new discrete fractional solutions of hydrogen atom type equations by using discrete fractional nabla operator ∇α(0 < α < 1). This operator is applied homogeneous and non-homogeneous hydrogen atom type equations. So, we obtain many particular solutions of these equations.
2019-12
On discrete fractional solutions of the hydrogen atom type equations
Thermal Science (Issue : 6) (Volume : 23)
Discrete fractional calculus deals with sums and differences of arbitrary orders. In this study, we acquire new discrete fractional solutions (dfs) of hydrogen atom type equations (HAEs) by using discrete fractional nabla operator α(0 < α < 1). This operator is applied homogeneous and nonhomogeneous HAEs. So, we obtain many particular solutions of these equations.
2019-11
WEISSENBERG AND WILLIAMSON MHD FLOW OVER A STRETCHING SURFACE WITH THERMAL RADIATION AND CHEMICAL REACTION
JP Journal of Heat and Mass Transfer (Issue : 1) (Volume : 18)
This survey investigates the characteristic flow, heat transfer and chemical reaction of Weissenberg and Williamson over a stretching sheet with thermal radiation and viscous dissipation. The nonlinear differential equation of the thermal boundary layer, which covers the physical problem, is derived and used to transform its equations into a system of higher nonlinear ordinary differential equations with similarity. The resulting non-monthly equation system is numerically solved using the explicit Runge-Kutta technique with the shooting technique. The behavior of the favorite physical parameters with different values in the speed profile, heat transfer, concentration of nanoparticles, skin friction, Nusselt number and Sherwood number has been discussed and graphically analyzed.
2019-07

2017

MHD casson flow over an unsteady stretching sheet
Advances and Applications in Fluid Mechanics (Issue : 4) (Volume : 20)
This research is presented to analyze and investigate the magnetohydrodynamic Casson flow of liquid film over the stretching surface. Continuity and Navier-Stokes equation that covers problem is transformed to the nonlinear ordinary differential equation, next arising governing equations are solved numerically using ADM-Pade in order to investigate variations of interesting parameters on the velocity profile, finally all physical parameters are presented graphically .
2017-10
MHD Casson fluid with heat transfer in a liquid film over unsteady stretching plate
International Journal of Advanced and Applied Sciences (Issue : 1) (Volume : 4)
In this research, we analyze heat transfer of MHD boundary layer flow of Casson fluid. Strong nonlinear ordinary differential equations are solved numerical using Shooting method with fourth order Runge-Kutta (RK4) integration technique. Variations of interesting different parameters on the velocity are showed graphically.
2017-01

2016

Numerical simulation using the homotopy perturbation method for a thin liquid film over an unsteady stretching sheet
International Journal of Pure and Applied Mathematics (Issue : 2) (Volume : 107)
In this article, the flow of liquid film over an unsteady elastic stretching surface is analyzed. Similarity transformations are used to transform the governing equations to a nonlinear ordinary differential equation. The differential equation reformulated to system of Voltera integral equations and solved analytically using the new technique of numerical solution called homotopy perturbation method. The results of the proposed method are compared with previously published work and the results are found to be in an excellent agreement. Also, discussed and presented graphically the effects of various parameters Darcy number and unsteadiness parameter.
2016-04

2013

Fluid Flow and Stability Analysis in Certain thin Liquid Films
M.Sc. Thesis, University of Zakho Duhok, Zakho, Kurdistan Region, Iraq
2013-10-06

2013

Methods of Teaching
University of Duhok, Duhok, Kurdistan Region, Iraq.
2013-12-22,2013-10-06
E-Learning System Design: Teacher-Student Websites
University of zakho
2013-09-16,2013-09-17

2020

Analytical Solutions to the Coupled Boussinesq–Burgers Equations via Sine-Gordon Expansion Method
In the current study, we investigate the coupled Boussinesq-Burgers equations through sine-Gordon expansion method. BBEs arises in the research of fluid flow and describes the spreading of shallow water waves. A traveling wave transformation has been applied to turn the governing equation into a nonlinear ordinary differential equation. As a result, we produce some novel analytical solutions, such as topological, non-topological, and kink-type soliton solutions. Furthermore, 2D, 3D and contour surfaces are also plotted for all obtaining solutions.
2020-01-11