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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 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 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 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 the Modified Burgers' Equation by Finite Difference Methods(De Gruyter Poland Sp Zoo, 2017) Ucar, Y.; Yagmurlu, N. M.; Tasbozan, O.In this study, a numerical solution of the modified Burgers' equation is obtained by the finite difference methods. For the solution process, two linearization techniques have been applied to get over the non-linear term existing in the equation. Then, some comparisons have been made between the obtained results and those available in the literature. Furthermore, the error norms L-2 and L-infinity are computed and found to be sufficiently small and compatible with others in the literature. The stability analysis of the linearized finite difference equations obtained by two different linearization techniques has been separately conducted via Fourier stability analysis method.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.
