Browsing by Author "Uncu, Sevket"

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An Efficient Method for Solving Second-Order Delay Differential Equation
(Univ Miskolc inst Math, 2021) Cimen, Erkan; Uncu, Sevket
In this paper, the initial-value problem for a linear second order delay differential equation is considered. To solve this problem numerically, an appropriate difference scheme is constructed by using the method of integral identities which contains basis functions and interpolating quadrature rules with weight and remainder term in integral form. Besides, the method is proved to be first-order convergent in discrete maximum norm. The numerical illustration provided support the theoretical results. Finally, the proposed method is compared with the im
A Novel Approximation on the Solution of Systems of Ordinary Differential Equations
(Rgn Publ, 2024) Uncu, Sevket; Cimen, Erkan
In this paper, the initial-value problem for the system of first-order differential equations is considered. To solve this problem, we construct a fitted difference scheme using the finite difference method, which is based on integral identities for the quadrature formula with integral term remainder terms. Next, we prove first-order convergence for the method in the discrete maximum norm. Although this scheme has the same rate of convergence, it has more efficiency and accuracy compared to the classical Euler scheme. Two test problems are solved by using the proposed method and the classical Euler method, which confirm the theoretical findings. The numerical results obtained from here show that the proposed method is reliable, efficient, and accurate.
On the Solution of the Delay Differential Equation Via Laplace Transform
(Rgn Publ, 2020) Cimen, Erkan; Uncu, Sevket
In this paper, we consider the initial-value problem for a linear second order delay differential equation. We use Laplace transform method for solving this problem. Furthermore, we present examples provided support the theoretical results.
Uniformly Convergent Method for Two Coupled Nonlinear Singularly Perturbed Systems Arising in Chemical Kinetics
(Belgian Mathematical Soc Triomphe, 2024) Cimen, Erkan; Uncu, Sevket
The initial value problem for the nonlinear system of singularly perturbed differential equations, which emerges as a model for chemical kinetics, is considered. In order to solve this problem numerically, a novel fitted difference scheme is constructed by the finite difference method on non-uniform meshes, like the Shishkin mesh and the Bakhvalov mesh, using quadrature rules with the remaining terms in integral form. The scheme is proven to achieve almost first-order convergence in the discrete maximum norm on the Shishkin mesh and first-order convergence on the Bakhvalov mesh. Two numerical examples are considered to illustrate the accuracy and performance of the method. In order to show the advantage of the proposed method we compare our results with those obtained by an implicit linear difference scheme. Comparison shows that the proposed method is fast convergent and highly accurate.