Browsing by Author "Sakar, Mehmet Giyas"

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Alternative Variational Iteration Method for Solving the Time-Fractional Fornberg-Whitham Equation
(Elsevier Science inc, 2015) Sakar, Mehmet Giyas; Ergoren, Hilmi
In this study, we apply an alternative variational iteration method to solve the time-fractional Fomberg-Whitham equation. A comparison between the results obtained using the alternative variational iteration method and known results obtained with the variational iteration method demonstrates that the proposed method is highly effective and convenient. The fractional derivatives are described in the Caputo sense. Numerical results are presented for different specific cases of the problem. (C) 2014 Elsevier Inc. All rights reserved.
Analytical Approximate Solutions of (n+1)-Dimensional Fractal Heat-Like and Wave-Like Equations
(Mdpi, 2017) Acan, Omer; Baleanu, Dumitru; Al Qurashi, Maysaa Mohamed; Sakar, Mehmet Giyas
In this paper, we propose a new type (n + 1)-dimensional reduced differential transform method (RDTM) based on a local fractional derivative (LFD) to solve (n + 1)-dimensional local fractional partial differential equations (PDEs) in Cantor sets. The presented method is named the (n + 1)-dimensional local fractional reduced differential transform method (LFRDTM). First the theories, their proofs and also some basic properties of this procedure are given. To understand the introduced method clearly, we apply it on the (n + 1)-dimensional fractal heat-like equations (HLEs) and wave-like equations (WLEs). The applications show that this new technique is efficient, simply applicable and has powerful effects in (n + 1)-dimensional local fractional problems.
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.
The Homotopy Analysis Method for Solving the Time-Fractional Fornberg-Whitham Equation and Comparison With Adomian's Decomposition Method
(Elsevier Science inc, 2013) Sakar, Mehmet Giyas; Erdogan, Fevzi
In this paper, we applied relatively new analytical techniques, the homotopy analysis method (HAM) and the Adomian's decomposition method (ADM) for solving time-fractional Fomberg-Whitham equation. The homotopy analysis method contains the auxiliary parameter, which provides us with a simple way to adjust and control the convergence region of solution series. The fractional derivatives are described in the Caputo sense. A comparison is made the between HAM and ADM results. The present methods performs extremely well in terms of efficiency and simplicity. Numerical results for different particular cases of the problem are presented. (C) 2013 Elsevier Inc. All rights reserved.
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.
Iterative Reproducing Kernel Hilbert Spaces Method for Riccati Differential Equations
(Elsevier, 2017) Sakar, Mehmet Giyas
This paper presents iterative reproducing kernel Hilbert spaces method (IRKHSM) to obtain the numerical solutions for Riccati differential equations with constant and variable coefficients. Representation of the exact solution is given in the W-2(2) [0, X] reproducing kernel space. Numerical solution of Riccati differential equations is acquired by interrupting the n-term of the exact solution. Also, the error of the numerical solution is monotone decreasing in terms of the norm of W-2(2)[0, X]. The outcomes from numerical examples show that the present iterative algorithm is very effective and convenient. (C) 2016 Elsevier B.V. All rights reserved.
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.
Novel Design of Morlet Wavelet Neural Network for Solving Second Order Lane-Emden Equation
(Elsevier, 2020) Sabir, Zulqurnain; Wahab, Hafiz Abdul; Umar, Muhammad; Sakar, Mehmet Giyas; Raja, Muhammad Asif Zahoor
In this study, a novel computational paradigm based on Morlet wavelet neural network (MWNN) optimized with integrated strength of genetic algorithm (GAs) and Interior-point algorithm (IPA) is presented for solving second order Lane-Emden equation (LEE). The solution of the LEE is performed by using modelling of the system with MWNNs aided with a hybrid combination of global search of GAs and an efficient local search of IPA. Three variants of the LEE have been numerically evaluated and their comparison with exact solutions demonstrates the correctness of the presented methodology. The statistical analyses are performed to establish the accuracy and convergence via the Theil's inequality coefficient, mean absolute deviation, and Nash Sutcliffe efficiency based metrics. (C) 2020 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.
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 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 Neutral Functional-Differential Equations With Proportional Delays
(2017) Sakar, Mehmet Giyas
In this paper, homotopy analysis method is improved with optimal determination of auxiliary parameter by use of residual error function for solving neutral functional-differential equations (NFDEs) with proportional delays. Convergence analysis and error estimate of method are given. Some numerical examples are solved and comparisons are made with the existing results. The numerical results show that the homotopy analysis method with residual error function is very effective and simple.
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.
Numerical Solution of Time-Fractional Nonlinear Pdes With Proportional Delays by Homotopy Perturbation Method
(Elsevier Science inc, 2016) Sakar, Mehmet Giyas; Uludag, Fatih; Erdogan, Fevzi
In this paper, homotopy perturbation method (HPM) is applied to solve fractional partial differential equations (PDEs) with proportional delay in t and shrinking in x. The method do not require linearization or small perturbation. The fractional derivatives are taken in the Caputo sense. The present method performs extremely well in terms of efficiency and simplicity. Numerical results for different particular cases of alpha are presented graphically. (C) 2016 Elsevier Inc. All rights reserved.
Numerical Solutions of Fractional Partial Differential Equations With Homotopy Analysis Method
(2019) Alkan, Aslı; Sakar, Mehmet Giyas
Bu yüksek lisans tezi 7 bölümden oluşmaktadır. Birinci bölümde, homotopi analiz metodunun tarihi ile literatür bildirişi verilmiştir. İkinci bölümde tez çalışmasında kullanılacak olan temel tanım, teorem ve ön bilgiler verilmiştir. Üçüncü bölümde, homotopi analiz metoduna ait bilgiler ve bu metodun gelişimi verilmiştir. Dördüncü bölümde Caputo anlamında kesir mertebeli türev içeren lineer olmayan Burgers denkleminin nümerik çözümleri Homotopi analiz metodu kullanılarak elde edilmiş ve bu probleme ait çizelge ve grafiklerle bölüm sonlandırılmıştır. Beşinci bölümde, Caputo anlamında kesir mertebeli türev içeren lineer olmayan Adveksiyon denkleminin nümerik çözümleri homotopi analiz metodu kullanılarak elde edilmiş ve bu probleme ait çizelge ve grafiklerle bölüm sonlandırılmıştır. Altıncı bölümde, Caputo anlamında kesir mertebeli türev içeren lineer ve lineer olmayan Klein-Gordon denklemlerinin nümerik çözümleri homotopi analiz metodu kullanılarak elde edilmiştir. Bu problemlere ait çizelge ve grafiklerle bölüm sonlandırılmıştır. Son bölüm ise, tez ile ilgili sonuç ve tartışma bölümünden oluşmaktadır.