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    Investigation of exact soliton solutions of nematicons in liquid crystals according to nonlinearity conditions
    (World Scientific, 2023) Durur, Hülya; Yokuş, Asıf; Duran, Serbay
    In this work, new traveling wave solutions are generated for the system that models nematicons in liquid crystals using the (1/G?)-expansion method. In the equation system, nonlinearity is taken into account for Kerr law and Power law. Also, the existence of exact solutions under restriction conditions is guaranteed. We suggest that the solutions produced are of a different type than the solutions in the literature. Figures representing the intensity of the produced traveling wave solutions are presented. In addition, the simulation of the solitary wave is made for different values of the parameter that affects the inclination angle of the molecules in the nematicons mechanism in liquid crystals. How classical solitary wave behavior translates into triangular wave behavior is discussed. We believe this paper will provide an important perspective on the problems encountered in various application areas such as fluid dynamics, chemical engineering, chaos and complex networks in terms of investigating different mechanisms by taking into account nonlinearity factors.
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    Modeling of dark solitons for nonlinear longitudinal wave equation in a magneto-electro-elastic circular rod
    (Tech Science Press, 2021) Durur, Hülya; Yokuş, Asıf; Kaya, Doğan; Ahmad, Hijaz
    In this paper, sub equation and ? ? 1=G’ expansion methods are proposed to construct exact solutions of a nonlinear longitudinal wave equation (LWE) in a magneto-electro-elastic circular rod. The proposed methods have been used to construct hyperbolic, rational, dark soliton and trigonometric solutions of the LWE in the magnetoelectro-elastic circular rod. Arbitrary values are given to the parameters in the solutions obtained. 3D, 2D and contour graphs are presented with the help of a computer package program. Solutions attained by symbolic calculations revealed that these methods are effective, reliable and simple mathematical tool for finding solutions of nonlinear evolution equations arising in physics and nonlinear dynamics.
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    Numerical comparison of Caputo and Conformable derivatives of time fractional Burgers-Fisher equation
    (Elsevier, 2021) Yokuş, Asıf; Durur, Hülya; Kaya, Doğan; Ahmad, Hijaz; Nofal, Taher A.
    In this paper, the sub-equation method is used to obtain new types of complex traveling wave solutions of the time-fractional Burgers-Fisher equation. In this work is to compare the exact complex traveling wave solutions of new types and the numerical solutions obtained by suitable transformations of Caputo and Conformable de-rivatives. Also, to discuss the advantages and disadvantages of those derivatives and a new initial condition was created by using the obtained solution and the numerical solutions of the equation were obtained by the finite difference method. A comparison of the numerical solutions with the obtained exact solution is made. L2 and L? norm errors, absolute error values, Von Neumann stability analysis supporting this comparison are investigated. To consolidate the accuracy of the numerical results some tables and graphs are presented. For drawing complex mathematical operations and graphs, computer package programs are used
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    Refraction simulation of internal solitary waves for the fractional Benjamin–Ono equation in fluid dynamics
    (World Scientific, 2021) Duran, Serbay; Yokuş, Asıf; Durur, Hülya; Kaya, Doğan
    In this study, the modified (1/G?)-expansion method and modified sub-equation method have been successfully applied to the fractional Benjamin–Ono equation that models the internal solitary wave event in the ocean or atmosphere. With both analytical methods, dark soliton, singular soliton, mixed dark-singular soliton, trigonometric, rational, hyperbolic, complex hyperbolic, complex type traveling wave solutions have been produced. In these applications, we consider the conformable operator to which the chain rule is applied. Special values were given to the constants in the solution while drawing graphs representing the stationary wave. By making changes of these constants at certain intervals, the refraction dynamics and physical interpretations of the obtained internal solitary waves were included. These physical comments were supported by simulation with 3D, 2D and contour graphics. These two analytical methods used to obtain analytical solutions of the fractional Benjamin–Ono equation have been analyzed in detail by comparing their respective states. By using symbolic calculation, these methods have been shown to be the powerful and reliable mathematical tools for the solution of fractional nonlinear partial differential equations.
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    Traveling wave and general form solutions for the coupled Higgs system
    (John Wiley & Sons Ltd., 2023) Duran, Serbay; Durur, Hülya; Yokuş, Asıf
    In this study, the coupled Higgs system, which is a special case of the coupledHiggs field equation, which is effective in energy transport in the sub-particlesof the atom, is discussed. With the help of the modified generalized exponen-tial rational function method, which is an important instrument in obtainingtraveling wave solutions, both the propagating wave solutions and generalform solutions of coupled Higgs system are presented. These solutions areexamined under some restrictive conditions as they are presented. It is arguedthat these solutions differ from the literature. The advantages and disadvan-tages of the method discussed in the conclusion and discussion section are dis-cussed. In addition, the changes in the behavior of the traveling wave solutionare discussed by giving physical meaning to some constants in the travelingwave solutions produced by the method. The effects on the traveling wavesolution are analyzed for different values of wave number, wave velocity, andwave frequency, which have physically important meanings. In addition, thebehaviors caused by these effects are supported with the help of simulation.