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    A collocation method for solving time fractional nonlinear Korteweg-de Vries-Burgers equation arising in shallow water waves
    (World Scientific Publ Co Pte Ltd, 2023) Karaagac, Berat; Esen, Alaattin; Owolabi, Kolade M. M.; Pindza, Edson
    This paper focuses on numerical solutions of time fractional nonlinear Korteweg-de Vries-Burgers equation formulated with Caputo's fractional derivative. For this purpose, a framework of combinations of collocation method with the finite-element method is provided using trigonometric quintic B-spline basis. The method consists of both spatial discretization and temporal discretization using approximate solution and Crank-Nicolson approach. Discretizing fractional derivative is made using L1(0 <= 1) algorithm which is derived from the definition of Caputo derivative using an approximate function. The stability analysis is established using von-Neumann stability technique. The numerical results obtained using the collocation method are presented via tables and graphics. The novel results demonstrate the efficiency and reliability of the method.
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    Dynamics of modified improved Boussinesq equation via Galerkin Finite Element Method
    (Wiley, 2020) Karaagac, Berat; Ucar, Yusuf; Esen, Alaattin
    The aim of this paper is to investigate numerical solutions of modified improved Boussinesq (MIBq) equationutt=uxx+alpha mml:mfenced close=) open=( separators=u3xx+uxxtt, which is a modified type of Boussinesq equations born as an art of modelling water-wave problems in weakly dispersive medium such as surface waves in shallow waters or ion acoustic waves. For this purpose, Lumped Galerkin finite element (LGFE) method, an effective, accurate, and cost-effective method, is applied to model equation by the aid of quadratic B-spline basis. The efficiency and accuracy of the method are tested with two problems, namely, propagation solitary wave and interaction of two solitary waves. The error normsL(2)andL(infinity)have been computed in order to measure how accurate the numerical solutions. Also, the stability analysis has been investigated.
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    Exact solutions of nonlinear evolution equations using the extended modified Exp(-?(?)) function method
    (Tbilisi Centre Math Sci, 2019) Karaagac, Berat; Kutluay, Selcuk; Yagmurlu, Nuri Murat; Esen, Alaattin
    Obtaining exact solutions of the evolution equation is one of the very important subjects in mathematics, science and technology. For this purpose, many different methods have been constructed and developed. In this article, a new technique which is called extended modified Exp(-Omega(xi)) function method is going to be studied for seeking new exact solutions of Burger-Fisher equation and Phi Four equation. The method is capable of deriving many number of solutions. With the aid of the method, various exact solutions including trigonometric, hyperbolic and rational solutions have been obtained and using a software the graphical representation of the solutions have been presented. In conclusion, we can say that the present method can also be used for the solutions of a wide range of problems.
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    A Fresh Look To Exact Solutions of Some Coupled Equations
    (E D P Sciences, 2018) Karaagac, Berat; Yagmurlu, Nuri Murat; Esen, Alaattin; Kutluay, Selcuk
    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.
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    The Hunter-Saxton Equation: A Numerical Approach Using Collocation Method
    (Wiley, 2018) Karaagac, Berat; Esen, Alaattin
    In this study, we are going to present an overview on the Hunter-Saxton equation which is a famous equation modelling waves in a massive director field of a nematic liquid crystal. The collocation finite element method is based on quintic B-spline basis for obtaining numerical solutions of the equation. Using this method, after discretization, solution of the equation expressed as linear combination of shape functions and B-spline basis. So, Hunter-Saxton equation converted to nonlinear ordinary differential equation system. With the aid of the error norms L-2 and L-infinity, some comparisons are presented between numeric and exact solutions for different step sizes. As a result, the authors observed that the method is a powerful, suitable and reliable numerical method for solving various kind of partial differential equations.
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    A Lumped Galerkin finite element method for the generalized Hirota-Satsuma coupled KdV and coupled MKdV equations
    (Tbilisi Centre Math Sci, 2019) Yagmurlu, Nuri Murat; Karaagac, Berat; Esen, Alaattin
    In the present study, a Lumped Galerkin finite element method using quadratic B-splines has been applied to the generalized Hirota-Satsuma coupled Korteweg de Vries (KdV) and coupled modified Korteweg-de Vries (mKdV) equations. The numerical solutions of discretized equations using Lumped Galerkin finite element method have been obtained using the fourth order Runge-Kutta method which is widely used for the solution of ordinary differential equation system. The numerical solutions obtained for various space and time values have been compared with exact ones using the error norms L-2 and L-infinity. Lumped Galerkin finite element method is an effective one which can be applied to a wide range of nonlinear evolution equations.
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    A new numerical approach to Gardner Kawahara equation in magneto-acoustic waves in plasma physics
    (Wiley, 2023) Ucar, Yusuf; Yagmurlu, Nuri Murat; Esen, Alaattin; Karaagac, Berat
    The basic idea of this article is to investigate the numerical solutions of Gardner Kawahara equation, a particular case of extended Korteweg-de Vries equation, by means of finite element method. For this purpose, a collocation finite element method based on trigono-metric quintic B-spline basis functions is presented. The standard finite difference method is used to discretize time derivative and Crank-Nicolson approach is used to obtain more accurate numerical results. Then, von Neumann stability analysis is performed for the numerical scheme obtained using collocation finite element method. Several numerical examples are presented and discussed to exhibit the feasibility and capability of the finite element method and trigonometric B-spline basis functions. More specifically, the error norms L-2 and L-infinity are reported for numerous time and space discretization values in tables. Graphical representations of the solutions describing motion of wave are presented.
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    A new outlook for analysis of Noyes-Field model for the nonlinear Belousov-Zhabotinsky reaction using operator splitting method
    (Pergamon-Elsevier Science Ltd, 2023) Karaagac, Berat; Esen, Alaattin; Ucar, Yusuf; Yagmurlu, Nuri Murat
    The main idea of this paper is to investigate numerical solutions of Noyes Field model for Belousov-Zhabotinsky reaction by implementing the combination of two well-known numerical techniques. The proposed methods are collocation method based on finite elements, which is a useful and very flexible approach for solving partial differential equations (PDE), and operator splitting method which is a widely used procedure in the numerical solution of initial and boundary value problems for PDEs. Especially, for this paper, the application of collocation methods are based on trigonometric cubic B-splines. With the help of two techniques discrete schemes are investigated. Next, we presented stability of discrete schemes with Von- Neumann stability analysis. Also, we present the result of applying methods to Noyes Field model and the error norms L-2 and L-infinity to show how accurate numerical solutions to exact ones and graphical representations associated numerical results are shown.
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    A New Perspective on The Numerical Solution for Fractional Klein Gordon Equation
    (Gazi Univ, 2019) Karaagac, Berat; Ucar, Yusuf; Yagmurlu, N. Murat; Esen, Alaattin
    In the present manuscript, a new numerical scheme is presented for solving the time fractional nonlinear Klein-Gordon equation. The approximate solutions of the fractional equation are based on cubic B-spline collocation finite element method and L2 algorithm. The fractional derivative in the given equation is handled in terms of Caputo sense. Using the methods, fractional differential equation is converted into algebraic equation system that are appropriate for computer coding. Then, two model problems are considered and their error norms are calculated to demonstrate the reliability and efficiency of the proposed method. The newly calculated error norms show that numerical results are in a good agreement with the exact solutions.
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    Novel Exact Solutions of the Extended Shallow Water Wave and the Fokas Equations
    (E D P Sciences, 2018) Duran, Serbay; Karaagac, Berat; Esen, Alaattin
    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.
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    Numerical solutions of Boussinesq equation using Galerkin finite element method
    (Wiley, 2021) Ucar, Yusuf; Esen, Alaattin; Karaagac, Berat
    In this study, Galerkin finite element method has been applied to good Boussinesq (GBq) and bad Boussinesq (BBq) equations which are examples of Boussinesq type equations. To apply the method, cubic B-spline basis functions are taken as element and weight functions. The solutions of the numerical schemes have been obtained using the fourth order Runge-Kutta method. The error norms L-2 and L-infinity have been used to test how compatible they obtained numerical solutions with those of exact ones. As numerical examples, solitary wave motion and interaction of solitary waves have been investigated for both GBq and BBq equation. Also blow-up solutions related to the interaction of two solitary waves are considered for GBq equation.
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    Numerical Solutions of the Improved Boussinesq Equation by the Galerkin Quadratic B-Spline Finite Element Method
    (Univ Nis, Fac Sci Math, 2018) Karaagac, Berat; Ucar, Yusuf; Esen, Alaattin
    In this paper, we are going to obtain numerical solutions of the improved Boussinesq equation with the aid of Galerkin quadratic B-spline finite element method. To test the accuracy and efficiency of the current method, four test problems have been used. These are solitary wave movement, interaction of two solitary waves, wave break-up and blow-up of solutions. Their results have been compared with those available in the literature for different values of space and time steps. Also, the error norms L-2 and L-infinity have been computed and presented in comparison.