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A 3-Scale Haar Wavelet Collocation Method for Numerical Solution of the Nonlinear Gardner Equation
(Springer Basel Ag, 2025) Bulut, Fatih; Oruc, Omer; Esen, Alaattin
In this paper, a 3-scale Haar wavelet collocation method was applied to the nonlinear Gardner equation which can be used to describe the large-amplitude inner waves in the ocean. We start the solution process with the time discretization of the Gardner equation, with the help of finite difference method. Then, we have used 3-scale Haar wavelets for the space discretization. These steps gave us a system of algebraic equations, by solving these equations we were able to get wavelet coefficients and used them to construct the numerical solution of the Gardner equation. We applied the proposed method to five different problems to test the accuracy and compared the obtained results with other studies in the literature. The results and comparisons are given in tables and solutions are depicted graphically. The results show that the method proposed in this manuscript is highly accurate even with a low number of collocation points.
A numerical aproach to dispersion-dissipation-reaction model: third order KdV-Burger-Fisher equation
(Iop Publishing Ltd, 2024) Esen, Alaattin; Karaagac, Berat; Yagmurlu, Nuri Murat; Ucar, Yusuf; Manafian, Jalil
In this study, an efficient numerical method is applied to KdV-Burger-Fisher equation which is one of the dispersion-dissipation-reaction model. The present method is based on the collocation method whose weight functions are taken from the family of the Dirac delta functions in finite element methods. The element functions are selected as quintic trigonometric B-spline basis. The error norms L 2 and L infinity are calculated to measure the efficiency of the method. Numerical solutions and error norms which are obtained via collocation method and trigonometric basis are presented in tables and simulations of the solutions are exhibited as well. Additionally, stability analysis is investigated.
A trigonometric quintic B-spline collocation technique for the fifth-order KdV-Burgers-Fisher equation
(Springer, 2025) Karaagac, Berat; Esen, Alaattin; Ucar, Yusuf; Yagmurlu, Nuri Murat
The paper investigates numerical solutions to the KdV-Burgers-Fisher (KBF) equation, which models a dispersion-dissipation-reaction phenomenon. The stated equation is a mathematical structure for describing physical, chemical, or biological systems in which the dynamics of the system are shaped by the interaction of dispersion, dissipation, and reaction processes. To solve the KBF equation, a collocation method based on the finite element approach is utilized. In order to construct the approximate solutions satisfying the governing equation at collocation points, the finite element shape functions have been selected as quintic trigonometric B-spline basis functions. The application of the collocation method to the equation yields an algebraic equation system that has a well-known penta-diagonal coefficient matrix. The resulting system allows us to calculate the error norms L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{2}$$\end{document} and L infinity\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{\infty }$$\end{document} and simulate space-time graphics of numerical solutions. As numerical examples of the KdV-Burgers-Fisher (KBF) equation, two test problems are presented to show the performance of the collocation method, while the error norms and graphs including comparison with exact solutions are used to prove the correctness and applicability of the method. Moreover, existence and uniqueness of the solutions are discussed via fixed-point theory, stability analysis which is investigated via von-Neumann technique are presented in this paper as well.
An application of Chebyshev wavelet method for the nonlinear time fractional Schrodinger equation
(Wiley, 2022) Esra Kose, G.; Oruc, Omer; Esen, Alaattin
In the present manuscript, we will deal with time fractional Schrodinger equation having appropriate initial and boundary conditions with Chebyshev wavelet method numerically. The Chebyshev wavelet method will be utilized successfully for two test problems. In order to find out efficiency and accuracy of this method, the widely used error norms L-2 and L-infinity of the newly found results have been compared with some of the other approximate results in the literature. The results have been given in tables and figures to show the compatibility between the new results and those in other articles.
AN APPLICATION OF FINITE ELEMENT METHOD FOR A MOVING BOUNDARY PROBLEM
(Vinca Inst Nuclear Sci, 2018) Aksan, Emine Nesligul; Karabenli, Hatice; Esen, Alaattin
The Stefan problems called as moving boundary problems are defined by the heat equation on the domain 0 < x < s(t). In these problems, the position of moving boundary s(t) is determined as part of the solution. As a result, they are non-linear problems and thus have limited analytical solutions. In this study, we are going to consider a Stefan problem described as solidification problem. After using variable space grid method and boundary immobilization method, collocation finite element method is applied to the model problem. The numerical solutions obtained for the position of moving boundary are compared with the exact ones and the other numerical solutions existing in the literature. The newly obtained numerical results are more accurate than the others for the time step Delta t = 0.0005, it is also seen from the tables, the numerical solutions converge to exact solutions for the larger element numbers.
Application of the Exp-function method to the two dimensional sine-Gordon equation
(Walter De Gruyter & Co, 2009) Esen, Alaattin; Kutluay, Selcuk
In this paper, the Exp-function method is used to obtain some new generalized solitary wave solutions of the two dimensional sine-Gordon equation. In solving some other nonlinear evolution equations arising in mathematical physics, Exp-function method provides a straightforward and powerful mathematical tool.
Bir boyutlu hareketli sınır değer (Stefan) problemleri için nümerik çözüm yöntemleri
(İnönü Üniversitesi, 1997) Esen, Alaattin
ÖZET Üç bölümden oluşan bu çalışmanın ilk bölümünde sonraki bölümlerde kullanılacak olan temel tanım ve teoremlere yer verildi. İkinci bölümde çalışmamızın temelini oluşturan Stefan(buzun erimesi) problemi hakkında genel bilgiler verilerek, nümerik metodların uygulanabilmesi için problem boyutsuz forma indirgendi. Üçüncü bölümde Stefan probleminin çözümü sırasıyla Variable Space Grid, Boundary Immobilisation, Isotherm Migration, Fully Implicit ve Entalpi yöntemleri ile elde edilerek, nümerik çözümlerin karşılaştırılmaları her yöntemin ardından tablolar halinde sunuldu.
Collocation Finite Element Method for the Fractional Fokker-Planck Equation
(Wiley, 2025) Karabenli, Hatice; Esen, Alaattin; Ucar, Yusuf
In this study, the approximate results of the fractional Fokker-Planck equations have been investigated. First, finite element schemes have been obtained using collocation finite element method based on the trigonometric quintic B-spline basis functions. Then, the present method is tested on two fundamental problems having appropriate initial conditions. The newly obtained numerical results contained the error norms L2$$ {L}_2 $$ and L infinity$$ {L}_{\infty } $$ for various temporal and spatial steps are compared with the exact ones and other solutions. More accurate results have been obtained for large numbers of spatial and temporal elements. This study explores approximate solutions for fractional Fokker-Planck equations using general finite element schemes developed via the collocation finite element method with trigonometric quintic B-spline basis functions. It validates these methods on two fundamental problems and compares numerical results, including L2$$ {L}_2 $$ and L infinity$$ {L}_{\infty } $$ error norms across different temporal and spatial steps, against exact solutions and alternative methods. The findings highlight strong agreement between approximate and exact solutions, particularly when employing higher numbers of spatial and temporal elements. image
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.
Double Exp-Function Method for Multisoliton Solutions of The Tzitzeica-Dodd-Bullough Equation
(Springer Heidelberg, 2016) Esen, Alaattin; Yagmurlu, N. Murat; Tasbozan, Orkun
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.
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.
An effective approach to numerical soliton solutions for the Schrodinger equation via modified cubic B-spline differential quadrature method
(Pergamon-Elsevier Science Ltd, 2017) Bashan, Ali; Yagmurlu, Nuri Murat; Ucar, Yusuf; Esen, Alaattin
In this study, an effective differential quadrature method (DQM) which is based on modified cubic B-spline (MCB) has been implemented to obtain the numerical solutions for the nonlinear Schrodinger (NLS) equation. After separating the Schrodinger equation into coupled real value differential equations,we have discretized using DQM and then obtained ordinary differential equation systems. For time integration, low storage strong stability-preserving Runge-Kutta method has been used. Numerical solutions of five different test problems have been obtained. The efficiency and accuracy of the method have been measured by calculating error norms L2 and Linfinity and two lowest invariants I1 and I2. Also relative changes of invariants are given. The newly obtained numerical results have been compared with the published numerical results and a comparison has shown that the MCB-DQM is an effective numerical scheme to solve the nonlinear Schrodinger equation. (C) 2017 Elsevier Ltd. All rights reserved.
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.
Exp-function Method for Solving the General Improved KdV Equation
(Freund Publishing House Ltd, 2009) Kutluay, Selcuk; Esen, Alaattin
This paper. applies He's Exp-function method to the one-dimensional general improved KdV (GIKdV) equation with nth 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.
Finite difference method combined with differential quadrature method for numerical computation of the modified equal width wave equation
(Wiley, 2021) Bashan, Ali; Yagmurlu, N. Murat; Ucar, Yusuf; Esen, Alaattin
The aim of this study is to improve the numerical solution of the modified equal width wave equation. For this purpose, finite difference method combined with differential quadrature method with Rubin and Graves linearizing technique has been used. Modified cubic B-spline base functions are used as base function. By the combination of two numerical methods and effective linearizing technique high accurate numerical algorithm is obtained. Three main test problems are solved for various values of the coefficients. To observe the performance of the present method, the error norms of the single soliton problem which has analytical solution are calculated. Besides these error norms, three invariants are reported. Comparison of the results displays that our algorithm produces superior results than those given in the literature.
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.
A Galerkin Finite Element Method to Solve Fractional Diffusion and Fractional Diffusion-Wave Equations
(Vilnius Gediminas Tech Univ, 2013) Esen, Alaattin; Ucar, Yusuf; Yagmurlu, Nuri; Tasbozan, Orkun
In the present study, numerical solutions of the fractional diffusion and fractional diffusion-wave equations where fractional derivatives are considered in the Caputo sense have been obtained by a Galerkin finite element method using quadratic B-spline base functions. For the fractional diffusion equation, the L1 discretizaton formula is applied, whereas the L2 discretizaton formula is applied for the fractional diffusion-wave equation. The error norms L2 and L1 are computed to test the accuracy of the proposed method. It is shown that the present scheme is unconditionally stable by applying a stability analysis to the approximation obtained by the proposed scheme.
A haar wavelet approximation for two-dimensional time fractional reaction-subdiffusion equation
(Springer, 2019) Oruc, Omer; Esen, Alaattin; Bulut, Fatih
In this study, we established a wavelet method, based on Haar wavelets and finite difference scheme for two-dimensional time fractional reaction-subdiffusion equation. First by a finite difference approach, time fractional derivative which is defined in Riemann-Liouville sense is discretized. After time discretization, spatial variables are expanded to truncated Haar wavelet series, by doing so a fully discrete scheme obtained whose solution gives wavelet coefficients in wavelet series. Using these wavelet coefficients approximate solution constructed consecutively. Feasibility and accuracy of the proposed method is shown on three test problems by measuring error in norm. Further performance of the method is compared with other methods available in literature such as meshless-based methods and compact alternating direction implicit methods.
A Haar wavelet collocation method for coupled nonlinear Schrodinger-KdV equations
(World Scientific Publ Co Pte Ltd, 2016) Oruc, Omer; Esen, Alaattin; Bulut, Fatih
In this paper, to obtain accurate numerical solutions of coupled nonlinear Schrodinger-Korteweg-de Vries (KdV) equations a Haar wavelet collocation method is proposed. An explicit time stepping scheme is used for discretization of time derivatives and nonlinear terms that appeared in the equations are linearized by a linearization technique and space derivatives are discretized by Haar wavelets. In order to test the accuracy and reliability of the proposed method L-2, L-infinity error norms and conserved quantities are used. Also obtained results are compared with previous ones obtained by finite element method, Crank-Nicolson method and radial basis function meshless methods. Error analysis of Haar wavelets is also given.
Higher order Haar wavelet method integrated with strang splitting for solving regularized long wave equation
(Elsevier, 2022) Bulut, Fatih; Oruc, Oemer; Esen, Alaattin
In this paper, we are going to utilize newly developed Higher Order Haar wavelet method (HOHWM) and classical Haar wavelet method (HWM) to numerically solve the Regularized Long Wave (RLW) equation. Spatial variable of the RLW equation is treated with HOHWM and HWM separately. On the other hand temporal variable is discretized by finite differences combined with Strang splitting approach. The presented methods applied to three different test problems and the obtained results are given in tables as well as depicted in figures. The obtained results are compared with analytical results wherever they exist. The error norms L-2 and L-infinity and invariants I-1, I-2 and I-3 are used to show the accuracy of the methods when comparing the present results with those in the literature. (C)& nbsp;2022 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.

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