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Öğe An algorithm for numerical solution of some nonlinear multi-dimensional parabolic partial differential equations(ELSEVIER, RADARWEG 29, 1043 NX AMSTERDAM, NETHERLANDS, 2021) Özdemir, Neslihan; Seçer, Aydın; Bayram, MustafaIn 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 Ultraspherical wavelet collocation method is used to get the space derivatives. Ultraspherical wavelets have good properties which make useful to carry out this aim. The presented method is developed for Burgers’ equation, Fisher–Kolmogorov–Petrovsky–Piscounov (Fisher–KPP) equation, and quasilinear parabolic equation. Three illustrative numerical problems are solved to demonstrate the efficiency, simplicity, and reliability of the presented method.Öğe Bright soliton of the perturbed Schrödinger–Hirota equation with cubic–quintic–septic law of self-phase modulation in the presence of spatiotemporal dispersion(SPRINGER HEIDELBERGTIERGARTENSTRASSE 17, D-69121 HEIDELBERG, GERMANY, 2024) Özdemir, Neslihan; Altun, Selvi; Seçer, Aydın; Özışık, Müslüm; Bayram, MustafaFor the first time, we intend to scrutinize both the bright optical soliton solutions of the perturbed Schrödinger–Hirota equation with cubic–quintic–septic law having the spatiotemporal dispersion and the influences of the considered equation parameters on the soliton structure. The simple version of the new extended auxiliary equation method is utilized to carry out the aims. Taking the suitable complex wave transformation, the investigated equation becomes a nonlinear ordinary differential equation. Then, a system consisting of equations in polynomial structure utilizing the technique was able to produce. The bright optical solution is generated by utilizing the presented method. Finally, numerous projections of the bright soliton are indicated to explain the propagation of optical pulses in optic fibers. Furthermore, some depictions describing the effect of the model parameter were added.Öğe Extraction of soliton waves from the longitudinal wave equation with local M‑truncated derivatives(SPRINGERVAN GODEWIJCKSTRAAT 30, 3311 GZ DORDRECHT, NETHERLANDS, 2023) Özdemir, Neslihan; Seçer, Aydın; Bayram, MustafaWe 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 unifed Riccati equation expansion and new Kudryashov methods have been utilized for the frst time. The presented methods have been productively implemented to the considered model. With the help of these two methods, new soliton waves of the M-LWE have been obtained successfully. For a better understanding of the subject and analysis of the results, 3D, contour, and 2D graphs of some soliton solutions have been presented. The interesting part of our work is that both methods, named unifed Riccati equation expansion and the new Kudryashov methods have successfully been applied for the frst time to get new soliton solutions of M-LWE.Öğe The Gegenbauer Wavelets-Based Computational Methods for the Coupled System of Burgers' Equations with Time-Fractional Derivative(MDPI, ST ALBAN-ANLAGE 66, CH-4052 BASEL, SWITZERLAND, 2019) Özdemir, Neslihan; Seçer, Aydın; Bayram, MustafaIn this study, Gegenbauer wavelets are used to present two numerical methods for solving the coupled system of Burgers' equations with a time-fractional derivative. In the presented methods, we combined the operational matrix of fractional integration with the Galerkin method and the collocation method to obtain a numerical solution of the coupled system of Burgers' equations with a time-fractional derivative. The properties of Gegenbauer wavelets were used to transform this system to a system of nonlinear algebraic equations in the unknown expansion coefficients. The Galerkin method and collocation method were used to find these coefficients. The main aim of this study was to indicate that the Gegenbauer wavelets-based methods is suitable and efficient for the coupled system of Burgers' equations with time-fractional derivative. The obtained results show the applicability and efficiency of the presented Gegenbaur wavelets-based methods.Öğe A Hermite Polynomial Approach for Solving the SIR Model of Epidemics(MDPIST ALBAN-ANLAGE 66, CH-4052 BASEL, SWITZERLAND, 2018) Seçer, Aydın; Özdemir, Neslihan; Bayram, MustafaIn this paper, the problem of the spread of a non-fatal disease in a population is solved by using the Hermite collocation method. Mathematical modeling of the problem corresponds to a three-dimensional system of nonlinear ODEs. The presented scheme reduces the problem to a nonlinear algebraic equation system by expanding the approximate solutions by using Hermite polynomials with unknown coefficients. These coefficients of the Hermite polynomials are computed by using the matrix operations of derivatives together with the collocation method. Maple software is used to carry out the computations. In addition, comparison of our method with the Homotopy perturbation method (HPM) and Laplece-Adomian decomposition method (LADM) proves accuracy of solution.Öğe M-truncated soliton solutions of the fractional (4+1)-dimensional Fokas equation(Prof. Dr. Ramazan Yaman, 2023) Özdemir, NeslihanThis article aims to examine M-truncated soliton solutions of the fractional (4 + 1)-dimensional Fokas equation (FE), which is a generalization of the Kadomtsev-Petviashvili (KP) and Davey-Stewartson (DS) equations. The fractional (4 + 1)-dimensional Fokas equation with the M-truncated derivative is also studied first time in this study. The generalized projective Riccati equations method (GPREM) is successfully implemented. In the application of the presented method, a suitable fractional wave transformation is chosen to convert the proposed model into a nonlinear ordinary differential equation. Then, a linear equation system is acquired utilizing the GPREM, the system is solved, and the suitable solution sets are obtained. Dark and singular soliton solutions are successfully derived. Under the selection of appropriate values of the parameters, 2D, 3D, and contour plots are also displayed for some solutions.Öğe On optical soliton solutions of the higher-order Lakshmanan-Porsezian-Daniel model having the cubic-quintic-septic law in the presence of spatio-temporal and chromatic dispersion(Iop Publishing Ltd, 2024) Özdemir, Neslihan; Seçer, Aydın; Ozisik, Müslüm; Bayram, Mustafa; Yüce, SalimThe higher-order Lakshmanan-Porsezian-Daniel equation (LPDE) with the cubic-quintic-septic (CQS) law having spatiotemporal and chromatic dispersion terms (STD-CD) is examined to derive new optical soliton solutions. To accomplish this aim, we operated on a simple version of the new extended auxiliary equation method (SAEM26). The optical soliton solutions of the LPDE with CQS as well as STD-CD are constructed in detail. Moreover, 3D-surface, contour, and 2D plots are presented for the bright and periodic singular soliton solutions. Additionally, the effects of diverse model parameters on the bright soliton structure are surveyed, and these effects are displayed with 2D graphics. The findings established in this work can positively contribute to research in nonlinear optics, while the SAEM26 can be effectively applied to similar nonlinear models.Öğe On the investigation of optical soliton solutions of cubic-quartic Fokas-Lenells and Schrodinger-Hirota equations(ELSEVIER GMBH, HACKERBRUCKE 6, 80335 MUNICH, GERMANY, 2023) Özışık, Müslüm; Önder, İsmail; Esen, Handenur; Çınar, Melih; Özdemir, Neslihan; Seçer, Aydın; Bayram, MustafaPurpose: 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 telecommunications regarding optical soliton transmission in the absence of chromatic dispersion. In this study, in order to examine the optical solitons, the Schrödinger–Hirota equation in the presence of the chromatic dispersion and the cubic–quartic Fokas–Lenells equation discarding the chromatic dispersion were investigated. For this intent, by obtaining certain soliton types using the unified Riccati equation expansion method (UREEM), optical soliton solutions were obtained for both models and graphical representations and comments were made. Methodology: By developing appropriate computer algorithms and applying UREEM in the following ways, symbolic calculation software was made and analytical optical soliton solutions were obtained. Findings: Through computer algebra software, we plotted the obtained results via 3D, 2D views and we also illustrated the investigation of wave behavior caused by parameter change on 2D graphics. Originality: Different soliton behavior under the parameters effect of the Schrödinger–Hirota equation having chromatic dispersion and the cubic–quartic Fokas-Lenells equation is investigated and the obtained results are reported.Öğe Optical solitons for Radhakrishnan–Kundu–Lakshmanan equation in the presence of perturbation term and having Kerr law(ELSEVIER GMBH, HACKERBRUCKE 6, 80335 MUNICH, GERMANY, 2022) Özdemir, NeslihanPurpose: In this research study, obtaining the analytical and soliton solutions to the perturbed Radhakrishnan–Kundu–Lakshmanan (RKL) equation with Kerr law nonlinearity is aimed via the generalized projective Riccati equations method (GPREM), a simple version of the new extended auxiliary equation method (SAEM26), and unified Riccati equation expansion method (UREEM). At the same time, the roles of some parameters included in the perturbed RKL equation on soliton dynamics are analyzed. Methodology: The presented methods are successfully employed to the perturbed RKL equation. In the application of the presented methods, to convert the perturbed RKL equation into a nonlinear ordinary differential equation, we choose suitable complex wave transformation for the proposed model. Later, a linear equation system is derived using the GPREM, SAEM26, and UREEM, the system is solved, the appropriate solution sets are obtained, and the soliton solutions are achieved, respectively. Findings: The singular, bright and dark soliton solutions are generated by choosing the suitable set and parameter values. To comprehend the physical dynamics of some solutions, 3D, contour, and 2D graphs are demonstrated. In addition, 2D graphs are drawn to show how some parameters in the main equation have an effect on soliton behaviors. The examination indicates that the model parameters have a substantial effect on the soliton dynamics. Depending on the soliton forms, the effect can be varied. The results presented in this paper will be useful for future works in soliton theory and the presented methods can be effectively implemented to such equations. Originality: The effects of the model parameters included in the perturbed RKL equation on soliton dynamics are analyzed for the first time in this study.Öğe Perturbation of dispersive optical solitons with Schrödinger–Hirota equation with Kerr law and spatio-temporal dispersion(ELSEVIER GMBH, HACKERBRUCKE 6, 80335 MUNICH, GERMANY, 2022) Özdemir, Neslihan; Seçer, Aydın; Özışık, Müslüm; Bayram, MustafaObjective: 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 the first time. Our aim is not only restricted to obtaining different and more soliton solutions by proposed method for the first time in this study, but also includes examining the effect of the coefficients of self-steepening and nonlinear dispersion terms to the soliton propagation in the investigated problem. Methodology: Utilizing a traveling wave transformation, the perturbed Schrödinger–Hirota equation with the effect of spatio-temporal dispersion and Kerr law nonlinearity can be transformed into an nonlinear ordinary differential equation (NODE). Then, the NODE is convert into a set of algebraic equations by taking account into the Riccati differential equation. Solving the set of algebraic equations, we acquire the analytical soliton solutions of the perturbed Schrödinger–Hirota equation with the effect of spatio-temporal dispersion and Kerr law nonlinearity. In the proposed method, the modified extended tanh function method is enhanced by presenting more solutions of Riccati differential equations with the direct algebraic form, is utilized. Results: The more solutions have been established to the literature with new significant physical properties of the perturbed Schrödinger–Hirota equation with the effect of spatiotemporal dispersion and Kerr law nonlinearity. We indicated that the presented method are effective, easily computable, and reliable in solving such nonlinear problems. Moreover, we demonstrate the dynamical behaviors and physical significance of some soliton solutions at appropriate values of parameters. Originality: A variety of soliton solutions to the perturbed Schrödinger–Hirota equation with Kerr law non-linearity by the direct algebraic form of enhanced modified extended tanh expansion method have been acquired. These solutions are dark–bright, trigonometric, hyperbolic, periodic, and singular soliton solutions. 3D, contour and 2D plots of some obtained solutions have been demonstrated to interpret the physical meaning of the equation. For some parameter values in the equation, the behavior of soliton solutions has been examined. The constraint conditions are established to confirm the existence of valid solutions. The obtained results can be effective in interpreting the physical meaning of this nonlinear system. We have seen that the proposed direct algebraic form of the enhanced modified extended tanh expansion method is a powerful mathematical technique which can be utilized to acquire the analytical solutions to different complex nonlinear mathematical models.Öğe Revealing optical soliton solutions of Schrödinger equation having parabolic law and anti-cubic law with weakly nonlocal nonlinearity(TAYLOR & FRANCIS LTD2-4 PARK SQUARE, MILTON PARK, ABINGDON OR14 4RN, OXON, ENGLAND, 2024) Özdemir, Neslihan; Altun, Selvi; Seçer, Aydın; Özışık, Müslüm; Bayram, MustafaIn this study, we purpose to ensure optical soliton solutions of the nonlinear Schrödinger equation having parabolic and anti-cubic (AC) laws with a weakly non-local nonlinearity by using the new Kudryashov method. As far as we know this model has not been presented and studied before. Furthermore, what differs this study from other studies is, not only obtains a variety of analytical solutions of the examined model but also substantiates the effects of the parabolic and anti-cubic laws with a weakly non-local nonlinearity on soliton behaviour, by choosing the particular soliton forms, which are dark, bright and W-like. Eventually, we depict some of the derived solutions in contour, 2D and 3D diagrams selecting the appropriate values of parameters by means of Matlab to demonstrate the importance of the given model. It is indicated that parabolic and AC parameters taking into consideration the weak non-local contribution have a very remarkable impact on the soliton structure, and the impact alters connected with the parameters and the soliton form. Besides, enabling and retaining the critical balance between the parameters and the soliton form and the interactive relation of the parameters with each other comprises major challenges.Öğe Traveling wave structures of some fourth-order nonlinear partial differential equations(ELSEVIER, RADARWEG 29, 1043 NX AMSTERDAM, NETHERLANDS, 2023) Esen, Handenur; Özdemir, Neslihan; Seçer, Aydın; Bayram, MustafaThis 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 algebraic equations. Solving the set of algebraic equations, we acquire the novel exact solutions of the integrable fourth-order equations presented in this research paper. The physical interpretation of the nonlinear models are also detailed through the exact solutions, which demonstrate the effectiveness of the presented method.The Bäcklund transformation can produce an infinite sequence of solutions of the given two fourth-order nonlinear partial differential equations. Finally, 3D graphs of some derived solutions in this paper are depicted through suitable parameter values.Öğe Two Analytical Schemes for the Optical Soliton Solution of the (2 + 1) Hirota–Maccari System Observed in Single-Mode Fibers(MDPI, ST ALBAN-ANLAGE 66, CH-4052 BASEL, SWITZERLAND, 2022) Özdemir, Neslihan; Seçer, Aydın; Özışık, Müslüm; Bayram, MustafaIn 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 the motion of isolated waves localized in a small part of space, and is used in such various fields as fiber optics telecommunication systems, nonlinear optics, plasma physics, and hydrodynamics. In addition, the Hirota–Maccari system defines the dynamical characters of femtosecond soliton pulse propagation in single-mode fibers. Analytical solutions of the model are successfully acquired with the assistance of symbolic computation utilizing these methods. Finally, 3D, 2D, and contour graphs of solutions are depicted at specific values of parameters. It is shown that the new Kudryashov method and the tanh-coth method are uncomplicated, very effective, easily applicable, reliable, and indeed vital mathematical tools in solving nonlinear models.Öğe Wavelet-based Numerical Approaches for Solving the Korteweg-de Vries (KdV) Equation(Matematikçiler Derneği, 2022) Özdemir, Neslihan; Seçer, AydınIn this research work, we examine the Korteweg-de Vries equation (KdV), which is utilized to formulate the propagation of water waves and occurs in different fields such as hydrodynamics waves in cold plasma acoustic waves in harmonic crystals. This research presents two efficient computational methods based on Legendre wavelets to solve the Korteweg-de Vries. The three-step Taylor method is first applied to the Korteweg-de Vries equation for time discretization. Then, the Galerkin and collocation methods are used for spatial discretization. With these approaches, bringing the approximate solutions of the Korteweg-de Vries equation turns into getting the solution of the algebraic equation system. The solution of this system gives the Legendre wavelet coefficients. The approximate solution can be obtained by substituting the obtained coefficients into the Legendre wavelet series expansion. The presented wavelet methods are tested by studying different problems at the end of this study.
