Operator splitting methods for non-autonomous differential equations

Thesis (Master)--Izmir Institute of Technology, Mathematics, Izmir, 2011Includes bibliographical references (leaves: 61-64)Text in English; Abstract: Turkish and Englishix, 83 leavesIn this thesis, convergency and stability analysis are studied for the non-autonomous differential equations. Not only classical operator splitting methods; Lie Trother splitting, symmetrically weighted splitting and Strang splitting but also iterative splitting method which is recent popular technique of operator splitting methods are considered. We concentrate on how to improve the operator splitting methods with the help of the Magnus expansion. In addition, we construct a new symmetric iterative splitting scheme. Then, we also study its convergence properties by using the concepts of stability, consistency and order. For this purpose, we use C0 semigroup techniques. Finally, several numerical examples are illustrated in order to confirm our theoretical results by comparing the new symmetric iterative splitting method with frequently used operator splitting methods

Lineer olmayan titreşim problemlerini çözmek için yeni yaklaşımlar

Thesis (Master)--Izmir Institute of Technology, Mathematics, Izmir, 2015Includes bibliographical references (leaves: 67-70)Text in English; Abstract: Turkish and Englishx, 115 leavesThis thesis proposes two different numerical methods for solving nonlinear oscillation problems which appear in engineering and physics. Thus, the study is conducted in two parts. The first part introduces and analyzes the new iterative splitting method. In the construction of this method I utilize both the iterative splitting process and nonlinear Magnus expansion. Due to the fact that the iterative splitting procedure is employed, the constructed method can also be considered as a kind of operator splitting method. The second part presents a new linearization technique, based on the Newton-Raphson method and the Fréchet derivatives, for oscillation systems. Duffing oscillator and damped oscillator are used for testing the methods, respectively. Moreover, the proposed iterative splitting method and the proposed linearization technique are applied to both Van-der Pol equation and cubic nonlinear Schrödinger equation. Although the examples considered are a small sample of nonlinear oscillation equations, it is believed that the methods are easily adapted to solve such problems numerically. ivBu tez, mühendislik ve fizik alanında karşılaşılan lineer olmayan titreşim problemlerinin çözümleri için iki farklı sayısal metot önermektir. Bu yüzden çalışma iki parça halinde yönetilmektedir. Birinci bölüm yeni iteratif ayırma metodunu tanıtmakta ve analiz etmektetir. Bu metodun oluşturulmasında hem iteratif ayırma sürecinden hem de lineer olmayan Magnus açılımından yararlanılmıştır. Metodun oluşturulmasında iteratif ayırma yönteminin kullanılmasından dolayı önerilen metod aynı zamanda operatör ayırma metodun bir çeşidi olarak da ele alınabilir. İkinci bölüm titreşim problemleri için, Newton-Raphson metodu ve Fréchet türevlerini baz alan, yeni lineerizasyon tekniği sunar. Metotları test edebilmek için sırasıyla Duffing osilatör ve sönümlü osilatör kullanılmıştır. Ayrıca, önerilen iteratif ayırma metodu ve önerilen lineerizasyon tekniği hem Van-der Pol denklemi hem de kübik lineer olmayan Schrödinger denklemine uygulanmıştır. Uygulamalarda lineer olmayan titreşim problemlerinin az sayıda örneklerinin düşünülmüş olmasına rağmen metodun bu şekildeki problemlere sayısal olarak kolayca adapte edilebileceğine inanılmaktadır

An efficient iterative algorithm for solving non-linear oscillation problems

A new iterative method is presented for numerical solution of nonlinear evolutionary problems. The convergence properties of the proposed method are analysed in abstract framework by using the concepts of consistency, stability and order. Both the ϕ-functions and semigroup properties are used to overcome the presence of unboundedness of the operator. In order to confirm the theoretical results, the method is applied to three benchmark problems from the literature. The numerical results are compared with traditional splitting methods and confirm that the proposed method is more accurate as well as more efficient than the traditional splitting methods

A conserved linearization approach for solving nonlinear oscillation problems

Nonlinear oscillation problems are extensively used in engineering and applied sciences. Due to non-availability of the analytic solutions, numerical approaches have been used for these equations. In this study, a numerical method which is based on Newton-Raphson linearization and Fréchet derivative is suggested. The convergence analysis is also studied locally. The present method is tested on three examples: damped oscillator, Van-der Pol equation and Schrödinger equation. It is shown that the obtained solutions via the present method are more accurate than those of the well-known second order Runge-Kutta method. When examining the present method, preservation of characteristic properties of these equations is also considered. The obtained results show that the present method is applicable with respect to the efficiency and the physical compatibility

A NONLINEAR REGRESSION MODEL, ANALYSIS AND SIMULATIONS FOR THE SECOND WAVE OF COVID-19: THE CASE STUDY OF TURKEY

COVID-19 pandemic disease gained major attention among scientists due to its high mortality/ infectiousness rate. Moreover,the analysis of this disease requires much attention by the Government to take precautions and construct strategies. This studyaims to develop a new nonlinear model for COVID-19. The main focus is the time when the number of daily infectedindividuals has begun to increase constantly. To this end, the time series from 1 August 2020 to 22 September 2020 isconducted. Moreover, the proposed model takes into account the disease characteristics. After the model parameters areobtained by detailed mathematical analysis by the trained data, the model is validated by the test/evaluation data set. The resultsand simulations show that the proposed model has a perfect match with the raw data. Furthermore, the calculated standarderrors when compared by the population of Turkey are evidence of how well the model fits the raw data. This study is importantnot only because it achieves good results but also because it is the first nonlinear regression model including its mathematicalanalysis for the COVID-19 pandemic

A dual role of proton pump inhibition on cancer: a critical review

Proton pump inhibitors (PPIs) are widely used to suppress gastric acid secretion. Proton pumps belong to the family of ATPase and among them P-ATPase and V-ATPase types regulate intracellular as well as extracellular acid equilibrium. The main aim of the current survey is to present the existing literature putting forth the relation between cancer with both the use of PPIs and proton pumps from positive and negative aspects. To perform an objective study, various types of proton pumps and their relation to cancer have been taken into account. Up to date, the studies have been considered in the time range from 2011 to 2021 via various databases (PubMed, Scopus, and Google Scholar). H+/K+ ATPase, located within the gastric parietal cells, is one of the most important examples of P-ATPases. The findings of the literature review along with criticism were presented as decreased P-ATPase expression can be used as a marker for gastric cancer diagnosis whereas the association of the proton pump with cancer may be mainly due to V-ATPase. In conclusion, molecular, epidemiological, and bioinformatic studies are required to enlighten the subject