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Öğe An approach to time fractional gas dynamics equation: Quadratic B-spline Galerkin method(Elsevier Science Inc, 2015) Esen, A.; Tasbozan, O.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. (C) 2015 Elsevier Inc. All rights reserved.Öğe Approximate Analytical Solution to Time-Fractional Damped Burger and Cahn-Allen Equations(Natural Sciences Publishing Corp-Nsp, 2013) Esen, A.; Yagmurlu, N. M.; Tasbozan, O.The aim of the present paper is to obtain the approximate analytical solutions of time-fractional damped burger and Cahn-Allen equations by means of the homotopy analysis method (HAM). In the HAM solution, there exists an auxiliary parameter (h) over bar which provides a convenient way to adjust and check the convergence region of the solution series. In the model problems, an appropriate choice of the auxiliary parameter has been examined for increasing values of time.Öğe A B-spline collocation method for solving fractional diffusion and fractional diffusion-wave equations(Tbilisi Centre Math Sci, 2015) Esen, A.; Tasbozan, O.; Ucar, Y.; Yagmurlu, N. M.In this paper,we have considered the fractional diffusion and fractional diffusion wave equations in which the time derivative is a fractional derivative in the Caputo form and have obtained their numerical solutions by collocation method using cubic B-spline base functions. In the solution process, for the fractional diffusion equation L1 discretizaton formula of the fractional derivative is applied, and for the fractional diffusion-wave equation L2 discretizaton formula of the fractional derivative is applied. Accuracy of the proposed method is discussed by computing the error norms L2 and L-infinity. A stability analysis of the approximation obtained by the scheme shows that the method is unconditionally stable.Öğe Chebyshev Wavelet Method for Numerical Solutions of Coupled Burgers' Equation(Hacettepe Univ, Fac Sci, 2019) Oruc, O.; Bulut, F.; Esen, A.This paper deals with the numerical solutions of one dimensional time dependent coupled Burgers' equation with suitable initial and boundary conditions by using Chebyshev wavelets in collaboration with a collocation method. The proposed method converts coupled Burgers' equations into system of algebraic equations by aid of the Chebyshev wavelets and their integrals which can be solved easily with a solver. Benchmarking of the proposed method with exact solution and other known methods already exist in the literature is made by three test problems. The feasibility of the proposed method is demonstrated by test problems and indicates that the proposed method gives accurate results in short cpu times. Computer simulations show that the proposed method is computationally cheap, fast and quite good even in the case of less number of collocation points.Öğe A Crank-Nicolson Approximation for the time Fractional Burgers Equation(Walter De Gruyter Gmbh, 2020) Onal, M.; Esen, A.In the present manuscript, Crank Nicolson finite difference method is going to be applied to get the approximate solutions for the fractional Burgers equation. The fractional derivative used in this equation is going to be taken into consideration in the Caputo sense. The L1 type discretization formula is going to be applied to this equation. For checking the efficiency of proposed methods, the error norms L-2 and L-infinity have at the same time been calculated. Those newly got solutions using the presented method illustrate the easy usage and efficiency of the approach presented in this manuscript.Öğe Cubic B-spline collocation method for solving time fractional gas dynamics equation(De Gruyter Open Ltd, 2015) Esen, A.; Tasbozan, O.To the present manuscript, a cubic B-spline finite element collocation 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 terms of time. It has been seen that the results of the present study are in agreement with the those of exact solution. Therefore, the present method can be used as an alternative and efficient one to find out the numerical solutions of both linear and nonlinear fractional differential equations available in the literature. In order to control the accuracy and efficiency of the present method, the error norms L-2, and L-infinity have been calculated. It is evident that, the newly obtained numerical solutions by the present method can he computed easily with the implementation and effectiveness of the approach used in the article.Öğe Does smoking increase the anesthetic requirement?(2019) Esen, A.; Bulut, F.; Oruç, Ö.Abstract: Background/aim: To examine the effects of active and passive smoking on perioperative anesthetic and analgesic consumption. Materials and methods: Patients were divided into three groups: group S, smokers; group PS, passive smokers; and group NS, individuals who did not have a history of smoking and were not exposed to smoke. All patients underwent the standard total intravenous anesthesia method. The primary endpoint of this study was determination of the total amount of propofol and remifentanil consumed. Results: The amount of propofol used in induction of anesthesia was significantly higher in group S compared to groups PS and NS. Moreover, the total consumption of propofol was significantly higher in group S compared to groups PS and NS. The total propofol consumption of group PS was significantly higher than that of group NS (P = 0.00). Analysis of total remifentanil consumption showed that remifentanil use was significantly higher in group S compared to group NS (P = 0.00). Conclusion: The amount of the anesthetic required to ensure equal anesthetic depth in similar surgeries was higher in active smokers and passive smokers compared to nonsmokers.Öğe A finite difference solution of the regularized long-wave equation(Hindawi Ltd, 2006) Kutluay, S.; Esen, A.A linearized implicit finite difference method to obtain numerical solution of the one-dimensional regularized long-wave (RLW) equation is presented. The performance and the accuracy of the method are illustrated by solving three test examples of the problem: a single solitary wave, two positive solitary waves interaction, and an undular bore. The obtained results are presented and compared with earlier work.Öğe The (G?/G)-expansion method for some nonlinear evolution equations(Elsevier Science Inc, 2010) Kutluay, S.; Esen, A.; Tasbozan, O.In this paper, the (G'G)-expansion method is applied to the Liouville, sine-Gordon and new coupled MKdV equations to obtain their some generalized exact travelling wave solutions. (C) 2010 Elsevier Inc. All rights reserved.Öğe A Haar wavelet-finite difference hybrid method for the numerical solution of the modified Burgers' equation(Springer, 2015) Oruc, O.; Bulut, F.; Esen, A.In this paper, we investigate the numerical solutions of one dimensional modified Burgers' equation with the help of Haar wavelet method. In the solution process, the time derivative is discretized by finite difference, the nonlinear term is linearized by a linearization technique and the spatial discretization is made by Haar wavelets. The proposed method has been tested by three test problems. The obtained numerical results are compared with the exact ones and those already exist in the literature. Also, the calculated numerical solutions are drawn graphically. Computer simulations show that the presented method is computationally cheap, fast, reliable and quite good even in the case of small number of grid points.Öğe A new approach on numerical solutions of the Improved Boussinesq type equation using quadratic B-spline Galerkin finite element method(Elsevier Science Inc, 2015) Ucar, Y.; Karaagac, B.; Esen, A.In the present manuscript, some numerical solutions of an Improved Boussinesq type equation are obtained by means of quadratic B-spline Galerkin finite element method. Then, error norms L-2 and L-infinity have been calculated to test the accuracy of the current method. In the manuscript, solitary wave movement and interaction of solitary-antisolitary waves are considered as test problems. (C) 2015 Elsevier Inc. All rights reserved.Öğe New mixed-dark soliton solutions to the hyperbolic generalization of the Burgers equation(2019) Dusuncelı, Faruk; Baskonus, Hacı Mehmet; Esen, A.; Bulut, HasanAbstract: In this paper, we apply the exponential function method to find mixed-dark, exponentialand singular soliton solutions in the hyperbolic generalization of the Burgers equation.We obtain some entirely new mixed singular and dark soliton solutions. Under thesuitable values of parameters, various dimensional simulations of results are plotted.Finally, we present a conclusion by giving novelties of paper.Öğe Numerical solution of the KdV equation by Haar wavelet method(Indian Acad Sciences, 2016) Oruc, O.; Bulut, F.; Esen, A.This paper aims to get numerical solutions of one-dimensional KdV equation by Haar wavelet method in which temporal variable is expanded by Taylor series and spatial variables are expanded with Haar wavelets. The performance of the proposed method is measured by four different problems. The obtained numerical results are compared with the exact solutions and numerical results produced by other methods in the literature. The comparison of the results indicate that the proposed method not only gives satisfactory results but also do not need large amount of CPU time. Error analysis of the proposed method is also investigated.Öğe Numerical solution of time fractional Burgers equation(De Gruyter Poland Sp Zoo, 2015) Esen, A.; Tasbozan, O.In this article, the time fractional order Burgers equation has been solved by quadratic B-spline Galerkin method. This method has been applied to three model problems. The obtained numerical solutions and error norms L-2 and L-infinity have been presented in tables. Absolute error graphics as well as those of exact and numerical solutions have been given.Öğe A Numerical Solution to Fractional Diffusion Equation for Force-Free Case(Hindawi Ltd, 2013) Tasbozan, O.; Esen, A.; Yagmurlu, N. M.; Ucar, Y.A collocation finite element method for solving fractional diffusion equation for force-free case is considered. In this paper, we develop an approximation method based on collocation finite elements by cubic B-spline functions to solve fractional diffusion equation for force-free case formulated with Riemann-Liouville operator. Some numerical examples of interest are provided to show the accuracy of the method. A comparison between exact analytical solution and a numerical one has been made.Öğe Numerical Solutions of Fractional System of Partial Differential Equations By Haar Wavelets(Tech Science Press, 2015) Bulut, F.; Oruc, O.; Esen, A.In this paper, time fractional one dimensional coupled KdV and coupled modified KdV equations are solved numerically by Haar wavelet method. Proposed method is new in the sense that it doesn't use fractional order Haar operational matrices. In the proposed method L1 discretization formula is used for time discretization where fractional derivatives are Caputo derivative and spatial discretization is made by Haar wavelets. L-2 and L-infinity error norms for various initial and boundary conditions are used for testing accuracy of the proposed method when exact solutions are known. Numerical results which produced by the proposed method for the problems under consideration confirm the feasibility of Haar wavelet method combined with L1 discretization formula.Öğe Numerical Solutions of Regularized Long Wave Equation By Haar Wavelet Method(Springer Basel Ag, 2016) Oruc, O.; Bulut, F.; Esen, A.In this paper, we are going to investigate numerical solutions of the regularized long wave (RLW) equation by using Haar wavelet (HW), combined with finite difference method. The motion of a single solitary wave, interaction of two solitary waves, Maxwellian initial condition and wave undulation are our test problems for measuring performance of the proposed method. The results of computations are compared with exact solutions and those already published. and error norms and the numerical conservation laws are computed for discussing the accuracy and efficiency of the proposed method.Öğe A Numerical Treatment Based on Haar Wavelets for Coupled KdV Equation(2017) Oruç, Ö.; Bulut, F.; Esen, A.Abstract:In this paper, numerical solutions of one dimensional coupled KdV equation has been investigated by Haar Wavelet method. Time derivatives given in this equation are discretized by finite differences and nonlinear terms appearing in the equations are linearized by some linearization techniques and space derivatives are discretized by Haar wavelets. For examining performance of the proposed method, single soliton solution and conserved quantities of some test problems are used. Also error analysis of numerical scheme is investigated and numerical results are compared with some results already existing in the literatureÖğe Solitary wave solutions of the modified equal width wave equation(Elsevier Science Bv, 2008) Esen, A.; Kutluay, S.In this paper we use a linearized numerical scheme based on finite difference method to obtain solitary wave solutions of the one-dimensional modified equal width (MEW) equation. Two test problems including the motion of a single solitary wave and the interaction of two solitary waves are solved to demonstrate the efficiency of the proposed numerical scheme. The obtained results show that the proposed scheme is an accurate and efficient numerical technique in the case of small space and time steps. A stability analysis of the scheme is also investigated. (c) 2006 Elsevier B.V. All rights reserved.Öğe SOLVING FRACTIONAL DIFFUSION AND FRACTIONAL DIFFUSION-WAVE EQUATIONS BY PETROV-GALERKIN FINITE ELEMENT METHOD(Turkic World Mathematical Soc, 2014) Esen, A.; Ucar, Y.; Yagmurlu, M.; Tasbozan, O.In the last few years, it has become highly evident that fractional calculus has been widely used in several areas of science. Because of this fact, their numerical solutions also have become urgently important. In this manuscript, numerical solutions of both the fractional diffusion and fractional diffusion-wave equations have been obtained by a Petrov-Galerkin finite element method using quadratic B-spline base functions as trial functions and linear B-spline base functions as the test functions. In those equations, fractional derivatives are used in terms of the Caputo sense. While the L1 discretizaton formula has been applied to fractional diffusion equation, the L2 discretizaton formula has been applied to the fractional diffusion-wave equation. Finally, the error norms L-2 and L-infinity have been calculated for testing the accuracy of the proposed scheme.
