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    An application of Chebyshev wavelet method for the nonlinear time fractional Schrodinger equation
    (Wiley, 2022) G., Esra Köse; Oruç, Ömer; Esen, Alaattin
    In the present manuscript, we will deal with time fractional Schrodinger equation having appropriate initial and boundary conditions with Chebyshev wavelet method numerically. The Chebyshev wavelet method will be utilized successfully for two test problems. In order to find out efficiency and accuracy of this method, the widely used error norms L-2 and L-infinity of the newly found results have been compared with some of the other approximate results in the literature. The results have been given in tables and figures to show the compatibility between the new results and those in other articles.
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    Higher order Haar wavelet method integrated with strang splitting for solving regularized long wave equation
    (Elsevier, 2022) Bulut, Fatih; Oruç, Ömer; Esen, Alaattin
    In this paper, we are going to utilize newly developed Higher Order Haar wavelet method (HOHWM) and classical Haar wavelet method (HWM) to numerically solve the Regularized Long Wave (RLW) equation. Spatial variable of the RLW equation is treated with HOHWM and HWM separately. On the other hand temporal variable is discretized by finite differences combined with Strang splitting approach. The presented methods applied to three different test problems and the obtained results are given in tables as well as depicted in figures. The obtained results are compared with analytical results wherever they exist. The error norms L-2 and L-infinity and invariants I-1, I-2 and I-3 are used to show the accuracy of the methods when comparing the present results with those in the literature. (C)& nbsp;2022 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.
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    Highly accurate numerical scheme based on polynomial scaling functions for equal width equation
    (Elsevier, 2021) Oruc, Omer; Esen, Alaattin; Bulut, Fatih
    In this paper we established a numerical method for Equal Width (EW) Equation using Polynomial Scaling Functions. The EW equation is a simpler alternative to well known Korteweg de Vries (KdV) and regularized long wave (RLW) equations which have many applications in nonlinear wave phenomena. According to Polynomial scaling method, algebraic polynomials are used to get the orthogonality between the wavelets and corresponding scaling functions with respect to the Chebyshev weight. First we introduce polynomial scaling functions, how are the functions are approximated according to these and Operational matrix of derivatives are given. For time discretization of the function we use finite difference method with Rubin Graves linearization and polynomial scaling functions are used for the space discretization. The method is applied to four different problem and the obtained results are compared with the results in the literature and with the exact results to give the efficiency of the method. (C) 2021 Elsevier B.V. All rights reserved.
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    Numerical Solution of the Rosenau-KdV-RLW equation via combination of a polynomial scaling function collocation and finite difference method
    (Wiley, 2025) Oruc, Omer; Esen, Alaattin; Bulut, Fatih
    In this paper, we established a polynomial scaling method to investigate the numerical solution of Rosenau-Korteweg De Vries-regularized long wave (Rosenau-KdV-RLW) equation. We start with discretization of the time variable of the equation using a finite difference approach equipped with a linearization. After the time discretization, we have used polynomial scaling functions for the discretization of the spatial variable. These two discretizations give us the desired discrete system of equations to obtain numerical solutions. We further derive an error estimate for the proposed method. We have applied the proposed method to Rosenau-KdV, Rosenau-RLW, and Rosenau-KdV-RLW equations and used error norms to examine the accuracy and reliability of the presented method. Also, to enhance accuracy of the results, we utilize Richardson extrapolation. The comparisons with the analytical solution and earlier studies that use different methods indicate that the proposed method is accurate and reliable.