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Öğe An application of Chebyshev wavelet method for the nonlinear time fractional Schrodinger equation(Wiley, 2022) G., Esra Köse; Oruç, Ömer; Esen, AlaattinIn 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.Öğe A combination of Lie group-based high order geometric integrator and delta-shaped basis functions for solving Korteweg-de Vries (KdV) equation(World Scientific Publication, 2021) Polat, Murat; Oruç, ÖmerIn this work, we develop a novel method to obtain numerical solution of well-known Korteweg-de Vries (KdV) equation. In the novel method, we generate differentiation matrices for spatial derivatives of the KdV equation by using delta-shaped basis functions (DBFs). For temporal integration we use a high order geometric numerical integrator based on Lie group methods. This paper is a first attempt to combine DBFs and high order geometric numerical integrator for solving such a nonlinear partial differential equation (PDE) which preserves conservation laws. To demonstrate the performance of the proposed method we consider five test problems. We reckon L-infinity, L-2 and root mean square (RMS) errors and compare them with other results available in the literature. Besides the errors, we also monitor conservation laws of the KDV equation and we show that the method in this paper produces accurate results and preserves the conservation laws quite good. Numerical outcomes show that the present novel method is efficient and reliable for PDEs.Öğe A composite method based on delta-shaped basis functions and Lie group high-order geometric integrator for solving Kawahara-type equations(Wiley, 2023) Oruç, Ömer; Polat, Murat; 0000-0002-6655-3543; 0000-0003-1846-0817In this paper, we devise a novel method to solve Kawahara-type equations numerically. In this novel method, for spatial discretization, we use delta-shaped basis functions and generate differentiation matrices for spatial derivatives of the Kawahara-type equations. For discretization of temporal variable, we utilize a high-order geometric numerical integrator based on Lie group methods. For illustration of efficiency of the suggested method, we consider some test problems. We calculate errors and make some comparisons with other results that exist in literature. We also report changes in conservation laws during numerical simulations, and we indicate that the suggested method can preserve the conservation laws pretty good. Outcomes of numerical simulations indicate that the suggested method in this paper is reliable and effective for nonlinear partial differential equations (PDEs).Öğe Higher order Haar wavelet method integrated with strang splitting for solving regularized long wave equation(Elsevier, 2022) Bulut, Fatih; Oruç, Ömer; Esen, AlaattinIn 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.Öğe An IMEX approach assembled with Radial Basis Function-Finite Difference (RBF-FD) method for numerical solution of Zakharov-Kuznetsov Modified Equal Width (ZKMEW) equation with power law nonlinearity arising in wave phenomena(World Scientific Publ Co Pte Ltd, 2025) Oruç, ÖmerThis study deals with numerical solutions of Zakharov-Kuznetsov modified equal width (ZKMEW) equation with power law nonlinearity which is used in modeling of wave phenomena. The ZKMEW equation is a two-dimensional (2D) nonlinear partial differential equation and for numerical solution of it we first use an implicit-explicit (IMEX) backward differentiation formula for discretization of temporal variable and obtain a semi-discrete system. The IMEX approach treats nonlinear terms explicitly and linear terms implicitly. In this way, we avoid of solving nonlinear system of equations which is a big advantage in sense of computational load. Then space variables of the semi-discrete system are discretized via a local meshless radial basis function-finite difference (RBF-FD) method. For RBF-FD method we employ polyharmonic splines (PHS) which are free of shape parameters. One advantage of using RBF-FD method is its local property. Owing to the local property of the RBF-FD method sparse matrices are used which is an advantage in sense of execution time. Some numerical simulations are carried out and comparisons with the generalized finite difference method and space-time cloud method are performed. Stability of the proposed method is examined numerically. Obtained results verify efficiency and accuracy of the proposed method.Öğe Integrated Chebyshev wavelets for numerical solution of nonlinear one-dimensional and two-dimensional Rosenau equations(Elsevier, 2023) Oruç, ÖmerIn this study, we focus on developing an efficient computational method for one-dimensional and two-dimensional nonlinear Rosenau equations. Suggested computational method in this study uses finite differences for discretization of time variable. For discretization of space variables, firstly unknown function with highest derivative which is appeared in the Rosenau equation is expanded to Chebyshev wavelets and then by successive integrations the unknown function itself is found in terms of truncated Chebyshev wavelets. Also, a linearization technique is applied to handle the nonlinearity. By doing so we obtain a linear system of equations for solving the Rosenau equation. We apply the suggested method to four test problems and compare obtained numerical solutions with exact solutions and with finite element method to assess how accurate are the numerical results. The obtained numerical results endorse the efficiency and practicality of the suggested method for both one-dimensional and two-dimensional nonlinear Rosenau equations.Öğe A local meshfree radial point interpolation method for Berger equation arising in modelling of thin plates(Elsevier Inc., 2023) Oruç, ÖmerIn this study, two-dimensional (2D) Berger equation arising in deflection of thin plates is solved numerically. A local meshfree collocation technique based on radial point interpolation is developed for the Berger equation. As radial basis function thin plate splines are used to avoid selecting optimal shape parameter. The method is used to solve both linear and non-linear Berger equation on regular and irregular geometries. Obtained results during numerical simulations, are compared with different methods exist in literatrue such as method of fundamental solutions, local Kansa’s method and Pascal polynomial based meshless method. The comparisons affirm the accuracy and reliability of the suggested method.Öğe A local radial basis function-finite difference (RBF-FD) method for solving 1D and 2D coupled Schrödinger-Boussinesq (SBq) equations(Elsevier, 2021) Oruç, ÖmerIn this study, one-dimensional (1D) and two-dimensional (2D) coupled Schrodinger-Boussinesq (SBq) equations are examined numerically. A local meshless method based on radial basis function-finite difference (RBF-FD) method for spatial approximation is devised. We use polyharmonic splines as radial basis function along with augmented polynomials. By using polyharmonic splines we avoid to choose optimal shape parameter which requires special algorithms in meshless methods. For temporal discretization, low-storage ten-stage fourth-order explicit strong stability preserving Runge Kutta method is used which gives more flexibility on temporal step width. L-infinity and L-2 error norms are calculated to show accuracy of the proposed method. Further, conserved quantities are monitoried during numerical simulations to see how good the proposed method preserves them. Stability of the proposed method is dicussed numerically. Some codes are developed in Julia programming language to achieve more speed up in numerical simulations. Obtained results and their comparison with some studies such as wavelet, difference schemes and Fourier spectral methods available in literature verify the efficiency and reliability of the proposed method.Öğe Numerical simulation of two-dimensional and three-dimensional generalized Klein-Gordon-Zakharov equations with power law nonlinearity via a meshless collocation method based on barycentric rational interpolation(Wiley, 2022) Oruç, ÖmerThis study presents numerical simulations of generalized two-dimensional (2D) and three-dimensional (3D) Klein-Gordon-Zakharov (KGZ) equations with power law nonlinearity, which are coupled nonlinear partial differential equations. A meshless collocation method based on barycentric rational interpolation is developed for space variable of the KGZ equations. For time discretization, an explicit low storage fourth order Runge Kutta method is proposed after transforming KGZ equations to system of ordinary differential equations by introducing auxiliary variables. L-infinity and L-2 error norms for some test problems are computed. Obtained numerical results and comparisons with finite element methods indicate that barycentric rational interpolation method is an efficient method for solving multidimensional generalized KGZ system numerically.Öğe A radial basis function finite difference (RBF-FD) method for numerical simulation of interaction of high and low frequency waves: Zakharov-Rubenchik equations (vol 394, 125787, 2021)
(Elsevier Science INC., 2022) Oruç, ÖmerA radial basis function finite difference (RBF-FD) method for numerical simulation of interaction of high and low frequency waves: Zakharov-Rubenchik equationsÖğe A strong-form meshfree computational method for plane elastostatic equations of anisotropic functionally graded materials via multiple-scale Pascal polynomials(Elsevier Ltd., 2023) Oruç, ÖmerA strong-form meshfree method is proposed for solving plane elastostatic equations of anisotropic functionally graded materials. Any general function may be the grading function and it is changing smoothly from location to location in the material. The proposed method is based on Pascal polynomial basis and multiple-scale technique and it is a genuinely meshfree method since no numerical integrations over domains and meshing processes are required for considered problems. Implementation of the proposed method is straightforward and the method gives very accurate results. Stability of the solutions are examined numerically in occurrence of random noise. Some certain test problems with known exact solutions are solved both on regular and irregular geometries. Acquired solutions by the suggested method are compared with the exact solutions as well as with solutions of some existing numerical techniques in literature, such as boundary element, meshless local Petrov–Galerkin and radial basis function based meshless methods, to show accuracy of the proposed method.
