Browsing by Author "Zeybek, Halil"

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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 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.
An Efficient Approach to Numerical Study of the MRLW Equation with B-Spline Collocation Method
(HINDAWI PUBLISHING CORPORATION, 2014) Karakoc, Seydi Battal Gazi; Ak, Turgut; Zeybek, Halil; 0000-0002-4596-0553; AGÜ, Mühendislik Fakültesi, Bilgisayar Mühendisliği Bölümü; Zeybek, Halil
A septic B-spline collocation method is implemented to find the numerical solution of the modified regularized long wave (MRLW) equation. Three test problems including the single soliton and interaction of two and three solitons are studied to validate the proposed method by calculating the error norms L-2 and L-infinity and the invariants I-1, I-2, and I-3. Also, we have studied the Maxwellian initial condition pulse. The numerical results obtained by the method show that the present method is accurate and efficient. Results are compared with some earlier results given in the literature. A linear stability analysis of the method is also investigated.
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 numerical investigation of the GRLW equation using lumped Galerkin approach with cubic B-spline
(Springer, 2016) Zeybek, Halil; Karakoç, Battal Gazi; AGÜ; Zeybek, Halil
In this work, we construct the lumped Galerkin approach based on cubic B-splines to obtain the numerical solution of the generalized regularized long wave equation. Applying the von Neumann approximation, it is shown that the linearized algorithm is unconditionally stable. The presented method is implemented to three test problems including single solitary wave, interaction of two solitary waves and development of an undular bore. To prove the performance of the numerical scheme, the error norms [Formula: see text] and [Formula: see text] and the conservative quantities [Formula: see text], [Formula: see text] and [Formula: see text] are computed and the computational data are compared with the earlier works. In addition, the motion of solitary waves is described at different time levels.
Numerical solutions of the kawahara equation by the septic B-spline collocation method
(International Academic Press, 2014) Karakoç, Battal Gazi; Zeybek, Halil; Ak, Turgut; AGÜ; Zeybek, Halil
In this article, a numerical solution of the Kawahara equation is presented by septic B-spline collocation method. Applying the Von-Neumann stability analysis, the present method is shown to be unconditionally stable. The accuracy of the proposed method is checked by two test problems. L2 and L∞ error norms and conserved quantities are given at selected times. The obtained results are found in good agreement with the some recent results.
A septic B-spline collocation method for solving the generalized equal width wave equation
(ACADEMIC PUBLICATION COUNCIL, 2016) Karakoc, Seydi B. G.; Zeybek, Halil; AGÜ; Zeybek, Halil
In this work, a septic B-spline collocation method is implemented to find the numerical solution of the generalized equal width (GEW) wave equation by using two different linearization techniques. Test problems including single soliton, interaction of solitons and Maxwellian initial condition are solved to verify the proposed method by calculating the error norms L2 and L∞ and the invariants I1, I2 and I3. Applying the Von-Neumann stability analysis, the proposed method is shown to be unconditionally stable. As a result, the obtained results are found in good agreement with the some recent results.
Solitary-wave solutions of the GRLW equation using septic B-spline collocation method
(ELSEVIER SCIENCE INC, 2016) Karakoc, Seydi; Zeybek, Halil; AGÜ; Zeybek, Halil
In this work, solitary-wave solutions of the generalized regularized long wave (GRLW) equation are obtained by using septic B-spline collocation method with two different linearization techniques. To demonstrate the accuracy and efficiency of the numerical scheme, three test problems are studied by calculating the error norms L2 and L∞ and the invariants I1, I2 and I3. A linear stability analysis based on the von Neumann method of the numerical scheme is also investigated. Consequently, our findings indicate that our numerical scheme is preferable to some recent numerical schemes.