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    A Finite Difference Approximation for Numerical Simulation of 2D Viscous Coupled Burgers Equations
    (2022) Yağmurlu, N. Murat; Gagir, Abdulnasır
    Many of the physical phenomena in nature are usually expressed in terms of algebraic, differential or integral equations.Several nonlinear phenomena playing a very important role in engineering sciences, physics and computational mathematics are usually modeled by those non-linear partial differential equations (PDEs). It is usually difficult and problematic to examine and find out nalytical solutions of initial-boundary value problems consisting of PDEs. In fact, there is no a certain method or technique working well for all these type equations. For this reason, their approximate solutions are usually preferred rather than analytical ones of such type equations. Thus, many researchers are concentrated on approximate methods and techniques to obtain numerical solutions of non-linear PDEs. In the present article, the numerical simulation of the two-dimensional coupled Burgers equation (2D-cBE) has been sought by finite difference method based on Crank-Nicolson type approximation. Widely used three test examples given with appropriate initial and boundary conditions are used for the simulation process. During the simulation process,the error norms $L_{2}$, $L_{infty}$ are calculated if the exact solutions are already known, otherwise the pointwise values and graphics are provided for comparison. The newly obtained error norms $L_{2}$, $L_{infty}$ by the presented schemes are compared with those of some of the numerical solutions in the literature. A good consistency and accuracy are observed both by numerical values and visual illustrations.
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    Numerical Investigation of Modified Fornberg Whitham Equation
    (2021) Yağmurlu, N. Murat; Uçar, Yusuf; Esen, Alaattin; Yıldız, Ersin
    The aim of this study is to obtain numerical solutions of the modified Fornberg Whitham equation via collocation finite element method combined with operator splitting method. The splitting method is used to convert the original equation into two sub equations including linear and nonlinear part of the equation as a slight modification of splitting idea. After splitting progress, collocation method is used to reduce the sub equations into algebraic equation systems. For this purpose, quintic B-spline base functions are used as a polynomial approximation for the solution. The effectiveness and efficiency of the method and accuracy of the results are measured with the error norms $L_{2}$ and $L_{infty}$. The presentations of the numerical results are shown by graphics as well.
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    Numerical Simulation of Two Dimensional Coupled Burgers Equations by Rubin-Graves Type Linearization
    (2021) Yağmurlu, N. Murat; Gagir, Abdulnasır
    In the present article, the numerical solution of the two-dimensional coupled Burgers equation has been sought by finite difference method based on Rubin-Graves type linearization. Three models with appropriate initial and boundary conditions are applied to the problem. In order to show the accuracy of the method, the error norms L2, L? are computed. The error norms L2, L? of the obtained numerical solutions are compared with the error norms of some of the numerical solutions in the literature.
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    Numerical Solution of Burger’s Type Equation Using Finite Element Collocation Method with Strang Splitting
    (2020) Uçar, Yusuf; Yağmurlu, N. Murat; Çelikkaya, İhsan
    Abstract: The nonlinear Burgers equation, which has a convection term, a viscosity term and a time dependent term in its structure, has been split according to the time term and then has been solved by finite element collocation method using cubic B-spline bases. By splitting the equation Ut + UUx = vUxx into two simpler sub problems Ut + UUx = 0 and Ut ? vUxx = 0 have been obtained. A discretization process has been performed for each of these sub-problems and the stability analyzes have been carried out by Fourier (von Neumann) series method. Then, both sub-problems have been solved using the Strang splitting technique to obtain numerical results. To see the effectiveness of the present method, which is a combination of finite element method and Strang splitting technique, we have calculated the frequently used error norms kek1 , L2 and L? in the literature and have made a comparison between exact and a numerical solution.