Browsing by Author "Saldir, Onur"

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An Effective Approach for Numerical Solution of Linear and Nonlinear Singular Boundary Value Problems
(Wiley, 2023) Saldir, Onur; Sakar, Mehmet Giyas
In this study, an effective approach is presented to obtain a numerical solution of linear and nonlinear singular boundary value problems. The proposed method is constructed by combining reproducing kernel and Legendre polynomials. Legendre basis functions are used to get the kernel function, and then the approximate solution is obtained as a finite series sum. Comparison of numerical results is made with the results obtained by other methods available in the literature. Furthermore, efficiency and accuracy of the method are demonstrated in tabulated results and plotted graphs. The numerical outcomes demonstrate that our method is very effective, applicable, and convenient.
A Finite Difference Method on Layer-Adapted Mesh for Singularly Perturbed Delay Differential Equations
(Walter de Gruyter Gmbh, 2020) Erdogan, Fevzi; Sakar, Mehmet Giyas; Saldir, Onur
The purpose of this paper is to present a uniform finite difference method for numerical solution of a initial value problem for semilinear second order singularly perturbed delay differential equation. A numerical method is constructed for this problem which involves appropriate piecewise-uniform Shishkin mesh on each time subinterval. The method is shown to uniformly convergent with respect to the perturbation parameter. A numerical experiment illustrate in practice the result of convergence proved theoretically.
Improving Variational Iteration Method With Auxiliary Parameter for Nonlinear Time-Fractional Partial Differential Equations
(Springer/plenum Publishers, 2017) Sakar, Mehmet Giyas; Saldir, Onur
In this research, we present a new approach based on variational iteration method for solving nonlinear time-fractional partial differential equations in large domains. The convergence of the method is shown with the aid of Banach fixed point theorem. The maximum error bound is specified. The optimal value of auxiliary parameter is obtained by use of residual error function. The fractional derivatives are taken in the Caputo sense. Numerical examples that involve the time-fractional Burgers equation, the time-fractional fifth-order Korteweg-de Vries equation and the time-fractional Fornberg-Whitham equation are examined to show the appropriate properties of the method. The results reveal that a new approach is very effective and convenient.
An Iterative Approximation for Time-Fractional Cahn-Allen Equation With Reproducing Kernel Method
(Springer Heidelberg, 2018) Sakar, Mehmet Giyas; Saldir, Onur; Erdogan, Fevzi
In this article, we construct a novel iterative approach that depends on reproducing kernel method for Cahn-Allen equation with Caputo derivative. Representation of solution and convergence analysis are presented theoretically. Numerical results are given as tables and graphics with intent to show efficiency and power of method. The results demonstrate that approximate solution uniformly converges to exact solution for Cahn-Allen equation with fractional derivative.
A New Reproducing Kernel Approach for Nonlinear Fractional Three-Point Boundary Value Problems
(Mdpi, 2020) Sakar, Mehmet Giyas; Saldir, Onur
In this article, a new reproducing kernel approach is developed for obtaining a numerical solution of multi-order fractional nonlinear three-point boundary value problems. This approach is based on a reproducing kernel, which is constructed by shifted Legendre polynomials (L-RKM). In the considered problem, fractional derivatives with respect to alpha and beta are defined in the Caputo sense. This method has been applied to some examples that have exact solutions. In order to show the robustness of the proposed method, some examples are solved and numerical results are given in tabulated forms.
A Novel Iterative Solution for Time-Fractional Boussinesq Equation by Reproducing Kernel Method
(Springer Heidelberg, 2020) Sakar, Mehmet Giyas; Saldir, Onur
In this study, iterative reproducing kernel method (RKM) will be applied in order to observe the effect of the method on numerical solutions of fractional order Boussinesq equation. Hilbert spaces and their kernel functions, linear operators and base functions which are necessary to obtain the reproducing kernel function are clearly explained. Iterative solution is constituted in a serial form by using reproducing kernel function. Then convergence of RKM solution is shown with lemma and theorem. Two problems, "good" Boussinesq and generalized Boussinesq equations, are examined by using RKM for different fractional values. Results are presented with tables and graphics.
A Novel Numerical Approach for the Third Order Emden-Fowler Type Equations
(Wiley-v C H verlag Gmbh, 2024) Sakar, Mehmet Giyas; Saldir, Onur; Aydin, Fatih; Rece, M. Yasin
This article aims to achieve robust numerical results by applying the Chebyshev reproducing kernel method without homogenizing the initial-boundary conditions of the Emden-Fowler (E-F) equation, thereby introducing a new perspective to the literature. A novel numerical approach is presented for solving the initial-boundary value problem of third-order E-F equations using Chebyshev reproducing kernel theory. Unlike previous applications, which were confined to homogeneous initial-boundary value problems or required homogenization, the proposed method is effective for both homogeneous and nonhomogeneous cases. To handle the initial-boundary conditions of the E-F equations, additional basis functions are introduced rather than imposing conditions on the reproducing kernel Hilbert space. The method's effectiveness is demonstrated through five examples, which validate the theoretical analysis. Overall, the results emphasize the method's efficiency.
A Novel Technique for Fractional Bagley-Torvik Equation
(Natl Acad Sciences india, 2019) Sakar, Mehmet Giyas; Saldir, Onur; Akgul, Ali
In this research, numerical solution of boundary value problem of fractional Bagley-Torvik equation is given in the reproducing kernel space. The central point of this approach is to set up a new reproducing kernel Hilbert space (RKHS) that satisfies the boundary conditions. Predicated on the properties of the RKHS, a new approach is applied to obtain precise numerical approximation. The results shows that a new approach is very effective and convenient for large interval.
Numerical Investigations To Design a Novel Model Based on the Fifth Order System of Emden-Fowler Equations
(Elsevier, 2020) Zulqurnain, System; Sakar, Mehmet Giyas; Yeskindirova, Manshuk; Saldir, Onur
The aim of the present study is to design a new fifth order system of Emden-Fowler equations and related four types of the model. The standard second order form of the Emden-Fowler has been used to obtain the new model. The shape factor that appear more than one time discussed in detail for every case of the designed model. The singularity at eta = 0 at one point or multiple points is also discussed at each type of the model. For validation and correctness of the new designed model, one example of each type based on system of fifth order Emden-Fowler equations are provided and numerical solutions of the designed equations of each type have been obtained by using variational iteration scheme. The comparison of the exact results and present numerical outcomes for solving one problem of each type is presented to check the accuracy of the designed model. (C) 2020 The Authors. Published by Elsevier Ltd on behalf of The Chinese Society of Theoretical and Applied Mechanics.
Numerical Solution of Fractional Integro-Differential Equations With Non-Local Conditions Using Reproducing Kernel Method Based on Chebyshev Polynomials
(Springer Heidelberg, 2025) Saldir, Onur
In this study, an improved version of the reproducing kernel method is presented, combined with shifted Chebyshev polynomials, to obtain numerical solutions of fractional order integro-differential equations with non-local conditions. One of the important steps of this method is the selection of a linear operator, and the selected linear operator consists of the fractional order derivative term of the considered problem. Another important step of this method is to use extra basis functions for non-homogeneous initial or boundary conditions of the considered problem and to define as many basis functions as the number of conditions of the problem. Although the construction of the approximate solution for both linear and non-linear cases of the considered problem is given separately, all numerical results are obtained by the iteration process. To show the robustness and efficiency of the method, the numerical results of six different problems considered in the literature are compared with alternative techniques, and also these results are presented by tables and graphs.
Numerical Solution of Fractional Order Burgers' Equation With Dirichlet and Neumann Boundary Conditions by Reproducing Kernel Method
(Mdpi, 2020) Saldir, Onur; Sakar, Mehmet Giyas; Erdogan, Fevzi
In this research, obtaining of approximate solution for fractional-order Burgers' equation will be presented in reproducing kernel Hilbert space (RKHS). Some special reproducing kernel spaces are identified according to inner products and norms. Then an iterative approach is constructed by using kernel functions. The convergence of this approach and its error estimates are given. The numerical algorithm of the method is presented. Furthermore, numerical outcomes are shown with tables and graphics for some examples. These outcomes demonstrate that the proposed method is convenient and effective.
Numerical Solution of Fractional Order Multi-Point Boundary Value Problems Using Reproducing Kernel Method With Shifted Legendre Polynomials
(Springer Int Publ Ag, 2025) Sakar, Mehmet Giyas; Saldir, Onur; Ata, Ayse
This article presents an efficient numerical method for obtaining numerical solutions to multi-point boundary problems with multi-fractional orders of Caputo derivatives.The novelty of the suggested method is based on the application of shifted Legendre polynomials in the reproducing kernel theory, as well as the use of extra basis functions for homogeneous or nonhomogeneous boundary conditions. The approximate solution for both linear and nonlinear cases of the considered problem is given in the form of a finite series sum. Convergence and error analysis of the presented method are given. The numerical outcomes obtained with the proposed method are compared with other results found in the literature. The obtained results for linear and nonlinear examples support the method's efficiency.
Numerical Solution of Time-Fractional Kawahara Equation Using Reproducing Kernel Method With Error Estimate
(Springer Heidelberg, 2019) Saldir, Onur; Sakar, Mehmet Giyas; Erdogan, Fevzi
We present a new approach depending on reproducing kernel method (RKM) for time-fractional Kawahara equation with variable coefficient. This approach consists of obtaining an orthonormal basis function on specific Hilbert spaces. In this regard, some special Hilbert spaces are defined. Kernel functions of these special spaces are given and basis functions are obtained. The approximate solution is attained as serial form. Convergence analysis, error estimation and stability analysis are presented after obtaining the approximate solution. To show the power and effect of the method, two examples are solved and the results are given as table and graphics. The results demonstrate that the presented method is very efficient and convenient for Kawahara equation with fractional order.