Browsing by Author "Sakar, M.G."

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Boundary Value Problems for Impulsive Fractional Differential Equations With Nonlocal Conditions
(Springer New York LLC, 2013) Ergören, H.; Sakar, M.G.
In this study, we discuss some existence results for the solutions to impulsive fractional differential equations with nonlocal conditions by using contraction mapping principle and Krasnoselskii's fixed point theorem. © Springer Science+Business Media New York 2013.
Finite and Infinite Dimensional Reproducing Kernel Hilbert Space Approach for Bagley–torvik Equation
(Springer, 2025) Ata, A.; Sakar, M.G.; Saldır, O.; Şenol, M.
In this paper, two different numerical approaches are presented in finite dimensional and infinite dimensional reproducing kernel Hilbert spaces for the fractional order Bagley–Torvik equation with boundary conditions. The reproducing kernel functions are obtained in finite dimensional Hilbert space Πρn[0,A] using Legendre polynomials, while they are obtained by a known classical method in infinite dimensional Sobolev-Hilbert space W23[0,A]. A comprehensive theoretical analysis is given for both approaches, which have different forms of reproducing kernel methods. Numerical results are calculated over wide intervals with both proposed approaches. In order to compare the efficiency of these proposed methods, the numerical results obtained for the considered six examples are presented through tabulated data and graphical representations. © The Author(s), under exclusive licence to Springer Nature India Private Limited 2024.
A Hybrid Method for Singularly Perturbed Convection–diffusion Equation
(Springer, 2019) Sakar, M.G.; Saldır, O.; Erdogan, F.
In this research, we present a hybrid method which combination of simplified reproducing kernel method (SRKM) and asymptotic expansion for solving singularly perturbed convection–diffusion problems. According to the hybrid method, firstly asymptotic expansion formed on boundary layer domain and then terminal value problem solved via SRKM on regular domain. To apply the SRKM, some special Hilbert spaces are defined and related reproducing kernel functions are given. Then, the approximate solution is obtained as serial form. A new algorithm of SRKM is presented for nonlinear problem. Also, convergence analysis of approximate solution is given. Some linear and nonlinear problems are solved by hybrid method. The obtained outcomes indicate that the hybrid method is reliable and very effective for solving singularly perturbed convection–diffusion equation. © 2019, Springer Nature India Private Limited.
Numerical Solution of Fractional Bratu Type Equations With Legendre Reproducing Kernel Method
(Springer, 2018) Sakar, M.G.; Saldır, O.; Akgül, A.
In this research, a new numerical method is proposed for solving fractional Bratu type boundary value problems. Fractional derivatives are taken in Caputo sense. This method is predicated on iterative approach of reproducing kernel Hilbert space theory with shifted Legendre polynomials. Construction of iterative process is shown by orthogonal projection operator. Numerical results show that our method is effective and convenient for fractional Bratu type problem. © 2018, Springer Nature India Private Limited.