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    An application of Chebyshev wavelet method for the nonlinear time fractional Schrodinger equation
    (Wiley, 2022) Esra Kose, G.; Oruc, Omer; 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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    A haar wavelet approximation for two-dimensional time fractional reaction-subdiffusion equation
    (Springer, 2019) Oruc, Omer; Esen, Alaattin; Bulut, Fatih
    In this study, we established a wavelet method, based on Haar wavelets and finite difference scheme for two-dimensional time fractional reaction-subdiffusion equation. First by a finite difference approach, time fractional derivative which is defined in Riemann-Liouville sense is discretized. After time discretization, spatial variables are expanded to truncated Haar wavelet series, by doing so a fully discrete scheme obtained whose solution gives wavelet coefficients in wavelet series. Using these wavelet coefficients approximate solution constructed consecutively. Feasibility and accuracy of the proposed method is shown on three test problems by measuring error in norm. Further performance of the method is compared with other methods available in literature such as meshless-based methods and compact alternating direction implicit methods.
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    A Haar wavelet collocation method for coupled nonlinear Schrodinger-KdV equations
    (World Scientific Publ Co Pte Ltd, 2016) Oruc, Omer; Esen, Alaattin; Bulut, Fatih
    In this paper, to obtain accurate numerical solutions of coupled nonlinear Schrodinger-Korteweg-de Vries (KdV) equations a Haar wavelet collocation method is proposed. An explicit time stepping scheme is used for discretization of time derivatives and nonlinear terms that appeared in the equations are linearized by a linearization technique and space derivatives are discretized by Haar wavelets. In order to test the accuracy and reliability of the proposed method L-2, L-infinity error norms and conserved quantities are used. Also obtained results are compared with previous ones obtained by finite element method, Crank-Nicolson method and radial basis function meshless methods. Error analysis of Haar wavelets is also given.
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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 investigation of dynamic Euler-Bernoulli equation via 3-Scale Haar wavelet collocation method
    (Hacettepe Univ, Fac Sci, 2021) Oruc, Omer; Esen, Alaattin; Bulut, Fatih
    In this study, we analyze the performance of a numerical scheme based on 3-scale Haar wavelets for dynamic Euler-Bernoulli equation, which is a fourth order time dependent partial differential equation. This type of equations governs the behaviour of a vibrating beam and have many applications in elasticity. For its solution, we first rewrite the fourth order time dependent partial differential equation as a system of partial differential equations by introducing a new variable, and then use finite difference approximations to discretize in time, as well as 3-scale Haar wavelets to discretize in space. By doing so, we obtain a system of algebraic equations whose solution gives wavelet coefficients for constructing the numerical solution of the partial differential equation. To test the accuracy and reliability of the numerical scheme based on 3-scale Haar wavelets, we apply it to five test problems including variable and constant coefficient, as well as homogeneous and non-homogeneous partial differential equations. The obtained results are compared wherever possible with those from previous studies. Numerical results are tabulated and depicted graphically. In the applications of the proposed method, we achieve high accuracy even with small number of collocation points.
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    A Strang Splitting Approach Combined with Chebyshev Wavelets to Solve the Regularized Long-Wave Equation Numerically
    (Springer Basel Ag, 2020) Oruc, Omer; Esen, Alaattin; Bulut, Fatih
    In this manuscript, a Strang splitting approach combined with Chebyshev wavelets has been used to obtain the numerical solutions of regularized long-wave (RLW) equation with various initial and boundary conditions. The performance of the proposed method measured with three different test problems. To measure the accuracy of the method, L-2 and L-infinity error norms and the I-1,I-2,I-3 invariants are computed. The results of the computations are compared with the existing numerical and exact solutions in the literature.
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    A UNIFIED FINITE DIFFERENCE CHEBYSHEV WAVELET METHOD FOR NUMERICALLY SOLVING TIME FRACTIONAL BURGERS' EQUATION
    (Amer Inst Mathematical Sciences-Aims, 2019) Oruc, Omer; Esen, Alaattin; Bulut, Fatih
    In this paper, we developed a unified method to solve time fractional Burgers' equation using the Chebyshev wavelet and L1 discretization formula. First we give the preliminary information about Chebyshev wavelet method, then we describe time discretization of the problems under consideration and then we apply Chebyshev wavelets for space discretization. The performance of the method is shown by three test problems and obtained results compared with other results available in literature.