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    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.
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    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.
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    Numerical Solution of Time Fractional Burgers Equation by Cubic B-spline Finite Elements
    (Springer Basel Ag, 2016) Esen, Alaattin; Tasbozan, Orkun
    We present some numerical examples which support numerical results for the time fractional Burgers equation with various boundary and initial conditions obtained by collocation method using cubic B-spline base functions. The aim of this paper is to show that the finite element method based on the cubic B-spline collocation method approach is also suitable for the treatment of the fractional differential equations. The results of numerical experiments are compared with analytical solution to confirm the accuracy and efficiency of the presented scheme.