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Solution approximations for a mathematical model of relativistic electrons with beta derivative

Applied Mathematics(A Journal of Chinese Universities) 2024年第3期39卷 469-485页

作者： Ibrahim Yalcinkaya orkun tasbozan Ali Kurt Hijaz Ahmad Department of Mathematics and Computer Sciences Faculty of ScienceNecmettin Erbakan UniversityKonyaTurkey Department of Mathematics Faculty of Science and LiteratureHatay Mustafa Kemal UniversityHatayTurkey Department of Mathematics Faculty of SciencePamukkale UniversityDenizliTurkey Near East University Operational Research Center in HealthcareTRNC Mersin 10Nicosia99138Turkey Department of Mathematics Faculty of ScienceIslamic University of MadinahMadinahSaudi Arabia

The main aim of this paper is to obtain the exact and semi-analytical solutions of the nonlinear Klein-Fock-Gordon(KFG)equation which is a model of relativistic electrons arising in the laser thermonuclear fusion with beta *** this purpose,both the modified extended tanh-function(mETF)method and the homotopy analysis method(HAM)are *** applying the mETF the chain rule for beta derivative and complex wave transform are used for obtaining the exact *** advantage of this procedure is that discretization or normalization is not *** applying the mETF,the exact solutions are ***,by applying the HAM semi-analytical results for the considered equation are *** HAM?curve gives us a chance to find the suitable value of the for the convergence of the solution ***,comparative graphical representations are given to show the effectiveness,reliability of the *** results show that the m ETF and HAM are reliable and applicable tools for obtaining the solutions of non-linear fractional partial differential equations that involve beta *** study can bring a new perspective for studies on fractional differential *** the other hand,it can be said that scientists can apply the considered methods for different mathematical models arising in physics,chemistry,engineering,social sciences and *** involves fractional *** the results may cause a new insight who studies on relativistic electron modelling.

关键词： Klein-Fock-Gordon(KFG)equation beta derivative semi-analytical solution exact solution

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Applied Mathematics(A Journal of Chinese Universities) 2021年第2期36卷 218-228页

作者： orkun tasbozan Department of Mathematics Hatay Mustafa Kemal University Hatay31000Turkey

The main objective of this article is to obtain the new analytical and numerical solutions of fractional Fitzhugh-Nagumo equation which arises as a model of reaction-ifusion systems,transmission of nerve impulses,circuit theory,biology and the area of population *** this aim conformable derivative with fractional order which is a well behaved,understandable and applicable definition is used as a *** analytical solutions were got by utilizing the fact that the conformable fractional derivative provided the chain *** the help of this feature which is not provided by other popular fractional derivatives,nonlinear fractional partial differential equation is turned into an integer order differential *** numerical solutions which is obtained with the aid of residual power series method are compared with the analytical results that obtained by performing sub equation *** comparison is made both with the help of three-dimensional graphical representations and tables for different values of theγ.

关键词： Fitzhugh-Nagumo equation reaction-diffusion model conformable fractional derivative sub equation method

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New Analytical Solutions for Time Fractional Benjamin–Ono Equation Arising Internal Waves in Deep Water

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China Ocean Engineering 2019年第5期33卷 593-600页

作者： orkun tasbozan Department of Mathematics Faculty of Science and Art Mustafa Kemal University

In this article, the author sets up the abundant traveling wave solutions for time fractional Benjamin–Ono equation which was introduced to describe internal waves in stratified fluids by using Jacobi elliptic function expansion method. The traveling wave solutions are expressed in terms of the hyperbolic functions, the trigonometric functions and the rational functions. It can be seen that the obtained results are found to be important for the statement of some physical demonstrations of problems in mathematical physics and ocean engineering. In ocean engineering Benjamin–Ono equations are generally used in computer simulation for the water waves in deep water and open seas.

关键词： Jacobi elliptic function expansion method Benjamin-Ono equation conformable fractional derivative

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Applying the New Extended Direct Algebraic Method to Solve the Equation of Obliquely Interacting Waves in Shallow Waters

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Journal of Ocean University of China 2020年第4期19卷 772-780页

作者： KURT Ali TOZAR Ali tasbozan orkun Department of Mathematics Faculty of Science and ArtPamukkale UniversityDenizli 20160Türkiye Department of Physic Faculty of Science and ArtMustafa Kemal UniversityHatay 31001Türkiye Department of Mathematics Faculty of Science and ArtMustafa Kemal UniversityHatay 31001Türkiye

In this study,the potential Kadomtsev-Petviashvili(pKP)equation,which describes the oblique interaction of surface waves in shallow waters,is solved by the new extended direct algebraic *** results of the study show that by applying the new direct algebraic method to the pKP equation,the behavior of the obliquely interacting surface waves in two dimensions can be *** article fairly clarifies the behaviors of surface waves in shallow *** the literature,several mathematical models have been developed in attempt to study these behaviors,with nonlinear mathematics being one of the most important steps;however,the investigations are still at a level that can be called‘baby steps’.Therefore,every study to be carried out in this context is of great ***,this study will serve as a reference to guide scientists working in this field.

关键词： conformable fractional derivative new extended direct algebraic method interacting wave equation shallow water waves

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New Wave Solutions of Time-Fractional Coupled Boussinesq–Whitham–Broer–Kaup Equation as A Model of Water Waves

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China Ocean Engineering 2019年第4期33卷 477-483页

作者： Emrah ATILGAN Mehmet SENOL Ali KURT orkun tasbozan Department of Management Informatics Systems Mustafa Kemal University Hatay Turkey Department of Mathematics Nevsehir Haci Bektas Veli University Nevsehir Turkey Department of Mathematics Pamukkale University Denizli Turkey Department of Mathematics Mustafa Kemal University Hatay Turkey

The main purpose of this paper is to obtain the wave solutions of conformable time fractional Boussinesq–Whitham–Broer–Kaup equation arising as a model of shallow water waves. For this aim, the authors employed auxiliary equation method which is based on a nonlinear ordinary differential equation. By using conformable wave transform and chain rule, a nonlinear fractional partial differential equation is converted to a nonlinear ordinary differential equation. This is a significant impact because neither Caputo definition nor Riemann–Liouville definition satisfies the chain rule. While the exact solutions of the fractional partial derivatives cannot be obtained due to the existing drawbacks of Caputo or Riemann–Liouville definitions, the reliable solutions can be achieved for the equations defined by conformable fractional derivatives.

关键词： time fractional coupled Boussinesq–Whitham–Broer–Kaup equation conformable fractional derivative auxiliary equation method

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Double Exp-Function Method for Multisoliton Solutions of The Tzitzeica-Dodd-Bullough Equation

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Acta Mathematicae Applicatae Sinica 2016年第2期32卷 461-468页

作者： Alaattin ESEN N.Murat YAGMURLU orkun tasbozan Department of Mathematics Faculty of Science and Art Inn University

In this work, it is aimed to find one- and two-soliton solutions to nonlinear Tzitzeica-Dodd-Bullough （TDB） equation. Since the double exp-function method has been widely used to solve several nonlinear evolution equations in mathematical physics, we have also used it with the help of symbolic computation for solving the present equation. The method seems to be easier and more accurate thanks to the recent developments in the field of symbolic computation.

关键词： double exp-function method one-soliton two-soliton Tzitzeica-Dodd-Bullough equation solitary waves

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Multiple-solitons for generalized(2+1)-dimensional conformable Korteweg-de Vries-Kadomtsev-Petviashvili equation

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Journal of Ocean Engineering and Science 2022年第6期7卷 536-542页

作者： Lanre Akinyemi Mehmet Senol orkun tasbozan Ali Kurt Department of Mathematics Lafayette CollegeEastonPennsylvaniaUSA Nevsehir HacıBektas Veli University Department of MathematicsNevsehirTurkey Mustafa Kemal University Department of MathematicsHatayTurkey Pamukkale University Department of MathematicsDenizliTurkey

This paper studied new class of integral equation called the Korteweg-de Vries-Kadomtsev-Petviashvili(KdV-KP)*** equation consist of the well-known fifth-order KdV equation in the context of the Kadomtsev-Petviashvili *** newly gathered class of sixth-order KdV-KP equation is studied using the sub-equation method to obtain several soliton-type solutions which consist of trigonometric,hyperbolic,and rational *** application of the sub-equation approach in this work draws attention to the outstanding characteristics of the suggested method and its ability to handle completely integrable ***,the obtained solutions have not been reported in the previous literature and might have significant impact on future research.

关键词： Conformable derivative Sub-equation method KdV-KP equations Multiple-soliton solutions

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Soliton solutions for time fractional ocean engineering models with Beta derivative

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Journal of Ocean Engineering and Science 2022年第5期7卷 444-448页

作者： Ibrahim Yalçınkaya Hijaz Ahmad orkun tasbozan Ali Kurt Department of Mathematics and Computer Sciences Faculty of ScienceNecmettin Erbakan UniversityKonyaTurkey Department of Basic Sciences University of Engineering and TechnologyPeshawarPakistan Department of Mathematics Faculty of Science and LiteratureHatay Mustafa Kemal UniversityHatayTurkey Department of Mathematics Faculty of Science and LiteraturePamukkale UniversityDenizliTurkey

In this study,the authors obtained the soliton and periodic wave solutions for time fractional symmetric regularized long wave equation(SRLW)and Ostrovsky equation(OE)both arising as a model in ocean *** this aim modified extended tanh-function(mETF)is *** using this method,chain rule is employed to turn fractional nonlinear partial differential equation into the nonlinear ordinary differential equation in integer *** to the chain rule,there is no further requirement for any normalization or *** derivative which involves fractional term is used in considered mathematical *** the exact solutions of these equations is very important for knowing the wave behavior in ocean engineering models.

关键词： Symmetric regularized long wave equation Beta derivative Ostrovsky equation Analytical solution Soliton solutions Periodic wave solution

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Two effective approaches for solving fractional generalized Hirota-Satsuma coupled KdV system arising in interaction of long waves

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Journal of Ocean Engineering and Science 2019年第1期4卷 24-32页

作者： Ali Kurt Hadi Rezazadeh Mehmet Senol Ahmad Neirameh orkun tasbozan Mostafa Eslami Mohammad Mirzazadeh Department of Mathematics Mustafa Kemal UniversityHatayTurkey Faculty of Engineering Technology Amol University of Special Modern TechnologiesAmolIran Department of Mathematics Nevsehir Haci Bektas Veli UniversityNevsehirTurkey Department of Mathematics Faculty of SciencesGonbad Kavous UniversityGonbadIran Department of Mathematics Faculty of Mathematical SciencesUniversity of MazandaranBabolsarIran Department of Engineering Sciences Faculty of Technology and EngineeringEast of GuilanUniversity of GuilanRudsar-VajargahIran

In this article,two different methods,namely sub-equation method and residual power series method,have been used to obtain new exact and approximate solutions of the generalized Hirota-Satsuma system of equations,which is a coupled KdV *** fractional derivative is taken in the conformable *** of the obtained exact solutions were checked by substituting them into the corresponding system with the help of Maple symbolic computation *** results indicate that both methods are easy to implement,effective and *** are therefore ready to apply for various partial fractional differential equations.

关键词： Hirota-Satsuma coupled KdV system Sub-equation method Power series method Conformable fractional derivative

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Approximate Analytical Solutions of Fractional Coupled mKdV Equation by Homotopy Analysis Method

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Open Journal of Applied Sciences 2012年第3期2卷 193-197页

作者： orkun tasbozan Alaattin Esen Nuri Murat Yagmurlu Department of Mathematics Inonu UniversityMalatyaTurkey

In this paper, the approximate analytical solutions of the fractional coupled mKdV equation are obtained by homotopy analysis method (HAM). The method includes an auxiliary parameter which provides a convenient way of adjusting and controlling the convergence region of the series solution. The suitable value of auxiliary parameter is determined and the obtained results are presented graphically.

关键词： Homotopy Analysis Method Approximate Analytical Solution Fractional Coupled mKdV Equation

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