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    Analytical and numerical approaches to nerve impulse model of fractional-order
    (Wiley, 2020) Yavuz, Mehmet; Yokus, Asif
    We consider a fractional-order nerve impulse model which is known as FitzHugh-Nagumo (F-N) model in this paper. Knowing the solutions of this model allows the management of the nerve impulses process. Especially, considering this model as fractional-order ensures to be able to analyze in detail because of the memory effect. In this context, first, we use an analytical solution and with the aim of this solution, we obtain numerical solutions by using two numerical schemes. Then, we demonstrate the walking wave-type solutions of the stated problem. These solutions include complex trigonometric functions, complex hyperbolic functions, and algebraic functions. In addition, the linear stability analysis is performed and the absolute error is occurred by comparing the numerical results with the analytical result. All of the results are depicted by tables and figures. This paper not only points out the exact and numerical solutions of the model but also compares the differences and the similarities of the stated solution methods. Therefore, the results of this paper are important and useful for either neuroscientists and physicists or mathematicians and engineers.
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    NOVEL COMPARISON OF NUMERICAL AND ANALYTICAL METHODS FOR FRACTIONAL BURGER-FISHER EQUATION
    (Amer Inst Mathematical Sciences-Aims, 2021) Yokus, Asif; Yavuz, Mehmet
    In this paper, we investigate some analytical, numerical and ap-proximate analytical methods by considering time-fractional nonlinear Burger- Fisher equation (FBFE). (1/G')-expansion method, finite difference method (FDM) and Laplace perturbation method (LPM) are considered to solve the FBFE. Firstly, we obtain the analytical solution of the mentioned problem via (1/G')-expansion method. Also, we compare the numerical method solutions and point out which method is more effective and accurate. We study trun-cation error, convergence, Von Neumann's stability principle and analysis of linear stability of the FDM. Moreover, we investigate the L-2 and Loo norm errors for the FDM. According to the results of this study, it can be concluded that the finite difference method has a lower error level than the Laplace per-turbation method. Nonetheless, both of these methods are totally settlement in obtaining efficient results of fractional order differential equations.