A study on oscillatory nonlinear ordinary differential equations of second and third order / Mastora Jaber Saad Hamid - Jaber Saad, Mastora (2013)

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Differential equations arise in many areas of science and technology. They can be used effectively to analyze the evolutionary trend of such systems, they also aid in the formulation of these systems and the qualitative examination of their stability and adaptability to external stimuli. In recent years, the qualitative theory and asymptotic behavior of differential equations and their applications have been and still are receiving intensive attention. As far as oscillation theory is concerned, most texts in differential equations, both elementary and advanced, deal with second-order equations. In fact, in the last few decades, a great number of research papers concerning oscillation theory have been presented. Although ordinary differential equations of second order have been studied extensively, the study of qualitative behavior of third order ordinary differential equations has received greatly less attention in the literature, however assured results for differential equations of third order have been known for a long time and their applications in mathematical modeling in biology and physics. In this thesis, oscillation of solutions of ordinary differential equations of second order and third order are considered. The obtained oscillation results are motivated extended and improved many previous oscillation results and new sufficient conditions are established which guarantee that our differential equations are oscillatory. Contradiction concept and Riccati technique are used in proofs of oscillation theorems. A number of theorems and illustrative examples for oscillation differential equation under study are given. Also, a number of numerical examples are given to illustrate the theoretical results. These numerical examples are computed by using Runge Kutta of fourth order iii in Matlab. Finally, the obtained results are compared with existing results to explain the motivation of proposed research work.