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Öğe A fractional-order mathematical model based on vaccinated and infected compartments of SARS-CoV-2 with a real case study during the last stages of the epidemiological event(PMC, 2023) Bilgil, Halis; Yousef, Ali; Erciyes, Ayhan; Erdinç, Ümmügülsüm; Öztürk, ZaferIn 2020 the world faced with a pandemic spread that affected almost everything of humans' social and health life. Regulations to decrease the epidemiological spread and studies to produce the vaccine of SARS-CoV-2 were on one side a hope to return back to the regular life, but on the other side there were also notable criticism about the vaccines itself. In this study, we established a fractional order differential equations system incorporating the vaccinated and re-infected compartments to a S I R frame to consider the expanded and detailed form as an S V I I v R model. We considered in the model some essential parameters, such as the protection rate of the vaccines, the vaccination rate, and the vaccine's lost efficacy after a certain period. We obtained the local stability of the disease-free and co-existing equilibrium points under specific conditions using the Routh-Hurwitz Criterion and the global stability in using a suitable Lyapunov function. For the numerical solutions we applied the Euler's method. The data for the simulations were taken from the World Health Organization (WHO) to illustrate numerically some scenarios that happenedÖğe A mathematical model with piecewise constant arguments of colorectal cancer with chemo-immunotherapy(Elsevier Ltd, 2023) Bozkurt, Fatma; Yousef, Ali; Bilgil, Halis; Baleanu, DumitruWe propose a new mathematical model with piecewise constant arguments of a system of ODEs to investigate the growth of colorectal cancer and its response to chemo-immunotherapy. Our main target in this paper is to analyze and represent the I.S.'s (immune system) efficiency during the chemotherapeutic process. Therefore, we proved and illustrated the necessity of IL-2 that supports the immune system, especially in early-detected cases of tumor density. Thus, the constructed model has been divided into sub-systems: the cell populations, the effects of the medications doxorubicin, and IL-2 concentration. Firstly, we analyze the stability of the equilibrium points (disease-free and co-existing) using the Routh-Hurwitz criteria. In addition, our study has shown that the system undergoes period-doubling, stationary and Neimark-Sacker bifurcations based on specific conditions. In the end, we illustrate some simulations to assist the theory of the manuscript.Öğe A note on exp-function method combined with complex transform method applied to fractional differential equations(Walter de Gruyter Gmbh, 2015) Güner, Özkan; Bekir, Ahmet; Bilgil, HalisIn this article, the fractional derivatives in the sense of modified Riemann-Liouville and the exp-function method are used to construct exact solutions for some nonlinear partial fractional differential equations via the nonlinear fractional Liouville equation and nonlinear fractional Zoomeron equation. These nonlinear fractional equations can be turned into another nonlinear ordinary differential equation by complex transform method. This method is efficient and powerful in solving wide classes of nonlinear fractional order equations. The exp-function method appears to be easier and more convenient by means of a symbolic computation system.Öğe A novel fractional forecasting model for time dependent real world cases(National Statistical Institute, 2024) Erdinç, Ümmügülsüm; Bilgil, Halis; Öztürk, ZaferThe grey modelling in the prediction of time series has been one of the interesting study fields recently due to its efficiency and convenience. Fractional grey models, on the other hand, have become preferable, despite the difficulty in calculations, since they give more effective results than standard models. Difficulties in fractional accumulation and difference calculations have begun to be overcome thanks to new definitions and theorems made in recent years. The new trend in grey modelling is to compose models that are more useful than the previous ones and give results with less error. In this paper, a new grey model derived from the conformable fractional order is defined and it is shown that more effective estimates are made compared to the models created in recent years. The verification of the method is shown with real data set. The results show that the proposed conformable fractional grey model is more effective than the existing models.Öğe An application of khuri’s biorthogonality condition on stokes flow in a rectangular cavity(Tsing Hua University, 2019) Bilgil, HalisThe paper present a different semi-analytical solutions for the Stokes flow in a rectangular cavity by using the biorthogonality condition of the eigenfunctions. Firstly, the biorthogonality condition for rectangular cavity is applied on Stokes Flow problem. Secondly, the biorthogonality condition and the new solution is verified against results available in the literature. Further, it has been found that this solution mechanism is effi cient and it can be successfully used to obtain flow structures and bifurcations.Öğe An optimized continuous fractional grey model for forecasting of the time dependent real world cases(Hacettepe University, 2022) Öztürk, Zafer; Bilgil, Halis; Erdinç, ÜmmügülsümThe new priority in the grey modelling is to build new models that have more accurate forecasting power than the previous ones. This paper aims to develop the prediction performance of the existing continuous grey models. Therefore, a novel continuous grey model (OCCFGM(1,1)) is proposed with conformable fractional derivative. The numerical results of three case studies show that the novel model's prediction accuracy is higher than other competitive models, and the proposed model is more reasonable for practical cases.Öğe Analytical solution and flow topology in a lid-driven S-shaped cavity(American Institute of Physics Inc., 2022) Bilgil, Halis; Şafak, İbrahim; Öztürk, ZaferIn this study, the Stokes flow problem in an S-shaped double lid-driven cavity filled with fluid was analyzed. Side edges of the cavity were considered as immovable walls. The flow region was divided into two sub-regions, and the streamfunction in each sub-region was considered as an extension of Papkovich-Faddle eigenfunctions. Parameters in the analytical solution were obtained using biorthogonality conditions. The Newton iteration method was used to obtain the eigenvalues of the problem, and integrals were calculated with the Gaussian quadrature method. It was ensured that solutions made separately for the two sub-regions converge on the interface, which is the intersection of these sub-regions. The two parameters controlling the flow structure were determined as the speed ratio of movable lids (S) and the aspect ratio of the cavity (A). The effects of these parameters on flow structures were shown. New eddy formation mechanisms and bifurcations were observed in the cavity by keeping the speed ratio of the lids constant and slowly changing the aspect ratio.Öğe Application of Fractional SPR Psychological Disease Model in Turkey and Stability Analysis(Mahmut Akyiğit, 2023) Öztürk, Zafer; Bilgil, Halis; Sorgun, SezerPsychological diseases and their treatment are problems related to public health. According to data from the World Health Organization, about a billion people have either mental illness or substance use disorder problems in 2017. Mental, neurological diseases and substance use disorders account for 30 percent of the global non-fatal disease burden and 10 percent of the global disease burden. It is noted that in the world Dec 2005 and 2015, the incidence of mental health diseases increased by about 16 percent. In this study, we have created a fractional-order mathematical modeling for the population of individuals suering from psychological diseases in a society. In this model, the total population was divided into three compartments: individuals who did not receive psychological treatment (S), individuals who received psychological support (P) and individuals who recovered after completing psychological treatment (R). As a fractional derivative, we used the Caputo derivative definitions. Numerical solutions were obtained with the help of Euler method by performing stability analysis related to the fractional SPR model created for the mathematical model of psychological patients. Thus, it was interpreted by creating dynamics for the number of individuals with psychological problems in a population.Öğe Bifurcations and eddy genesis of Stokes flow within a sectorial cavity(Gauthier-Villars/Editions Elsevier, 2013) Gürcan, Fuat; Bilgil, HalisThis work analyzes the Stokes flow generated in a sectorial driven cavity formed by a pair of curved stationary side walls capped by straight translating lids. The flow is governed by two physical control parameters: the cavity aspect ratio, A = r(2)/r(1) (where r1 and r2 are the radii of the inner and outer curved side walls, respectively) and the ratio (S = U-1/U-2) of the upper to the lower lid speed. A boundary value problem is formulated, which is solved analytically for the streamfunction, psi, expressed as an infinite series of Papkovich-Faddle eigenfunctions. The streamfunction solution is then expanded about any stagnation point to reveal changes in the local flow structure as A and S are varied. At critical values of A and S, stagnation point bifurcations arise and the local flow topology is transformed. For S is an element of [-1, 0) and A is an element of [1.6, 6.5), the various flow transformations are tracked as A is decreased and hence the means is identified by which new eddies appear and become fully developed. The key results are shown in an (5, A) control space diagram which exhibits an intricate structure due to the intersection and confluence of eight bifurcation curves representing flow bifurcations at degenerate critical points. There are eight points where two critical curves intersect and the flow bifurcations are described. It is shown that, for S not equal 0, the number of eddies increases from 1 to 3 via several key flow transformations, which become more complicated as vertical bar S vertical bar is reduced. For S = 0, as A is decreased slowly, the number of eddies also increases, in this case, from 1 to 2 via the development of corner eddies.Öğe Bifurcations and eddy genesis of Stokes flow within a sectorial cavity PART II: Co-moving lids(Elsevier Science Bv, 2016) Gurcan, Fuat; Bilgil, Halis; Sahin, A.This paper is a complementary study to earlier work (Gurcan, 2013), concerning eddy genesis and bifurcations in a sectorial driven cavity for Stokes flow, consisting of a pair of curved stationary side walls and two straight translating lids moving in the same radial direction with speeds U-1 and U-2. The flow is governed by two physical control parameters: the cavity aspect ratio A = r(2)/r(1) where r(1) and r(2) are the radii of the inner and outer curved side-walls, respectively and the ratio of the upper and lower lid speeds (S = U-1/U-2). In previous work (Gurcan, 2013), focus was on lids moving in the opposite direction or a single moving lid. By considering different A values for each S, the streamline patterns and their bifurcations are studied; occurrence of stagnation point bifurcations and transformation of the local flow topology are observed at critical values of A and S. The key results are shown in an (S, A) control space diagram which exhibits an intricate structure due to the intersection and confluence of eight bifurcation curves representing flow bifurcations at degenerate critical points. Six different eddy genesis mechanisms are observed for S is an element of (0, 1] and A is an element of (1.45. 20). In each case the flow structures and eddy genesis mechanisms are illustrated in detail with figures and bifurcation diagrams. The study also serves as a benchmark problem for testing new numerical algorithms. (C) 2015 Elsevier Masson SAS. All rights reserved.Öğe Comparision of Conformable and Caputo fractional grey models(Elsevier B.V., 2025) Bilgil, Halis; Yüksel, SimgeIn recent years, fractional order derivatives have been encountered in various fields of science, particularly in applied mathematics. Although there are many fractional derivative definitions in the literature, there are very few studies on which derivative definition works better in a mathematical model. In applications, it is seen that calculations are easier with the model using the Conformable derivative operator due to the simplicity of the derivative definition. However, the Caputo derivative operator, which is considered to be more effective in models related to time series due to its memory property, leads to more complex calculations. In this article, two fractional grey models were created in the same structure with Conformable and Caputo derivative operators and their applications were implemented on the same data sets to a performance comparison of the fractional operators. The working mechanisms of fractional grey models constructed with both Caputo and Conformable derivative operators were demonstrated in detail. Solution of the whitening differential equation in the Caputo fractional grey model was obtained using Laplace transforms. Here, Conformable and Caputo fractional grey models were applied to the forecast of three real time series and their forecast performances were compared. Data on China's annual domestic energy consumption, annual wind energy consumption, and areas affected by drought disasters were utilized as real-time series. It has been observed that Conformable fractional grey models provide more accurate predictions with lower errors for certain datasets, while Caputo fractional grey models demonstrate better performance for others. This study is the first study in the literature that compared the Conformable and Caputo derivative operators on a grey model.Öğe Effect of the Reynolds number on flow bifurcations and eddy genesis in a lid-driven sectorial cavity(Sprınger Japan Kk, 2016) Bilgil, Halis; Gürcan, FuatThis paper presents the two-dimensional (2D) steady incompressible flow in a lid-driven sectorial cavity. In order to analyze the flow structures, the 2D Navier-Stokes equations are solved by using the finite element method. Different cases of the cavity aspect ratio A and three cases of the speed ratios of the upper and the lower lids are considered. The finite element formulation for the governing equations is adopted via the velocity-pressure formulation. By varying A for each S, the effect of the Reynolds number on the streamline patterns and their bifurcations are investigated in range . A comparison between the obtained results and some earlier studies is presented.Öğe GM (1,1) ve EXGM (1,1) tahmin modellerinin Türkiye’nin Ar-Ge harcamalarına uygulanması(Aksaray Üniversitesi, 2022) Akyüz, Levent; Bilgil, HalisDisiplinler arası bir bilim alanı olan Gri sistem Teorisi az sayıda veri kullanarak bilinmeyen sistemlerin davranışını tahmin etmektedir ve birçok alana başarı ile uygulanmıştır. Bu çalışmada ise Gri Sistem Teorisi Türkiye’nin Araştırma ve Geliştirme (Ar-Ge) harcamalarını tahmin etmek için kullanılmıştır. 2011-2020 yılları arasındaki Ar-Ge harcamaları verileri Türkiye İstatistik Kurumu’ndan alınarak 2021-2030 yıllarının tahmini Standart Gri Model (GM (1,1)) ve Üstel Gri Model (EXGM (1,1)) kullanılarak yapılmıştır. Her iki modelin tahmin değerlerindeki hata ve korelasyon sonuçları karşılaştırılmıştır. 2011-2020 yılları arasındaki verileri kullanarak, gerçek ve tahmin değeri arasındaki yüzde bağıl hata (RPE) ve ortalama yüzde bağıl hata (MAPE) belirlenmiştir.Öğe Gri tahmin modeli ve uygulaması(Aksaray Üniversitesi, 2019) Bilgil, Halis; Öztürk, Zafer; Özgül, EmineGri teori, zayıf bilgi veya bilgilerin eksik olduğu sistemlerle ilgilenen etkili bir teoridir. Bu teoride az sayıda veriden faydalanarak gelecek zamanlar için etkin ve oldukça yaklaşık tahminler oluşturulabilmektedir. Gri modelleme yöntemi, gri sistemler teorisinin bir alt dalı olup, modelleme işlemi ilgili fark denklemleri ve diferansiyel denklemler kullanılarak yapılır. En küçük kareler yaklaşımı sonuçların uygunluğu üzerinde önemli bir rol oynamaktadır. Kesinliği ve kullanışlılığı ile gri tahmin modellerinin temelini oluşturan GM(1,1) modelleme yöntemi kullanılarak, Aksaray ilinden ileriki yıllarda elde edilecek vergi gelirlerinin tahmini yapılmıştır. Bu tahminler başta yerel yönetimler olmak üzere ekonomik politikalar için oldukça faydalıdır.Öğe Mathematical estimation of expenditures in the health sector in Turkey with grey modeling(Erciyes Üniversitesi, 2019) Öztürk, Zafer; Bilgil, HalisGrey theory is an effective theory that deals with systems that lack weak information and / or information. With this theory, effective and very accurate estimates can be created for future times by utilizing a small number of data. The grey modeling method is a sub-branch of the theory of grey systems and the modeling process is carried out with the help of differences equations and differential equations. The least squares approach plays a role in the precise results of the method. Using the GM (1,1) modeling method, which is the basis of grey prediction models with its accuracy and usefulness, health expenditures in Turkey that will be achieved in the coming years were estimated.These estimates are particularly useful for health and economic policies.Öğe New grey forecasting model with its application and computer code(American Institute of Mathematical Sciences, 2021) Bilgil, HalisGrey theory is an approach that can be used to construct a model with limited samples to provide better forecasting advantage for short-term problems. In some cases, a grey forecasting model may yield unacceptable forecasting errors. In this work, a new exponential grey prediction model, which is called as EXGM (1,1), is proposed. By using this model, new cases, deaths and recovered cases of COVID-19 in Turkey is forecast. Numerical results show that EXGM (1,1) is a model that performs more accurately than the comparison models.Öğe Optical soliton solutions for the variable coefficient modified Kawahara equation(Elsevier GmbH, 2015) Bekir, Ahmet; Güner, Özkan; Bilgil, HalisIn this paper, we obtain the 1-soliton solutions of the variable-coefficient modified Kawahara equation (VCMKE). The dark optical as well as bright optical soliton solutions were found related to the model considered in this study. The solitary wave ansatz method is used to carry out the integration.
