Browsing by Author "Sezer, Mehmet"

Now showing 1 - 7 of 7

Citation - WoS: 1
Citation - Scopus: 1
An Accurate and Novel Numerical Simulation With Convergence Analysis for Nonlinear Partial Differential Equations of Burgers-Fisher Type Arising in Applied Sciences
(Walter De Gruyter Gmbh, 2022) Kürkçü, Ömür Kuvanc; Sezer, Mehmet
In this study, the second-order nonlinear partial differential equations of Burgers-Fisher type are considered under a unique formulation by introducing a novel highly accurate numerical method based on the Norlund rational polynomial and matrix-collocation computational system. The method aims to obtain a sustainable approach since it contains the rational structure of the Norlund polynomial. A unique computer program module, which involves very few routines, is constructed to discuss the precision and efficiency of the method and these few steps are described via an algorithm. A residual function is employed in both the error and convergence analyses with mean value theorem for double integrals. The considered equations in the numerical tests stand for model phenomena arising widely in applied sciences. Graphical and numerical comparisons provide a clear observation about the consistency of the method. All results prove that the method is highly accurate, eligible, and provides the ultimate operation for aforementioned problems.
Citation - WoS: 6
Citation - Scopus: 8
A Directly Convergent Numerical Method Based on Orthoexponential Polynomials for Solving Integro-Differential Equations With Variable Coefficients and Infinite Boundary on Half-Line
(ELSEVIER, 2021) Kürkçü, Ömür Kıvanç; Sezer, Mehmet
In this study, main concern is focused on numerically solving the integro-differentialdelay equations with variable coefficients and infinite boundary on half-line, proposing a matrix-collocation method based on the orthoexponential polynomials. The method is equipped with the collocation points and the hybridized matrix relations between the orthoexponential and Taylor polynomials, which enable us to convert an integral form with infinite boundary into a mathematical formulation. The method also directly establishes the verification of the existence and uniqueness of this integral form through a convergent result. In order to observe the validity of the method versus its computation limit, an error bound analysis is performed by using the upper bound of the orthoexponential polynomials. A computer module containing main infrastructure of the method is specifically designed and run for providing highly precise results. Thus, the numerical and graphical implementations are completely monitored in table and figures, respectively. Based on the comparisons and findings, one can state that the method is remarkable, dependable, and accurate for approaching the aforementioned equations. (C) 2020 Elsevier B.V. All rights reserved.
Citation - WoS: 10
Citation - Scopus: 12
A Fast Numerical Method for Fractional Partial Integro-Differential Equations With Spatial-Time Delays
(ELSEVIER, 2021) Aslan, Ersin; Kürkçü, Ömür Kıvanç; Sezer, Mehmet
This study aims to efficiently solve the space-time fractional partial integro-differential equations with spatial-time delays, employing a fast numerical methodology dependent upon the matching polynomial of complete graph and matrix-collocation procedure. This methodology provides a sustainable approach for each computation limit since it arises from the durable graph structure of complete graph and fractional matrix relations. The convergence analysis is established using the residual function of mean value theorem for double integrals. An error estimation is also implemented. All computations are performed with the aid of a unique computer program, which returns the desired results in seconds. Some specific numerical problems are tested to discuss the applicability of the method in tables and figures. It is stated that the method stands for fast, simple and highly accurate computation. (c) 2020 IMACS. Published by Elsevier B.V. All rights reserved.
A Generic Numerical Method for Treating a System of Volterra Integro-Differential Equations With Multiple Delays and Variable Bounds
(Emerald Group Publishing Ltd, 2024) Kürkçü, Ömür Kıvanç; Sezer, Mehmet
PurposeThis study aims to treat a novel system of Volterra integro-differential equations with multiple delays and variable bounds, constituting a generic numerical method based on the matrix equation and a combinatoric-parametric Charlier polynomials. The proposed method utilizes these polynomials for the matrix relations at the collocation points.Design/methodology/approachThanks to the combinatorial eligibility of the method, the functional terms can be transformed into the generic matrix relations with low dimensions, and their resulting matrix equation. The obtained solutions are tested with regard to the parametric behaviour of the polynomials with $\alpha$, taking into account the condition number of an outcome matrix of the method. Residual error estimation improves those solutions without using any external method. A calculation of the residual error bound is also fulfilled.FindingsAll computations are carried out by a special programming module. The accuracy and productivity of the method are scrutinized via numerical and graphical results. Based on the discussions, one can point out that the method is very proper to solve a system in question.Originality/valueThis paper introduces a generic computational numerical method containing the matrix expansions of the combinatoric Charlier polynomials, in order to treat the system of Volterra integro-differential equations with multiple delays and variable bounds. Thus, the method enables to evaluate stiff differential and integral parts of the system in question. That is, these parts generates two novel components in terms of unknown terms with both differentiated and delay arguments. A rigorous error analysis is deployed via the residual function. Four benchmark problems are solved and interpreted. Their graphical and numerical results validate accuracy and efficiency of the proposed method. In fact, a generic method is, thereby, provided into the literature.
Citation - WoS: 5
Citation - Scopus: 5
A Matched Hermite-Taylor Matrix Method To Solve the Combined Partial Integro-Differential Equations Having Nonlinearity and Delay Terms
(SPRINGER HEIDELBERG, 2020) Yalçın, Elif; Kürkçü, Ömür Kıvanç; Sezer, Mehmet
In this study, a matched numerical method based on Hermite and Taylor matrix-collocation techniques is developed to obtain the numerical solutions of a combination of the partial integro-differential equations (PIDEs) under Dirichlet boundary conditions, which involve the nonlinearity, delay and Volterra integral terms. These type equations govern wide variety applications in physical sense. The present method easily constitutes the matrix relations of the linear and nonlinear terms in a considered PIDE, using the eligibilities of the Hermite and Taylor polynomials. It thus directly produces a polynomial solution by eliminating a matrix system of nonlinear algebraic functions gathered from the matrix relations. Besides, the validity and precision of the method are tested on stiff examples by fulfilling several error computations. One can state that the method is fast, validate and productive according to the numerical and graphical results
Citation - WoS: 2
Citation - Scopus: 2
A New Characteristic Numerical Approach With Evolutionary Residual Error Analysis To Nonlinear Boundary Value Problems Occurring in Heat and Mass Transfer Via Combinatoric Mittag-Leffler Polynomial
(Taylor & Francis Inc, 2022) Kürkçü, Ömür Kıvanç; Sezer, Mehmet
This study focuses on new numerical approach to the solutions of nonlinear boundary value problems occurring in heat and mass transfer, constructing a matrix-combinatorial method collocated by the Chebyshev-Lobatto points and based on the Mittag-Leffler polynomial. For the first time, a matrix-collocation method is coupled with a combinatoric polynomial. In view of this combination, the method converts the linear and nonlinear terms to the matrix forms and then gathers them to a fundamental matrix equation. In addition to the novelty, an inventive nonlinear residual error analysis of general type is firstly theorized and adapted for improving the solutions to the problems in question and also, it allows to regard the nonlinear terms as an operator in calculations. The obtained solutions are thereby corrected. Numerical and graphical illustrations are provided to scrutinize the accuracy, productivity and comparability of the method. Upon evaluations of all these tasks, one can admit that the method is comprehensible, consistent and easily programmable.
Citation - WoS: 12
Citation - Scopus: 13
Pell-Lucas Series Approach for a Class of Fredholm-Type Delay Integro-Differential Equations With Variable Delays
(SPRINGER HEIDELBERG, 2021) Demir, Duygu Dönmez; Lukonde, Alpha Peter; Kürkçü, Ömür Kıvanç; Sezer, Mehmet
In this study, a Pell-Lucas matrix-collocation method is used to solve a class of Fredholm-type delay integro-differential equations with variable delays under initial conditions. The method involves the basic matrix structures gained from the expansions of the functions at collocation points. Therefore, it performs direct and immediate computation. To test its advantage on the applications, some numerical examples are evaluated. These examples show that the method enables highly accurate solutions and approximations. Besides, the accuracy of the solutions and the validity of the method are checked via the residual error analysis and the upper bound error, respectively. Finally, the numerical results, such as errors and computation time, are compared in the tables and figures.