Browsing by Author "Konuralp A."
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Item Two-dimensional differential transform method, Adomian's decomposition method, and variational iteration method for partial differential equations(2006) Bildik N.; Konuralp A.The implementation of the two-dimensional differential transform method (DTM), Adomian's decomposition method (ADM), and the variational iteration method (VIM) in the mathematical applications of partial differential equations is examined in this paper. The VIM has been found to be particularly valuable as a tool for the solution of differential equations in engineering, science, and applied mathematics. The three methods are compared and it is shown that the VIM is more efficient and effective than the ADM and the DTM, and also converges to its exact solution more rapidly. Numerical solutions of two examples are calculated and the results are presented in tables and figures.Item Solution of different type of the partial differential equation by differential transform method and Adomian's decomposition method(2006) Bildik N.; Konuralp A.; Bek F.O.; Küçükarslan S.In this paper, the definitions and operations of the differential transform method [J.K. Zhou, Differential Transformation and Its Applications for Electrical Circuits, Huarjung University Press, Wuuhahn, China, 1986] and Adomian's decomposition method which is given by George Adomian for approximate solution of linear and non-linear differential equations are expressed [G. Adomian, Convergent series solution of nonlinear equation, Comput. Appl. Math. 11 (1984) 113-117]. Different partial differential equations are solved under the view of these methods and compared with the approximate solution and analytic solution. At the end, these solutions are illustrated by tables and figures. © 2005 Elsevier Inc. All rights reserved.Item The use of variational iteration method, differential transform method and adomian decomposition method for solving different types of nonlinear partial differential equations(Freund Publishing House Ltd, 2006) Bildik N.; Konuralp A.In this paper, the Variational Iteration Method (VIM), Differential Transform Method (DTM) and Adomian's Decomposition Method (ADM) are implemented to investigate different types of partial differential equations. The analytical solutions of nonlinear partial differential equations are obtained. On the other hand, comparison of the three methods shows that the variational iteration method is more powerful, reliable and effective than the other twos. Some examples are presented to further show the ability of the variational iteration method for nonlinear partial differential equations. © Freund Publishing House Ltd.Item The steady temperature distributions with different types of nonlinearities(2009) Konuralp A.The nonlinear two-point boundary value problems are solved and the steady temperature distributions in a rod are found by considering different types of the nonlinear parts of the problems, particularly in the polynomial, exponential and trigonometric forms. In this paper, with the aid of some transformations the variational iteration method's scheme is reproduced for the nonlinear problems including two-point boundary value problems. The illustrative related problems are solved by means of the method scheme. © 2009 Elsevier Ltd. All rights reserved.Item Numerical solution to the van der Pol equation with fractional damping(2009) Konuralp A.; Konuralp Ç.; Yildirim A.In this study, the van der Pol equation with fractional damping is investigated and the numerical solution of the problem is obtained by means of the variational iteration method. For this purpose, specific α values are considered and the emerged fractional differential equations are solved approximately. Furthermore, these solutions are compared and the relations between them are figured out. © 2009 The Royal Swedish Academy of Sciences.Item The approximate solution of steady temperature distribution in a rod: Two-point boundary value problem with higher order nonlinearity(2010) Konuralp A.In this paper, two-point boundary value problems have been solved by the well-known variational iteration method. Considering the situation in which the nonlinear part is a polynomial function with degree of ≥ 2, the steady temperature distribution in a rod has been computed. The strongly nonlinear differential equation has been become a reduced differential equation by the aid of a proper transformation and variational iteration method has been applied to the boundary value problem. © 2009 Elsevier Ltd. All rights reserved.Item Comparison of Legendre polynomial approximation and variational iteration method for the solutions of general linear Fredholm integro-differential equations(2010) Bildik N.; Konuralp A.; Yalçinbaş S.In this study it is shown that the numerical solutions of linear Fredholm integro-differential equations obtained by using Legendre polynomials can also be found by using the variational iteration method. Furthermore the numerical solutions of the given problems which are solved by the variational iteration method obviously converge rapidly to exact solutions better than the Legendre polynomial technique. Additionally, although the powerful effect of the applied processes in Legendre polynomial approach arises in the situations where the initial approximation value is unknown, it is shown by the examples that the variational iteration method produces more certain solutions where the first initial function approximation value is estimated. In this paper, the Legendre polynomial approximation (LPA) and the variational iteration method (VIM) are implemented to obtain the solutions of the linear Fredholm integro-differential equations and the numerical solutions with respect to these methods are compared. © 2009 Elsevier Ltd. All rights reserved.Item Variational Iteration Method for Volterra Functional Integrodifferential Equations with Vanishing Linear Delays(Hindawi Limited, 2014) Konuralp A.; Sorkun H.H.Application process of variational iteration method is presented in order to solve the Volterra functional integrodifferential equations which have multi terms and vanishing delays where the delay function θ(t) vanishes inside the integral limits such that θ(t)=qt for 0Item Fractional variational iteration method for time-fractional non-linear functional partial differential equation having proportional delays(Serbian Society of Heat Transfer Engineers, 2018) Dogan Durgun D.; Konuralp A.In this paper, time-fractional non-linear partial differential equation with proportional delays are solved by fractional variational iteration method taking into account modified Riemann-Liouville fractional derivative. The numerical solutions which are calculated by using this method are better than those obtained by homotopy perturbation method and differential transform method with same data set and approximation order. On the other hand, to improve the solutions obtained by fractional variational iteration method, residual error function is used. With this additional process, the resulting approximate solutions are getting closer to the exact solutions. The results obtained by taking into account different values of variables in the domain are supported by compared tables and graphics in detail. © 2018 Society of Thermal Engineers of Serbia.Item FAST APPROXIMATION OF ALGEBRAIC AND LOGARITHMIC HYPERSINGULAR TYPE SINGULAR INTEGRALS WITH HIGHLY OSCILLATORY KERNEL(Etamaths Publishing, 2020) Kayijuka I.; Ege S.M.; Konuralp A.; Topal F.S.Herein, highly oscillatory integrals with hypersingular type singularities are studied. After transforming the original integral into a sum of line integrals over a positive semi-infinite interval, a Gauss-related quadrature rule is constructed. The vehicle utilized is the moment's information. The comparison of two algorithms (Chebyshev and its modified one) to produce the recursion coefficients that satisfy orthogonal polynomial with respect to Gautschi logarithmic weight function, is investigated. Lastly, numerical examples are given to substantiate the effectiveness of the proposed method. © 2020, Etamaths Publishing. All rights reserved.Item Optimal perturbation iteration method for solving fractional model of damped burgers' equation(MDPI AG, 2020) Deniz S.; Konuralp A.; la Sen M.D.The newly constructed optimal perturbation iteration procedure with Laplace transform is applied to obtain the new approximate semi-analytical solutions of the fractional type of damped Burgers' equation. The classical damped Burgers' equation is remodeled to fractional differential form via the Atangana-Baleanu fractional derivatives described with the help of the Mittag-Leffler function. To display the efficiency of the proposed optimal perturbation iteration technique, an extended example is deeply analyzed. © 2020 by the authors.Item Numerical Solutions Based on a Collocation Method Combined with Euler Polynomials for Linear Fractional Differential Equations with Delay(De Gruyter Open Ltd, 2020) Konuralp A.; Öner S.In this study, a method combined with both Euler polynomials and the collocation method is proposed for solving linear fractional differential equations with delay. The proposed method yields an approximate series solution expressed in the truncated series form in which terms are constituted of unknown coefficients that are to be determined according to Euler polynomials. The matrix method developed for the linear fractional differential equations is improved to the case of having delay terms. Furthermore, while putting the effect of conditions into the algebraic system written in the augmented form in which the coefficients of Euler polynomials are unknowns, the condition matrix scans the rows one by one. Thus, by using our program written in Mathematica there can be obtained more than one semi-analytic solutions that approach to exact solutions. Some numerical examples are given to demonstrate the efficiency of the proposed method. © 2020 Walter de Gruyter GmbH, Berlin/Boston 2020.Item Fast gauss-related quadrature for highly oscillatory integrals with logarithm and cauchy-logarithmic type singularities(University of Alberta, 2021) Kayijuka I.; Ege S.M.; Konuralp A.; Topal F.S.This paper presents an efficient method for the computation of two highly oscillatory integrals having logarithmic and Cauchy-logarithmic singularities. This approach first requires the transformation of the original oscillatory integrals into a sum of line integrals with semi-infinite intervals. Afterwards, the coefficients of the three-term recurrence relation that satisfy the orthogonal polynomial are obtained by using the method based on moments, where classical Laguerre and Gautschi’s logarithmic weight functions are employed. The algorithm reveals that with fixed n, the method is capable of achieving significant figures within a short time. Furthermore, the approach yields higher accuracy as the frequency increases. The results of numerical experiments are given to substantiate our theoretical analysis. © 2021 Institute for Scientific Computing and Information.Item Reconstruction of potential function in inverse Sturm-Liouville problem via partial data(Balikesir University, 2021) Açil M.; Konuralp A.In this paper, three different uniqueness data are investigated to reconstruct the potential function in the Sturm-Liouville boundary value problem in the normal form. Taking account of Röhrl’s objective function, the steepest descent method is used in the computation of potential functions. To decrease the volume of computation, we propose a theorem to precalculate the minimization parameter that is required in the optimization. Further, we propose a novel time-saving algorithm in which the obligation of using the asymptotics of eigenvalues and eigenfunctions and the appropriateness of selected boundary conditions are also eliminated. As partial data, we take two spectra, the set of the jth elements of the infinite numbers of spectra obtained by changing boundary conditions in the problem, and one spectrum with the set of terminal velocities. In order to show the efficiency of the proposed method, numerical results are given for three test potentials which are smooth, nonsmooth continuous, and noncontinuous, respectively. © 2021 Balikesir University. All rights reserved.Item Clenshaw–Curtis algorithms for an efficient numerical approximation of singular and highly oscillatory Fourier transform integrals(Elsevier B.V., 2021) Kayijuka I.; Ege Ş.M.; Konuralp A.; Topal F.S.This paper investigates the implementation of Clenshaw–Curtis algorithms on singular and highly oscillatory integrals for efficient evaluation of the finite Fourier-type transform of integrands with endpoint singularities. In these methods, integrands are truncated by orthogonal polynomials and special function series term by term. Then their singularity types are computed using third and fourth-order homogeneous recurrence relations. The first approach reveals its efficiency for low, moderate and very high frequencies, whereas the second one, is more efficient for small values of frequencies. Moreover, all the results were found quite satisfactory. Algorithms and programming code in MATHEMATICA® 9.0 are provided for the implementation of methods for automatic computation on a computer. Lastly, illustrative numerical experiments and comparison of the proposed Clenshaw–Curtis algorithms to the steepest descent method are mentioned in support of our theoretical analysis in the examples section. © 2020 Elsevier B.V.Item AN EFFICIENT ALGORITHM FOR EVALUATION OF OSCILLATORY INTEGRALS HAVING CAUCHY AND JACOBI TYPE SINGULARITY KERNELS(Korean Society for Computational and Applied Mathematics, 2022) Kayijuka I.; Ege Ş.M.; Konuralp A.; Topal F.S.Herein, an algorithm for efficient evaluation of oscillatory Fourier-integrals with Jacobi-Cauchy type singularities is suggested. This method is based on the use of the traditional Clenshaw-Curtis (CC) algorithms in which the given function is approximated by the truncated Cheby-shev series, term by term, and the oscillatory factor is approximated by using Bessel function of the first kind. Subsequently, the modified moments are computed efficiently using the numerical steepest descent method or special functions. Furthermore, Algorithm and programming code in MATHEMATICA® 9.0 are provided for the implementation of the method for automatic computation on a computer. Finally, selected numerical ex-amples are given in support of our theoretical analysis. © 2022 KSCAM.Item Gegenbauer wavelet solutions of fractional integro-differential equations(Elsevier B.V., 2023) Özaltun G.; Konuralp A.; Gümgüm S.The aim of this study is to use Gegenbauer wavelets in the solution of fractional integro-differential equations. The method is applied to several problems with different values of resolution parameter and the degree of the truncated polynomial. The results are compared with those obtained from other numerical methods. We observe that the current method is very effective and gives accurate results. One of the reasons for that is it enables us to improve accuracy by increasing resolution parameter, while keeping the degree of polynomial fixed. Another reason is nonlinear terms do not require linearization. Hence the method can be directly implemented and results in the system of algebraic equations which solved by Wolfram Mathematica. It can be asserted that this is the first application of the Gegenbauer wavelet method to the aforementioned types of problems. © 2022 Elsevier B.V.