Browsing by Author "Sezer, Mehmet"

Now showing 1 - 13 of 13

Citation - WoS: 12
Citation - Scopus: 10
An Advanced Method With Convergence Analysis for Solving Space-Time Fractional Partial Differential Equations With Multi Delays
(Springer Heidelberg, 2019) Kürkçü, ÖmÜr Kıvanç; Aslan, Ersin; Sezer, Mehmet
This study considers the space-time fractional partial differential equations with multi delays under a unique formulation, proposing a numerical method involving advanced matrix system. This matrix system is made up of the matching polynomial of complete graph together with fractional Caputo and Jumarie derivative types. Also, the derivative types are scrutinized to determine which of them is more proper for the method. Convergence analysis of the method is established via an average value of residual function using double integrals. The obtained solutions are improved with the aid of a residual error estimation. A general computer program module, which contains few steps, is developed. Tables and figures prove the efficiency and simplicity of the method. Eventually, an algorithm is given to illustrate the basis of the method.
Citation - WoS: 8
Citation - Scopus: 8
An Integrated Numerical Method With Error Analysis for Solving Fractional Differential Equations of Quintic Nonlinear Type Arising in Applied Sciences
(Wiley, 2019) Kürkçü, ÖmÜr Kıvanç; Aslan, Ersin; Sezer, Mehmet
In this study, fractional differential equations having quintic nonlinearity are considered by proposing an accurate numerical method based on the matching polynomial and matrix-collocation system. This method provides an integration between matrix and fractional derivative, which makes it fast and efficient. A hybrid computer program is designed by making use of the fast algorithmic structure of the method. An error analysis technique consisting of the fractional-based residual function is constructed to scrutinize the precision of the method. Some error tests are also performed. Figures and tables present the consistency of the approximate solutions of highly stiff model problems. All results point out that the method is effective, simple, and eligible.
Citation - WoS: 10
Citation - Scopus: 11
An Inventive Numerical Method for Solving the Most General Form of Integro-Differential Equations With Functional Delays and Characteristic Behavior of Orthoexponential Residual Function
(Springer Heidelberg, 2019) Kürkçü, Ömür Kıvanç; Aslan, Ersin; Sezer, Mehmet
In this study, we constitute the most general form of functional integro-differential equations with functional delays. An inventive method based on Dickson polynomials with the parameter- along with collocation points is employed to solve them. The stability of the solutions is simulated according to an interval of the parameter-. A useful computer program is developed to obtain the precise values from the method. The residual error analysis is used to improve the obtained solutions. The characteristic behavior of the residual function is established with the aid of the orthoexponential polynomials. We compare the present numerical results of the method with those obtained by the existing methods in tables.
The Legendre Matrix-Collocation Approach for Some Nonlinear Differential Equations Arising in Physics and Mechanics
(2019) Kürkçü, Ömür Kıvanç; Dönmez Demir, Duygu; Sezer, Mehmet; Çınardalı, Tuğçe
In this study, the Legendre operational matrix method based on collocation points is introduced to solve high order ordinary differentialequations with some nonlinear terms arising in physics and mechanics. This technique transforms the nonlinear differential equationinto a matrix equation with unknown Legendre coefficients via mixed conditions. This solution of this matrix equation yields theLegendre coefficients of the solution function. Thus, the approximate solution is obtained in terms of Legendre polynomials. Some testproblems together with residual error estimation are given to show the usefulness and applicability of the method and the numericalresults are compared.
Lucas Polynomial Approach for Second Order Nonlinear Differential Equations
(2020) Gümgüm, Sevin; Kürkçü, Ömür Kıvanç; Sezer, Mehmet; Bayku S Sava Saner Il, Nurcan
This paper presents the Lucas polynomial solution of second-order nonlinearordinary differential equations with mixed conditions. Lucas matrix method is based oncollocation points together with truncated Lucas series. The main advantage of the methodis that it has a simple structure to deal with the nonlinear algebraic system obtained frommatrix relations. The method is applied to four problems. In the first two problems, exactsolutions are obtained. The last two problems, Bratu and Duffing equations are solvednumerically; the results are compared with the exact solutions and some other numericalsolutions. It is observed that the application of the method results in either the exact oraccurate numerical solutions.
Citation - WoS: 17
Citation - Scopus: 20
Lucas Polynomial Solution of Nonlinear Differential Equations With Variable Delays
(Hacettepe Univ, Fac Sci, 2020) Gumgum, Sevin; Savasaneril, Nurcan Baykus; Kürkçü, ÖmÜr Kıvanç; Sezer, Mehmet
In this study, a novel matrix method based on Lucas series and collocation points has been used to solve nonlinear differential equations with variable delays. The application of the method converts the nonlinear equation to a matrix equation which corresponds to a system of nonlinear algebraic equations with unknown Lucas coefficients. The method is tested on three problems to show that it allows both analytical and approximate solutions.
Citation - WoS: 9
Morgan-Voyce Matrix Method for Generalized Functional Integro-Differential Equations of Volterra-Type
(Editura Bibliotheca-Bibliotheca Publ House, 2019) Ozel, Mustafa; Kürkçü, ÖmÜr Kıvanç; Sezer, Mehmet
This study deals with the generalized linear Volterra-type functional integro-differential equations with mixed delays. A combination between matrix-collocation method and Morgan-Voyce polynomials is developed to solve these type equations. In addition, an error analysis technique is given to improve the obtained solutions. Numerical examples are performed to confirm the efficiency and validity of the method. The comparisons are made in tables and figures. The discussions show that the method is fast and precise.
Citation - WoS: 11
Citation - Scopus: 11
A Novel Graph-Operational Matrix Method for Solving Multidelay Fractional Differential Equations With Variable Coefficients and a Numerical Comparative Survey of Fractional Derivative Types
(Scientific Technical Research Council Turkey-Tubitak, 2019) Kürkçü, ÖmÜr Kıvanç; Aslan, Ersin; Sezer, Mehmet
In this study, we introduce multidelay fractional differential equations with variable coefficients in a unique formula. A novel graph-operational matrix method based on the fractional Caputo, Riemann-Liouville, Caputo-Fabrizio, and Jumarie derivative types is developed to efficiently solve them. We also make use of the collocation points and matrix relations of the matching polynomial of the complete graph in the method. We determine which of the fractional derivative types is more appropriate for the method. The solutions of model problems are improved via a new residual error analysis technique. We design a general computer program module. Thus, we can explicitly monitor the usefulness of the method. All results are scrutinized in tables and figures. Finally, an illustrative algorithm is presented.
Citation - WoS: 4
Citation - Scopus: 3
A Novel Hybrid Method for Solving Combined Functional Neutral Differential Equations With Several Delays and Investigation of Convergence Rate Via Residual Function
(Univ Tabriz, 2019) Kürkçü, ÖmÜr Kıvanç; Aslan, Ersin; Sezer, Mehmet
In this study, we introduce a novel hybrid method based on a simple graph along with operational matrix to solve the combined functional neutral differential equations with several delays. The matrix relations of the matching polynomials of complete and path graphs are employed in the matrix-collocation method. We improve the obtained solutions via an error analysis technique. The oscillation of them on time interval is also estimated by coupling the method with Laplace-Pade technique. We develop a general computer program and so we can efficiently monitor the precision of the method. We investigate a convergence rate of the method by constructing a formula based on the residual function. Eventually, an algorithm is described to show the easiness of the method.
Citation - WoS: 2
A Numerical Approach for Solving Pantograph-Type Functional Differential Equations With Mixed Delays Using Dickson Polynomials of the Second Kind
(Editura Bibliotheca-Bibliotheca Publ House, 2018) Yildizhan, Iclal; Kürkçü, ÖmÜr Kıvanç; Sezer, Mehmet
In this study, a hybrid matrix-collocation method based on Dickson polynomials of the second kind along with Taylor polynomials is proposed to solve pantograph type functional differential equations with mixed delays under the initial conditions. The parameter-alpha in Dickson polynomials is interpreted for obtaining the optimum solutions. An error estimation related with the residual function and the mean-value theorem is implemented and also some illustrative examples are presented. It is observed that the proposed method is easy to be applied.
Citation - WoS: 18
Citation - Scopus: 17
A Numerical Approach Technique for Solving Generalized Delay Integro-Differential Equations With Functional Bounds by Means of Dickson Polynomials
(World Scientific Publ Co Pte Ltd, 2018) Kürkçü, ÖmÜr Kıvanç; Aslan, Ersin; Sezer, Mehmet; Ilhan, Ozgul
In this study, we have considered the linear classes of differential-(difference), integro-differential-(difference) and integral equations by constituting a generalized form, which contains proportional delay, difference, differentiable difference or delay. To solve the generalized form numerically, we use the efficient matrix technique based on Dickson polynomials with the parameter-a along with the collocation points. We also encode the useful computer program for susceptibility of the technique. The residual error analysis is implemented by using the residual function. The consistency of the technique is analyzed. Also, the numerical results illustrated in tables and figures are compared.
A Numerical Technique Based on Lucas Polynomials Together With Standard and Chebyshev-Lobatto Collocation Points for Solving Functional Integro-Differential Equations Involving Variable Delays
(2018) Gümgüm, Sevin; Sezer, Mehmet; Savaşaneril, Nurcan Baykuş; Kürkçü, Ömür Kıvanç
In this paper, a new numerical matrix-collocation technique is considered to solve functional integrodifferentialequations involving variable delays under the initial conditions. This technique is basedessentially on Lucas polynomials together with standard and Chebyshev-Lobatto collocation points. Somedescriptive examples are performed to observe the practicability of the technique and the residual erroranalysis is employed to improve the obtained solutions. Also, the numerical results obtained by using thesecollocation points are compared in tables and figures.
Citation - WoS: 5
Citation - Scopus: 7
On the Numerical Solution of Fractional Differential Equations With Cubic Nonlinearity Via Matching Polynomial of Complete Graph
(Springer India, 2019) Kürkçü, Ömür Kıvanç; Aslan, ErsIn; Sezer, Mehmet
This study deals with a generalized form of fractional differential equations with cubic nonlinearity, employing a matrix-collocation method dependent on the matching polynomial of complete graph. The method presents a simple and efficient algorithmic infrastructure, which contains a unified matrix expansion of fractional-order derivatives and a general matrix relation for cubic nonlinearity. The method also performs a sustainable approximation for high value of computation limit, thanks to the inclusion of the matching polynomial in matrix system. Using the residual function, the convergence and error estimation are investigated via the second mean value theorem having a weight function. In comparison with the existing results, highly accurate results are obtained. Moreover, the oscillatory solutions of some model problems arising in several applied sciences are simulated. It is verified that the proposed method is reliable, efficient and productive.