Bilgisayar Bilimleri Fakültesi

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APPLICATION OF THE COLLOCATION METHOD WITH B-SPLINES TO THE GEW EQUATION
(KENT STATE UNIVERSITYETNA, DEPT MATHEMATICS & COMPUTER SCIENCE, KENT, OH 44242-0001, 2017) Zeybek, Halil; Karakoc, S. Battal Gazi; AGÜ, Mühendislik Fakültesi, Bilgisayar Mühendisliği Bölümü
In this paper, the generalized equal width (GEW) wave equation is solved numerically by using a quintic B-spline collocation algorithm with two different linearization techniques. Also, a linear stability analysis of the numerical scheme based on the von Neumann method is investigated. The numerical algorithm is applied to three test problems consisting of a single solitary wave, the interaction of two solitary waves, and a Maxwellian initial condition. In order to determine the performance of the numerical method, we compute the error in the L-2- and L-infinity- norms and in the invariants I-1, I-2, and I-3 of the GEW equation. These calculations are compared with earlier studies. Afterwards, the motion of solitary waves according to different parameters is designed.
A collocation algorithm based on quintic B-splines for the solitary wave simulation of the GRLW equation
(SHARIF UNIV TECHNOLOGY, PO BOX 11155-8639, TEHRAN, 00000, IRAN, 2019) Zeybek, H.; Karakoc, S. Battal Gazi; AGÜ, Mühendislik Fakültesi, Bilgisayar Mühendisliği Bölümü
In this article, a collocation algorithm based on quintic B-splines is proposed to find a numerical solution to the nonlinear Generalized Regularized Long Wave (GRLW) equation. Moreover, to analyze the linear stability of the numerical scheme, the von-Neumann technique is used. The numerical approach to three test examples consisting of a single solitary wave, the collision of two solitary waves, and the growth of an undular bore is discussed. The accuracy of the method is demonstrated by calculating the error in L-2 and L-infinity norms and the conservative quantities l(1) , l(2) and l(3). The findings are compared with those previously reported in the literature. Finally, the motion of solitary waves is graphically plotted according to different parameters. (C) 2019 Sharif University of Technology. All rights reserved.
Comparative assessment of smooth and non-smooth optimization solvers in HANSO software
(Balikesir University, 2022) Tor, Ali Hakan; AGÜ, Mühendislik Fakültesi, Mühendislik Bilimleri Bölümü; Tor, Ali Hakan
The aim of this study is to compare the performance of smooth and nonsmooth optimization solvers from HANSO (Hybrid Algorithm for Nonsmooth Optimization) software. The smooth optimization solver is the implementation of the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method and the nonsmooth optimization solver is the Hybrid Algorithm for Nonsmooth Optimization. More precisely, the nonsmooth optimization algorithm is the combination of the BFGS and the Gradient Sampling Algorithm (GSA). We use well-known collection of academic test problems for nonsmooth optimization containing both convex and nonconvex problems. The motivation for this research is the importance of the comparative assessment of smooth optimization methods for solving nonsmooth optimization problems. This assessment will demonstrate how successful is the BFGS method for solving nonsmooth optimization problems in comparison with the nonsmooth optimization solver from HANSO. Performance profiles using the number iterations, the number of function evaluations and the number of subgradient evaluations are used to compare solvers. © 2021 Balikesir University. All rights reserved.
A cubic B-spline Galerkin approach for the numerical simulation of the GEW equation
(International Academic Press, 2016) Battal Gazi Karakoç S.; Zeybek, Halil; AGÜ; Zeybek, Halil
The generalized equal width (GEW) wave equation is solved numerically by using lumped Galerkin approach with cubic B-spline functions. The proposed numerical scheme is tested by applying two test problems including single solitary wave and interaction of two solitary waves. In order to determine the performance of the algorithm, the error norms L2 and L∞ and the invariants I1, I2 and I3 are calculated. For the linear stability analysis of the numerical algorithm, von Neumann approach is used. As a result, the obtained findings show that the presented numerical scheme is preferable to some recent numerical methods.
Eigenvalue problems for nonlinear third-order m-point p-Laplacian dynamic equations on time scales
(WILEY111 RIVER ST, HOBOKEN 07030-5774, NJ, 2016) Dogan, Abdulkadir; 0000-0002-7532-1920; AGÜ, Mühendislik Fakültesi, Bilgisayar Mühendisliği Bölümü; Dogan, Abdulkadir
This work deals with the existence and uniqueness of a nontrivial solution for the third-order p-Laplacian m-point eigenvalue problems on time scales. We find several sufficient conditions of the existence and uniqueness of nontrivial solution of eigenvalue problems when lambda is in some interval. The proofs are based on the nonlinear alternative of Leray-Schauder. To illustrate the results, some examples are included. Copyright (C) 2014 John Wiley & Sons, Ltd.
EIGENVALUE PROBLEMS FOR SINGULAR MULTI-POINT DYNAMIC EQUATIONS ON TIME SCALES
(TEXAS STATE UNIV, 601 UNIVERSTITY DRIVE, SAN MARCOS, TX 78666 USA, 2017) Dogan, Abdulkadir; AGÜ, Mühendislik Fakültesi, Bilgisayar Mühendisliği Bölümü; Dogan, Abdulkadir
In this article, we study a singular multi-point dynamic eigenvalue problem on time scales. We find existence of positive solutions by constructing the Green's function and studying its positivity eigenvalue intervals. Two examples are given to illustrate our results.
Existence of Countably Many Positive Solutions for Nonlinear Boundary Value Problems on Time Scales
(NATURAL SCIENCES PUBLISHING CORP-NSPPO BOX 32038, ISA TOWN 00000, BAHRAIN, 2014) Dogan, Abdulkadir; AGÜ, Mühendislik Fakültesi, Bilgisayar Mühendisliği Bölümü; Dogan
In this paper, we consider the existence of countably many positive solutions for nonlinear singular boundary value problem on time scales. By using the fixed-point index theory and a new fixed-point theorem in cones, the sufficient conditions for the existence of countably many positive solutions are established.
Existence of multiple positive solutions for p-Laplacian multipoint boundary value problems on time scales
(SPRINGEROPEN, CAMPUS, 4 CRINAN ST, LONDON, N1 9XW, ENGLAND, 2020) Dogan, Abdulkadir; AGÜ, Mühendislik Fakültesi, Bilgisayar Mühendisliği Bölümü; Dogan, Abdulkadir
In this paper, we consider p-Laplacian multipoint boundary value problems on time scales. By using a generalization of the Leggett-Williams fixed point theorem due to Bai and Ge, we prove that a boundary value problem has at least three positive solutions. Moreover, we study existence of positive solutions of a multipoint boundary value problem for an increasing homeomorphism and homomorphism on time scales. By using fixed point index theory, sufficient conditions for the existence of at least two positive solutions are provided. Examples are given to illustrate the results.
THE EXISTENCE OF POSITIVE SOLUTIONS FOR A SEMIPOSITONE SECOND-ORDER m-POINT BOUNDARY VALUE PROBLEM
(DYNAMIC PUBLISHERS, INC, 2015) Dogan, Abdulkadir; AGÜ; Dogan, Abdulkadir
In this paper, we study the existence of positive solutions to boundary value problem {u '' + lambda f(t,u)=0, t is an element of(0,1); u(0)=Sigma(m-2)(i-1) alpha (i)u(xi(i)), u'(1) = Sigma (m-2)(i=1) beta(i) u'(xi(i)), where xi(i) is an element of(0, 1), 0 < xi(1) < xi(2) < ... < xi(m-2) < 1, alpha(i), beta(i) is an element of[0,infinity), lambda is positive parameter. By using Krasnoserskii's fixed point theorem, we provide sufficient conditions for the existence of at least one positive solution to the above boundary value problem.
Existence of positive solutions for nonlinear multipoint p-Laplacian dynamic equations on time scales
(SCIENTIFIC TECHNICAL RESEARCH COUNCIL TURKEY-TUBITAK, ATATURK BULVARI NO 221, KAVAKLIDERE, TR-06100 ANKARA, TURKEY, 2020) Dogan, Abdulkadir; AGÜ, Mühendislik Fakültesi, Mühendislik Bilimleri Bölümü
In this paper, we investigate the existence of positive solutions for nonlinear multipoint boundary value problems for p-Laplacian dynamic equations on time scales with the delta derivative of the nonlinear term. Sufficient assumptions are obtained for existence of at least twin or arbitrary even positive solutions to some boundary value problems. Our results are achieved by appealing to the fixed point theorems of Avery-Henderson. As an application, an example to demonstrate our results is given.
EXISTENCE OF POSITIVE SOLUTIONS FOR p-LAPLACIAN AN m-POINT BOUNDARY VALUE PROBLEM INVOLVING THE DERIVATIVE ON TIME SCALES
(TEXAS STATE UNIV, 601 UNIVERSTITY DRIVE, SAN MARCOS, TX 78666 USA, 2014) Dogan, Abdulkadir; AGÜ, Mühendislik Fakültesi, Bilgisayar Mühendisliği Bölümü; Dogan, Abdulkadir
We are interested in the existence of positive solutions for the p-Laplacian dynamic equation on time scales (phi(P)(u(Delta)(t)))(V) a(t)f (t, u Delta(t), (t)) = 0, t is an element of (0,T)(T), subject to the multipoint boundary condition, u(0) = Sigma(m-2)(i=1) alpha iu, (xi(i)) = u Delta(T) = 0, where phi(p)(S) = vertical bar s vertical bar p(-2) s, p > 1, xi i is an element of [0, T](T,) 0 < xi 1 < xi 2 < . . . < xi m-2 < p(T). By using fixed point theorems, we prove the existence of at least three nonnegatvie solutions, two of them positive, to the above boundary value problem. The interesting point is the nonlinear term f is involved with the first order derivative explicitly. An example is given to illustrate the main result.
Existence of positive solutions to multi-point third order problems with sign changing nonlinearities
(UNIV TARTU PRESS, 1 W STRUVE ST, TARTU, 50091, ESTONIA, 2020) Dogan, Abdulkadir; Graef, John R; 0000-0002-7532-1920; 0000-0002-8149-4633; AGÜ, Mühendislik Fakültesi, Bilgisayar Mühendisliği Bölümü
In this paper, the authors examine the existence of positive solutions to a third-order boundary value problem having a sign changing nonlinearity. The proof makes use of fixed point index theory. An example is included to illustrate the applicability of the results.
EXISTENCE OF THREE POSITIVE SOLUTIONS FOR AN m-POINT BOUNDARY-VALUE PROBLEM ON TIME SCALES
(TEXAS STATE UNIV, 601 UNIVERSTITY DRIVE, SAN MARCOS, TX 78666 USA, 2013) Dogan, Abdulkadir; AGÜ, Mühendislik Fakültesi, Bilgisayar Mühendisliği Bölümü; Dogan, Abdulkadir
We study an m-point boundary-value problem on times scales. By using a fixed point theorem, we prove the existence of at least three positive solutions, under suitable growth conditions imposed on the nonlinear term. An example is given to illustrate our results.
Existence results for a class of boundary value problems for fractional differential equations
(SCIENTIFIC TECHNICAL RESEARCH COUNCIL TURKEY-TUBITAKATATURK BULVARI NO 221, KAVAKLIDERE, TR-06100 ANKARA, TURKEY, 2021) Dogan, Abdulkadir; AGÜ, Mühendislik Fakültesi, Bilgisayar Mühendisliği Bölümü; Dogan, Abdulkadir
By application of some fixed point theorems, that is, the Banach fixed point theorem, Schaefer's and the LeraySchauder fixed point theorem, we establish new existence results of solutions to boundary value problems of fractional differential equations. This paper is motivated by Agarwal et al. (Georgian Math. J. 16 (2009) No.3, 401-411).
An introduction to non-smooth convex analysis via multiplicative derivative
(TAYLOR & FRANCIS LTD, 2-4 PARK SQUARE, MILTON PARK, ABINGDON OR14 4RN, OXON, ENGLAND, 2019) Tor, Ali Hakan; AGÜ, Mühendislik Fakültesi, Bilgisayar Mühendisliği Bölümü; Tor, Ali Hakan
In this study, *-directional derivative and *-subgradient are defined using the multiplicative derivative, making a new contribution to non-Newtonian calculus for use in non-smooth analysis. As for directional derivative and subgradient, which are used in the non-smooth optimization theory, basic definitions and preliminary facts related to optimization theory are stated and proved, and the *-subgradient concept is illustrated by providing some examples, such as absolute value and exponential functions. In addition, necessary and sufficient optimality conditions are obtained for convex problems.
A Modified Multiple Shooting Algorithm for Parameter Estimation in ODEs Using Adjoint Sensitivity Analysis
(ELSEVIER SCIENCE INCSTE 800, 230 PARK AVE, NEW YORK, NY 10169, 2021) Aydogmus, Ozgur; Tor, Ali Hakan; 0000-0003-3193-5004; 0000-0002-9463-7197; AGÜ, Bilgisayar Bilimleri Fakültesi, Bilgisayar Bilimleri Fakültesi/Faculty of Computer Sciences
To increase the predictive power of a model, one needs to estimate its unknown parameters. Almost all parameter estimation techniques in ordinary differential equation models suffer from either a small convergence region or enormous computational cost. The method of multiple shooting, on the other hand, takes its place in between these two extremes. The computational cost of the algorithm is mostly due to the calculation of directional derivatives of objective and constraint functions. Here we modify the multiple shooting algorithm to use the adjoint method in calculating these derivatives. In the literature, this method is known to be a more stable and computationally efficient way of computing gradients of scalar functions. A predator-prey system is used to show the performance of the method and supply all necessary information for a successful and efficient implementation. (C) 2020 Elsevier Inc. All rights reserved.
Multiple positive solutions of nonlinear m-point dynamic equations for p-Laplacian on time scales
(SCIENTIFIC TECHNICAL RESEARCH COUNCIL TURKEY-TUBITAK, ATATURK BULVARI NO 221, KAVAKLIDERE, TR-06100 ANKARA, TURKEY, 2016) Dogan, Abdulkadir; AGÜ, Mühendislik Fakültesi, Mühendislik Bilimleri Bölümü; Dogan, Abdulkadir
In this paper, we study the existence of positive solutions of a nonlinear m-point p-Laplacian dynamic equation (phi(p) (x(Delta)(t)))(del) w(t)f (t,x(t), x(Delta)(t)) = 0, t(1) < m-1 X(ti) - B-0 (Sigma m-1 i=2 a(i)x(Delta)(t(i))) = 0, x(Delta) (tm) = 0, or x(Delta)(t(1)) - 0, x(t(m)) + B-1(Sigma m-1 i=2 b(i)s(Delta)(t(i))) -0, where phi(p)(s) =vertical bar s vertical bar(P-2) s, p > 1. Sufficient conditions for the existence of at least three positive solutions of the problem are obtained by using a fixed point theorem. The interesting point is the nonlinear term f is involved with the first order derivative explicitly. As an application, an example is given to illustrate the result.
A NEW APPROACH FOR NUMERICAL SOLUTION OF LINEAR AND NON-LINEAR SYSTEMS
(KOREAN SOC COMPUTATIONAL & APPLIED MATHEMATICS & KOREAN SIGCAM, 2017) Zeybek, Halil; Dolapci, Ihsan Timucin; AGÜ; Zeybek, Halil
In this study, Taylor matrix algorithm is designed for the approximate solution of linear and non-linear differential equation systems. The algorithm is essentially based on the expansion of the functions in differential equation systems to Taylor series and substituting the matrix forms of these expansions into the given equation systems. Using the Mathematica program, the matrix equations are solved and the unknown Taylor coefficients are found approximately. The presented numerical approach is discussed on samples from various linear and non-linear differential equation systems as well as stiff systems. The computational data are then compared with those of some earlier numerical or exact results. As a result, this comparison demonstrates that the proposed method is accurate and reliable.
A New Semi-supervised Classification Method Based on Mixture Model Clustering for Classification of Multispectral Data
(SPRINGER, 233 SPRING ST, NEW YORK, NY 10013 USA, 2018) Gogebakan, Maruf; Erol, Hamza; AGÜ, Mühendislik Fakültesi, Bilgisayar Mühendisliği Bölümü
A new method for semi-supervised classification of remotely-sensed multispectral image data is developed in this study. It consists of unsupervised-clustering for data labelling and supervised-classification of clusters in multispectral image data (MID) using spectral signatures. Mixture model clustering, based on model selection, is proposed for finding the number and determining the structures of clusters in MID. The best mixture model, for the best clustering of data, finds the number and determines the structure of clusters in MID. The number of elements in the best mixture model fits to the number of clusters in MID. The elements of the best mixture model fits to the structure of clusters in MID. Clusters in MID is supervised-classified using spectral signatures. Euclidean distance is used as the discrimination function for the supervised-classification method. The values of Euclidean distances are used as decision rule for the supervised-classification method.
Normal Mixture Model-Based Clustering of Data Using Genetic Algorithm
(SPRINGER INTERNATIONAL PUBLISHING AG, 2020) Gogebakan, Maruf; Erol, Hamza; AGÜ; Gogebakan, Maruf
In this study, a new algorithm was developed for clustering multivariate big data. Normal mixture distributions are used to determine the partitions of variables. Normal mixture models obtained from the partitions of variables are generated using Genetic Algorithms (GA). Each partition in the variables corresponds to a clustering center in the normal mixture model. The best model that fits the data structure from normal mixture models is obtained by using the information criteria obtained from normal mixture distributions.

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