Şengül, Süleyman; Bekiryazici, Zafer; Merdan, Mehmet

In this study, the optimal q-Homotopy Analysis Method (optimal q-HAM) has been used to investigate fractional Abel differential equations. This article is designed as a case study, where several forms of Abel equations, containing Bernoulli and Riccati equations, are given with ordinary derivatives and fractional derivatives in the Caputo sense to present the application of the method. The optimal q-HAM is an improved version of the Homotopy Analysis Method (HAM) and its modification q-HAM and focuses on finding the optimal value of the convergence parameters for a better approximation. Numerical applications are given where optimal values of the convergence control parameters are found. Additionally, the correspondence of the approximate solutions obtained for these optimal values and the exact or numerical solutions are shown with figures and tables. The results show that the optimal q-HAM improves the convergence of the approximate solutions obtained with the q-HAM. Approximate solutions obtained with the fractional Differential Transform Method, q-HAM and predictor–corrector method are also used to highlight the superiority of the optimal q-HAM. Analysis of the results from various methods points out that optimal q-HAM is a strong tool for the analysis of the approximate analytical solution in Abel-type differential equations. This approach can be used to analyze other fractional differential equations arising in mathematical investigations.

Şengül, Süleyman; Beki̇Ryazici, Zafer; Merdan, Mehmet

In this study, Wong-Zakai approximation method has been used to obtain approximate solutions for two compartmental models of smoking dynamics. Stratonovich stochastic differential equation systems are obtained for these two stochastic models for the application of Wong-Zakai method. Wong-Zakai method is used together with the predictor-corrector deterministic approximation method where Adams-Bashforth method is used as the predictor pair and Adams-Moulton method is used as the corrector pair. Stochastic Runge-Kutta IV, Euler-Maruyama and stochastic Runge-Kutta strong order 1.0 schemes are also used to investigate the models and the results are compared to the results from Wong-Zakai approximation. The comparison shows that Wong-Zakai method is a reliable tool for the analysis of stochastic models and can be considered as an alternative investigation method for modeling studies. Solution graphs, error graphs and numerical results have been given as evidence to show that Wong-Zakai method can also be a reliable method for analyzing various models. An alternate technique for parallelizing the algorithm has also been given to decrease CPU times for Wong-Zakai method. This technique is suggested to overcome the extra calculation load that comes with Wong-Zakai method.

Okcu, Melih; Şengül, Süleyman

Objective of the present study is to identify the species belonging to the genus Onobrychis (Fabaceae) growing native in and around Erzurum, and to determine their overall prevalence and morphological and quality characteristics. Species samples were collected from different locations in flowering periods of 2007, 2008 and 2009. At the end of the study, totally 10 genotypes were found representing 8 species and 2 sub-species. It was found that leaf length and leaflet frequency were the largest in O. atropatana var. grandiflora, the number of leaflets was the highest in O. viciifolia, the length and width of leaflets were maximum in O. radiata, the number of main branches and the number of fascicles in main branches were in O. hajastana, crude ash rate was highest in O. stenostachya, plant fresh and dry weight, ADF and NDF rate were highest in O. cornuta, where as crude protein rate was slightly higher in O. stenostachya subsp. sosnowskyi than other taxa.