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**Abstract**

Name: Abdulla Sulaiman Alshams Title: Riemann – Hilbert Boundary Value Problems Major: Mathematics Date: May 2005 Assistant Professor Mohamed Saad M. Akel, Chair The interest within the theory of Riemann–Hilbert Boundary Value Problems for high order elliptic differential equations and systems was revived in nineties of the 20th century. When the theory of analytic functions and harmonic functions and the theory of two-dimensional singular integral equations could be applied to investigate the solvability of high order elliptic differential equations and systems with boundary conditions are more general than these classical conditions of Dirichlet and Neumann. Also because the important applications, this field became in the most interest of the Mathematicians In the first chapter of this thesis we introduce the definitions and concepts are needed for the subsequent chapters of the thesis. The second chapter introduces an application of Riemann – Hilbert boundary value problems for analytic functions in solving one-dimensional singular integral equation In the third chapter, we study the solvability of Riemann – Hilbert boundary value problems for second order complex elliptic differential equation in sobolev spaces. The results in this chapter generalize and improve some recent result in this field

**Abstract**

Name: Yousra Al-Kattan Title: Non-linear transient gravity waves due to an oscillating external pressure applied to the free-surface of a fluid layer over a topography Major: Mathematics Master of Science in Mathematics in Fluid Mechanics (Mathematics) by Yousra S. Al-Kattan Major: Mathematics Professor Moustafa S. Abou-Dina, Chair Date: July 2009 In the present thesis, we investigate the behavior of nonlinear, two-dimensional, transient gravity waves inside an incompressible, inviscous and homogeneous fluid. These waves are the response of an oscillating external pressure applied to the free surface of a fluid layer occupying an infinite channel with a general topography. The mathematical model simulating the considered phenomenon is a free non-linear boundary value problem subject to certain initial conditions. A classical perturbation technique is used for the investigation of such model. In contrast to the limitations made on the horizontal extent of the topography by the shallow-water theory, the present approach necessitate small vertical extent of the topography relative to its arbitrary finite horizontal extent and to the mean water’s depth as well. Solutions up to the second order are obtained, discussed and illustrated. In the frame of the linear theory of motion, the steady harmonic case is deduced from the transient state solution and also the same phenomenon in a closed basin is considered. The resonance phenomenon is discovered and discussed. The validity of the obtained solution is tested by comparing the lowest streamline with the bottom topography and the proposed model is shown to be strongly efficient in simulating the considered phenomenon specially if the slope of the topography is not sharp. The difference between the linear and the nonlinear theories, and the effect of the bottom topography on the resulting flow are illustrated. The features of the phenomenon under consideration are revealed and discussed.

**Abstract**

Name: Kholood Salih Abdullah Alkulaibi. Title : New Results on The Existence of Monotone Solutions and Extremal Solutions for Second Order Differential Inclusions. Major : Mathematics. Date : June, 2003. The interest in set-valued differential equations was revived in the early sixties from the 20th century, when mathematicians become attracted to a new domain: Control Theory. Indeed many control systems can be modeled by a differential inclusion. So, many forms of differential inclusions were investigated and therefore the existence theorems of solutions for first and second order differential inclusions were studied extensively in many papers in the literature. In chapter one of this thesis, we gathered definitions and some known and new facts about the properties of set-valued functions ( continuity, measurability, integrability and differentiability ), differential inclusions and functional differential inclusions. In chapter two, we found the necessary conditions that guarantee the existence of monotone solutions for a second order nonconvex differential inclusions. This result generalize some new results in the literature. In chapter three, we gave new existence results for a solutions or extremal solutions for a second order functional differential inclusion with three boundary points. These results generalized And improved many recent papers.

**Abstract**

Name: Reem Alomeer In this thesis we are interested in finding the sufficient conditions that guarantee the existence of mild solutions for some forms of differential inclusions with nonlocal conditions. The thesis is divided into three chapters. In chapter one we gathered definitions and some known facts about the properties of set-valued functions (continuity, measurability, integrability), differential inclusions and functional differential inclusions. Also we give some facts about the measure of noncompactness and semigroups of linear operators. In the second chapter we present a new result concerning the existence of mild solutions for a differential inclusion with nonlocal condition. This result generalizes some recent results, for example ([21,32,42,45]). This work is published in the Electronic Journal of Differential Equations ([3]). In the third chapter we give the second result in this thesis. Indeed, we generalize a recant result due to Ibrahim and Soliman [43], concerning the existence of mild solutions of functional evolution differential inclusion, from the case with local condition to the case with nonlocal condition. This result will submit in a Mathematical journal.

**Abstract**

**Mathematics**