Alaattin Esen - Academia.edu (original) (raw)

Papers by Alaattin Esen

Mathematical Sciences and Applications E-Notes

The aim of this study is to obtain numerical solutions of the modified Fornberg Whitham equation ... more The aim of this study is to obtain numerical solutions of the modified Fornberg Whitham equation via collocation finite element method combined with operator splitting method. The splitting method is used to convert the original equation into two sub equations including linear and nonlinear part of the equation as a slight modification of splitting idea. After splitting progress, collocation method is used to reduce the sub equations into algebraic equation systems. For this purpose, quintic B-spline base functions are used as a polynomial approximation for the solution. The effectiveness and efficiency of the method and accuracy of the results are measured with the error norms L 2 and L ∞. The presentations of the numerical results are shown by graphics as well.

In this work, our aim is to obtain a numerical solution to some fractional differential equations... more In this work, our aim is to obtain a numerical solution to some fractional differential equations. In the solution proces s, we have used fractional derivatives in Caputo sense. The fundamental characteristics of the present method is the fact t hat it converts complex problems into those requiring the solution of algebraic ones, which is obviously more easy for computational processing. The obtained approximate values show the accuracy and suitability of the present scheme for applying a wide range of fractional partial differential equations. Finally, the error norms L2 and L∞ are computed and found to be sufficiently small.

The present article focuses on obtaining numerical solutions of time fractional KdV-Burger-Kuramo... more The present article focuses on obtaining numerical solutions of time fractional KdV-Burger-Kuramoto equation (KBK) with the finite element collocation method. The finite element collocation methods are common and effective tool for solving nonlinear problems because of their reasonable computational costs. The idea underlying the method is seeking the numerical solutions in a form of a linear combination of unknown functions with basis at nodal points by avoid of integration. Thus, in this article, we achieve more accurate numerical results are obtained with the application of the method to KBK equation. Additionally, we show the efficiency and effectiveness of the method using comparisons of numerical results with exact solutions via error norms and their simulations.

Mediterranean Journal of Mathematics, 2020

In this manuscript, a Strang splitting approach combined with Chebyshev wavelets has been used to... more In this manuscript, a Strang splitting approach combined with Chebyshev wavelets has been used to obtain the numerical solutions of regularized long-wave (RLW) equation with various initial and boundary conditions. The performance of the proposed method measured with three different test problems. To measure the accuracy of the method, L2 and L∞ error norms and the I1, I2, I3 invariants are computed. The results of the computations are compared with the existing numerical and exact solutions in the literature.

Wave Motion, 2021

Abstract In this paper we established a numerical method for Equal Width (EW) Equation using Poly... more Abstract In this paper we established a numerical method for Equal Width (EW) Equation using Polynomial Scaling Functions. The EW equation is a simpler alternative to well known Korteweg de Vries (KdV) and regularized long wave (RLW) equations which have many applications in nonlinear wave phenomena. According to Polynomial scaling method, algebraic polynomials are used to get the orthogonality between the wavelets and corresponding scaling functions with respect to the Chebyshev weight. First we introduce polynomial scaling functions, how are the functions are approximated according to these and Operational matrix of derivatives are given. For time discretization of the function we use finite difference method with Rubin Graves linearization and polynomial scaling functions are used for the space discretization. The method is applied to four different problem and the obtained results are compared with the results in the literature and with the exact results to give the efficiency of the method.

Proceedings (International Youth Science Forum “Litteris et Artibus”).2018.№1, 2018

In this paper, new singular soliton solution is found to the hyperbolic generalization of the Bur... more In this paper, new singular soliton solution is found to the hyperbolic generalization of the Burgers equation. 2D and 3D graphs are also presented. At the end of paper, a conclusion is introduced as well by mentioning novel aspects of paper.

ITM Web of Conferences, 2018

In this study, a Sine-Gordon expansion method for obtaining novel exact solutions of extended sha... more In this study, a Sine-Gordon expansion method for obtaining novel exact solutions of extended shallow water wave equation and Fokas equation is presented. All of the equations which are under consideration consist of three or four variable. In this method, first of all, partial differential equations are reduced to ordinary differential equations by the help of variable change called as travelling wave transformation, then Sine Gordon expansion method allows us to obtain new exact solutions defined as in terms of hyperbolic trig functions of considered equations. The newly obtained results showed that the method is successful and applicable and can be extended to a wide class of nonlinear partial differential equations.

ITM Web of Conferences, 2018

This manuscript is going to seek travelling wave solutions of some coupled partial differential e... more This manuscript is going to seek travelling wave solutions of some coupled partial differential equations with an expansion method known as Sine-Gordon expansion method. Primarily, we are going to employ a wave transformation to partial differential equation to reduce the equations into ordinary differential equations. Then, the solution form of the handled equations is going to be constructed as polynomial of hyperbolic trig or trig functions. Finally, with the aid of symbolic computation, new exact solutions of the partial differentials equations will have been found.

International Journal of Modern Physics C, 2018

In the present paper, a novel perspective fundamentally focused on the differential quadrature me... more In the present paper, a novel perspective fundamentally focused on the differential quadrature method using modified cubic B-spline basis functions are going to be applied for obtaining the numerical solutions of the complex modified Korteweg–de Vries (cmKdV) equation. In order to test the effectiveness and efficiency of the present approach, three test problems, that is single solitary wave, interaction of two solitary waves and interaction of three solitary waves will be handled. Furthermore, the maximum error norm [Formula: see text] will be calculated for single solitary wave solutions to measure the efficiency and the accuracy of the present approach. Meanwhile, the three lowest conservation quantities will be calculated and also used to test the efficiency of the method. In addition to these test tools, relative changes of the invariants will be calculated and presented. In the end of these processes, those newly obtained numerical results will be compared with those of some o...

Discrete & Continuous Dynamical Systems - S, 2019

In this paper, we developed a unified method to solve time fractional Burgers' equation using the... more In this paper, we developed a unified method to solve time fractional Burgers' equation using the Chebyshev wavelet and L1 discretization formula. First we give the preliminary information about Chebyshev wavelet method, then we describe time discretization of the problems under consideration and then we apply Chebyshev wavelets for space discretization. The performance of the method is shown by three test problems and obtained results compared with other results available in literature.

Optik, 2018

This study reveals the dark, bright, mixed dark-bright, singular and mixed singular optical solit... more This study reveals the dark, bright, mixed dark-bright, singular and mixed singular optical solitons to the (1+1)-dimensional coupled nonlinear nonlinear Schrödinger equation by using the extended sinh-Gordon equation expansion method. The constraint conditions for the existence of valid solitons are given. Under the choice of suitable values of the parameters and the fractional values of α and β, the 2-and 3-dimensional graphs to some of the reported solutions are plotted.

Bulletin of Mathematical Sciences and Applications, 2017

In this study, the authors employed the quadratic B-spline Galerkin method to solve time fraction... more In this study, the authors employed the quadratic B-spline Galerkin method to solve time fractional order telegraph equations. Three model problems are consideredto implement the method.L2,L∞error norms and numerical results have been presented in tables. Absolute error graphics for all the exact and numerical solutionshave been given

New Trends in Mathematical Science, 2016

The initial version of a Stefan problem is the melting of a semi-infinite sheet of ice. This prob... more The initial version of a Stefan problem is the melting of a semi-infinite sheet of ice. This problem is described by a parabolic partial differential equation along with two boundary conditions on the moving boundary which are used to determine the boundary itself and complete the solution of the differential equation. In this paper firstly, we use variable space grid method, boundary immobilisation method and isotherm migration method to get rid of the trouble of the Stefan problem. Then, collocation finite element method based on cubic B-spline bases functions is applied to model problem. The numerical schemes of finite element methods provide a good numerical approximation for the model problem. The numerical results show that the present results are in good agreement with the exact ones.

Acta Mathematicae Applicatae Sinica, English Series, 2016

In this work, it is aimed to find one- and two-soliton solutions to nonlinear Tzitzeica-Dodd-Bull... more 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.

Sohag Journal of Mathematics, 2016

In the present study, a B-spline collocation method has been applied to obtain a numerical soluti... more In the present study, a B-spline collocation method has been applied to obtain a numerical solution of the sine-Gordon equation. Then, the obtained numerical results have been compared with those given in the literature. The error norms L 2 and L ∞ are computed and they have been found out small enough to be accepted.

Applied Mathematics and Computation, 2015

In the present article, a quadratic B-spline finite element Galerkin method has been used to obta... more In the present article, a quadratic B-spline finite element Galerkin method has been used to obtain numerical solutions of the nonlinear time fractional gas dynamics equation. While the Caputo form is used for the time fractional derivative appearing in the equation, the L1 discretization formula is applied to the equation in time. A numerical example is given and the obtained results show the accuracy and efficiency of the method. Therefore, the present method can be used as an efficient alternative one to find out the numerical solutions of other both linear and nonlinear fractional differential equations available in the literature.

In this paper, we construct exact travelling wave solutions for the generalized (2+1)-dimensional... more In this paper, we construct exact travelling wave solutions for the generalized (2+1)-dimensional ZK-MEW equation by using the solutions of an auxiliary ordinary differential equation given by Sirendaoreji [1]. It is shown that some solutions obtained in this study are new solutions which have not been reported yet.

In this paper, the Homotopy Analysis Method (HAM) is applied to the damped Burgers and Boussinesq... more In this paper, the Homotopy Analysis Method (HAM) is applied to the damped Burgers and Boussinesq-Burgers equations to obtain their approximate analytical solutions. The HAM solution includes an auxiliary parameterh which provides a convenient way to adjust and control the convergence region of the solution series. An appropriate choice of the auxiliary parameter in the model problems for increasing time is investigated.

International Journal of Nonlinear Sciences and Numerical Simulation, 2009

In this paper, the Exp-function method is used to obtain generalized travelling wave solutions wi... more In this paper, the Exp-function method is used to obtain generalized travelling wave solutions with free parameters of the MKdV-sine-Gordon and Boussinesq-double sine-Gordon equations. It is shown that the Exp-function method, with the help of any symbolic computation packages, provides an effective mathematical tool for nonlinear evolution equations arising in mathematical physics.

International Journal of Nonlinear Sciences and Numerical Simulation, 2009

This paper applies He's Exp-function method to the one-dimensional general improved KdV (GIKdV) e... more This paper applies He's Exp-function method to the one-dimensional general improved KdV (GIKdV) equation with n th power nonlinear term to obtain some new generalized solitary solutions and periodic solutions. It is shown that the Exp-function method, with the help of any symbolic computation packages, provides a straightforward and powerful mathematical tool for solving many generalized nonlinear evolution equations arising in mathematical physics.

Mathematical Sciences and Applications E-Notes

The aim of this study is to obtain numerical solutions of the modified Fornberg Whitham equation ... more The aim of this study is to obtain numerical solutions of the modified Fornberg Whitham equation via collocation finite element method combined with operator splitting method. The splitting method is used to convert the original equation into two sub equations including linear and nonlinear part of the equation as a slight modification of splitting idea. After splitting progress, collocation method is used to reduce the sub equations into algebraic equation systems. For this purpose, quintic B-spline base functions are used as a polynomial approximation for the solution. The effectiveness and efficiency of the method and accuracy of the results are measured with the error norms L 2 and L ∞. The presentations of the numerical results are shown by graphics as well.

In this work, our aim is to obtain a numerical solution to some fractional differential equations... more In this work, our aim is to obtain a numerical solution to some fractional differential equations. In the solution proces s, we have used fractional derivatives in Caputo sense. The fundamental characteristics of the present method is the fact t hat it converts complex problems into those requiring the solution of algebraic ones, which is obviously more easy for computational processing. The obtained approximate values show the accuracy and suitability of the present scheme for applying a wide range of fractional partial differential equations. Finally, the error norms L2 and L∞ are computed and found to be sufficiently small.

The present article focuses on obtaining numerical solutions of time fractional KdV-Burger-Kuramo... more The present article focuses on obtaining numerical solutions of time fractional KdV-Burger-Kuramoto equation (KBK) with the finite element collocation method. The finite element collocation methods are common and effective tool for solving nonlinear problems because of their reasonable computational costs. The idea underlying the method is seeking the numerical solutions in a form of a linear combination of unknown functions with basis at nodal points by avoid of integration. Thus, in this article, we achieve more accurate numerical results are obtained with the application of the method to KBK equation. Additionally, we show the efficiency and effectiveness of the method using comparisons of numerical results with exact solutions via error norms and their simulations.

Mediterranean Journal of Mathematics, 2020

In this manuscript, a Strang splitting approach combined with Chebyshev wavelets has been used to... more In this manuscript, a Strang splitting approach combined with Chebyshev wavelets has been used to obtain the numerical solutions of regularized long-wave (RLW) equation with various initial and boundary conditions. The performance of the proposed method measured with three different test problems. To measure the accuracy of the method, L2 and L∞ error norms and the I1, I2, I3 invariants are computed. The results of the computations are compared with the existing numerical and exact solutions in the literature.

Wave Motion, 2021

Abstract In this paper we established a numerical method for Equal Width (EW) Equation using Poly... more Abstract In this paper we established a numerical method for Equal Width (EW) Equation using Polynomial Scaling Functions. The EW equation is a simpler alternative to well known Korteweg de Vries (KdV) and regularized long wave (RLW) equations which have many applications in nonlinear wave phenomena. According to Polynomial scaling method, algebraic polynomials are used to get the orthogonality between the wavelets and corresponding scaling functions with respect to the Chebyshev weight. First we introduce polynomial scaling functions, how are the functions are approximated according to these and Operational matrix of derivatives are given. For time discretization of the function we use finite difference method with Rubin Graves linearization and polynomial scaling functions are used for the space discretization. The method is applied to four different problem and the obtained results are compared with the results in the literature and with the exact results to give the efficiency of the method.

Proceedings (International Youth Science Forum “Litteris et Artibus”).2018.№1, 2018

In this paper, new singular soliton solution is found to the hyperbolic generalization of the Bur... more In this paper, new singular soliton solution is found to the hyperbolic generalization of the Burgers equation. 2D and 3D graphs are also presented. At the end of paper, a conclusion is introduced as well by mentioning novel aspects of paper.

ITM Web of Conferences, 2018

In this study, a Sine-Gordon expansion method for obtaining novel exact solutions of extended sha... more In this study, a Sine-Gordon expansion method for obtaining novel exact solutions of extended shallow water wave equation and Fokas equation is presented. All of the equations which are under consideration consist of three or four variable. In this method, first of all, partial differential equations are reduced to ordinary differential equations by the help of variable change called as travelling wave transformation, then Sine Gordon expansion method allows us to obtain new exact solutions defined as in terms of hyperbolic trig functions of considered equations. The newly obtained results showed that the method is successful and applicable and can be extended to a wide class of nonlinear partial differential equations.

ITM Web of Conferences, 2018

This manuscript is going to seek travelling wave solutions of some coupled partial differential e... more This manuscript is going to seek travelling wave solutions of some coupled partial differential equations with an expansion method known as Sine-Gordon expansion method. Primarily, we are going to employ a wave transformation to partial differential equation to reduce the equations into ordinary differential equations. Then, the solution form of the handled equations is going to be constructed as polynomial of hyperbolic trig or trig functions. Finally, with the aid of symbolic computation, new exact solutions of the partial differentials equations will have been found.

International Journal of Modern Physics C, 2018

In the present paper, a novel perspective fundamentally focused on the differential quadrature me... more In the present paper, a novel perspective fundamentally focused on the differential quadrature method using modified cubic B-spline basis functions are going to be applied for obtaining the numerical solutions of the complex modified Korteweg–de Vries (cmKdV) equation. In order to test the effectiveness and efficiency of the present approach, three test problems, that is single solitary wave, interaction of two solitary waves and interaction of three solitary waves will be handled. Furthermore, the maximum error norm [Formula: see text] will be calculated for single solitary wave solutions to measure the efficiency and the accuracy of the present approach. Meanwhile, the three lowest conservation quantities will be calculated and also used to test the efficiency of the method. In addition to these test tools, relative changes of the invariants will be calculated and presented. In the end of these processes, those newly obtained numerical results will be compared with those of some o...

Discrete & Continuous Dynamical Systems - S, 2019

In this paper, we developed a unified method to solve time fractional Burgers' equation using the... more In this paper, we developed a unified method to solve time fractional Burgers' equation using the Chebyshev wavelet and L1 discretization formula. First we give the preliminary information about Chebyshev wavelet method, then we describe time discretization of the problems under consideration and then we apply Chebyshev wavelets for space discretization. The performance of the method is shown by three test problems and obtained results compared with other results available in literature.

Optik, 2018

This study reveals the dark, bright, mixed dark-bright, singular and mixed singular optical solit... more This study reveals the dark, bright, mixed dark-bright, singular and mixed singular optical solitons to the (1+1)-dimensional coupled nonlinear nonlinear Schrödinger equation by using the extended sinh-Gordon equation expansion method. The constraint conditions for the existence of valid solitons are given. Under the choice of suitable values of the parameters and the fractional values of α and β, the 2-and 3-dimensional graphs to some of the reported solutions are plotted.

Bulletin of Mathematical Sciences and Applications, 2017

In this study, the authors employed the quadratic B-spline Galerkin method to solve time fraction... more In this study, the authors employed the quadratic B-spline Galerkin method to solve time fractional order telegraph equations. Three model problems are consideredto implement the method.L2,L∞error norms and numerical results have been presented in tables. Absolute error graphics for all the exact and numerical solutionshave been given

New Trends in Mathematical Science, 2016

The initial version of a Stefan problem is the melting of a semi-infinite sheet of ice. This prob... more The initial version of a Stefan problem is the melting of a semi-infinite sheet of ice. This problem is described by a parabolic partial differential equation along with two boundary conditions on the moving boundary which are used to determine the boundary itself and complete the solution of the differential equation. In this paper firstly, we use variable space grid method, boundary immobilisation method and isotherm migration method to get rid of the trouble of the Stefan problem. Then, collocation finite element method based on cubic B-spline bases functions is applied to model problem. The numerical schemes of finite element methods provide a good numerical approximation for the model problem. The numerical results show that the present results are in good agreement with the exact ones.

Acta Mathematicae Applicatae Sinica, English Series, 2016

In this work, it is aimed to find one- and two-soliton solutions to nonlinear Tzitzeica-Dodd-Bull... more 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.

Sohag Journal of Mathematics, 2016

In the present study, a B-spline collocation method has been applied to obtain a numerical soluti... more In the present study, a B-spline collocation method has been applied to obtain a numerical solution of the sine-Gordon equation. Then, the obtained numerical results have been compared with those given in the literature. The error norms L 2 and L ∞ are computed and they have been found out small enough to be accepted.

Applied Mathematics and Computation, 2015

In the present article, a quadratic B-spline finite element Galerkin method has been used to obta... more In the present article, a quadratic B-spline finite element Galerkin method has been used to obtain numerical solutions of the nonlinear time fractional gas dynamics equation. While the Caputo form is used for the time fractional derivative appearing in the equation, the L1 discretization formula is applied to the equation in time. A numerical example is given and the obtained results show the accuracy and efficiency of the method. Therefore, the present method can be used as an efficient alternative one to find out the numerical solutions of other both linear and nonlinear fractional differential equations available in the literature.

In this paper, we construct exact travelling wave solutions for the generalized (2+1)-dimensional... more In this paper, we construct exact travelling wave solutions for the generalized (2+1)-dimensional ZK-MEW equation by using the solutions of an auxiliary ordinary differential equation given by Sirendaoreji [1]. It is shown that some solutions obtained in this study are new solutions which have not been reported yet.

In this paper, the Homotopy Analysis Method (HAM) is applied to the damped Burgers and Boussinesq... more In this paper, the Homotopy Analysis Method (HAM) is applied to the damped Burgers and Boussinesq-Burgers equations to obtain their approximate analytical solutions. The HAM solution includes an auxiliary parameterh which provides a convenient way to adjust and control the convergence region of the solution series. An appropriate choice of the auxiliary parameter in the model problems for increasing time is investigated.

International Journal of Nonlinear Sciences and Numerical Simulation, 2009

In this paper, the Exp-function method is used to obtain generalized travelling wave solutions wi... more In this paper, the Exp-function method is used to obtain generalized travelling wave solutions with free parameters of the MKdV-sine-Gordon and Boussinesq-double sine-Gordon equations. It is shown that the Exp-function method, with the help of any symbolic computation packages, provides an effective mathematical tool for nonlinear evolution equations arising in mathematical physics.

International Journal of Nonlinear Sciences and Numerical Simulation, 2009

This paper applies He's Exp-function method to the one-dimensional general improved KdV (GIKdV) e... more This paper applies He's Exp-function method to the one-dimensional general improved KdV (GIKdV) equation with n th power nonlinear term to obtain some new generalized solitary solutions and periodic solutions. It is shown that the Exp-function method, with the help of any symbolic computation packages, provides a straightforward and powerful mathematical tool for solving many generalized nonlinear evolution equations arising in mathematical physics.