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    An accurate computational method for two-dimensional (2D) fractional Rayleigh-Stokes problem for a heated generalized second grade fluid via linear barycentric interpolation method
    (Pergamon-Elsevier Science Ltd, 2022) Oruc, Omer
    This paper deals with the development and analysis of an efficient method for the numerical solution of the two-dimensional (2D) fractional Rayleigh-Stokes problem for a heated generalized second grade fluid. The fractional time derivative that occurs in the problem is in the Riemann-Liouville sense and is discretized using a finite difference approach that is convergent with order of convergence O(tau(1+beta)) and unconditionally stable. After discretization of time variable, linear barycentric interpolation method is used to discretize space variables. In this way, a fully discrete scheme is obtained, which can be used to solve the problem numerically. The numerical solution is compared with exact solution to see how accurate it is. Also L-infinity error norm is reckoned and compared with those of other numerical methods in the literature to see performance of the method. Numerical simulations verify that the suggested method is accurate and workable for similar 2D fractional problems.
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    A generalized Gegenbauer wavelet collocation method for solving p-type fractional neutral delay differential and delay partial differential equations
    (Springer Heidelberg, 2022) Faheem, Mo; Khan, Arshad; Oruc, Omer
    In this work, we have investigated p-type fractional neutral delay differential equations (p-FNDDE) and p-type fractional neutral delay partial differential equations (p-FNDPDE) via generalized Gegenbauer wavelet. Generalized Gegenbauer scaling function fractional integral operator (GGSFIO) is constructed using the Riemann-Liouville definition of fractional integral to handle the fractional derivatives present in p-FNDDE and p-FNDPDE. The operation of Gegenbauer wavelet basis and GGSFIO to p-FNDDE and p-FNDPDE returns a system of equations which is later solved by Newton's method for unknown wavelet coefficients. With the help of these coefficients, we get the approximate solution. We have established the convergence analysis to assure the theoretical authenticity of the present method. The developed scheme is tested on several examples of p-FNDDE and p-FNDPDE to ensure computational convergence which validated the theoretical findings. The comparison of the numerical results of our method with the existing methods concludes the superiority of the proposed method.
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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.
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    A strong-form local meshless approach based on radial basis function-finite difference (RBF-FD) method for solving multi-dimensional coupled damped Schrodinger system appearing in Bose-Einstein condensates
    (Elsevier, 2022) Oruc, Omer
    In this work, one-dimensional (1D), two-dimensional (2D) and three-dimensional (3D) coupled damped Schrodinger system is solved numerically. A strong-form local meshless approach established on radial basis function-finite difference (RBF-FD) method for spatial approximation is developed. Polyharmonic splines are used as radial basis function with augmented polynomials. The use of the polyharmonic splines saves us from choosing an optimum shape parameter which is not a simple task for infinitely smooth RBFs such as multiquadrics or Gaussians. For time discretization classical fourth-order Runge Kutta method is utilized. L-infinity error norm and conserved quantities are computed to indicate performance of the proposed method. Stability of the proposed method is examined numerically. Some computer codes are devised in Julia programming language for obtaining numerical results. Acquired numerical results and their comparison with other studies available in literature such as cubic B-spline Galerkin method and direct meshless local Petrov-Galerkin (DMLPG) method endorse the performance and reliability of the proposed method. (C) 2021 Elsevier B.V. All rights reserved.