Theses supervised by Sonuç Zorlu Oğurlu
11 theses · Eastern Mediterranean University
Quantum Integral Inequalities on Finite Intervals
The Integral Inequalities can be used for the study of qualitative and quantitative properties of integrals and they perform an important role in the theory of differential equations. The study of the fractional q-integral inequalities is also of great importance. The purpose of this thesis is to study q-calculus analogs of some classical integral inequalities. In particular, some of the greatest significant integral inequalities of analysis are extended to Quantum calculus. We will work on the q-generalization of the Hölder, Hermite-Hadamard, Trapezoid, Ostrowski, Cauchy-BunyakovskySchwarz, Grüss, and Grüss-Chebysev integral inequalities. The analysis is based on the notions of q-derivative and q-integral on finite intervals presented recently by the author in [9]. Keywords: Quantum Integral Inequalities; Hölder’s inequality, Hermite-Hadamard’s inequality, Ostrowski's Inequality, Grüss-Chebysev integral inequality
Martingale Theory
From ages to ages there had been expectation of individuals on a specific predictions and future occurrences. So also in a game, different participant that involves in those specified game have their various expectations of the results or the output of the game they are involved in. That is why we need a mathematical theory that helps in prediction of the future expectations in our day to day activities. Therefore the Martingale Theory is a very good theory that explains and dissects the expectation of a gamer in a given game of chance. So in this thesis, we shall talk about the Martingale Theory expressing the expectations of a gamer in a game of chance, and also discuss the gaming strategies so as to enlighten everyone involved in a specific game their required expectation after proper understanding of the Martingale Theory. Keywords: Martingale, Game of chance, Random walk, Stopping time.
Important Relations of Classical Orthogonal Polynomials
In this thesis, the theory of classical orthogonal polynomials which are Hermite, Laguerre and Jacobi polynomials will be studied. To begin with, we will supply an outline regarding the special functions. Followed by examples of properties for orthogonal polynomials in Chapter 2. In the third chapter, we begin classical orthogonal polynomials. To start with, we collate the orthogonal relation, Rodrigues formulas followed by the norm of the classical orthogonal polynomials. In the same chapter, the division of the collected examples of classical orthogonal polynomials into three chapters and assign them the weight function, intermission of the orthogonality, followed by differential equations, hypergeometric representation. To finalise we explain limit relations between polynomials. Keywords: Classical orthogonal polynomials, hypergeometric functions, second order differential equations, Rodrigues formula.
Fractional Mixed Volttera - Fredhol Integrodifferential Equation
In mathematics, an integrodifferential equation is an equation that involves both integral and derivative of a function. These equations model many situation ranging from science and engineering. A particular rich source is electrical circuit analysis. Different techniques have been evolved for finding the solution of these differential equations under certain conditions. One of them is to prove the existence and uniqueness of mixed Volteraa-Fredholm type integral equation with the integral boundary conditions in Banach Space. This has been worked on by some authors such as S A Murad from Iraq, H J Zekri from Iraq, S Hadid from UAE. Keywords: fixed point theorems; sequential fractional derivative; integral boundary conditions; fractional differential equation
Markov Chains and Markov Processes
Markov chain, which was named after Andrew Markov is a mathematical system that transfers a state to another state. Many real world systems contain uncertainty. This study helps us to understand the basic idea of a Markov chain and how is been useful in our daily lives. For some times there had been suspense on distinct predictions and future existences. Also in different games there had been different expectations or results involved. That is the reason why we need Markov chains to predict our expectation for the future. In this thesis we specifically talk about Markov Chains and how it has been processed, the gaming tactics which gives us a clue in a game that requires expectation. Also, we gave some applications of Markov chains such as Random walk, Games of chance, Queuing chain etc. Keywords: Stochastic Process, Conditional Expectation, Markov chain, Random Walk, Birth and Death Chains
Bernstein-Type Operators
In this work we are interested in the approximation of some type of operators called Bernstein-type. For this purpose, the operator , 0, n L f x f C called the Bernstein-type approximation operator is considered. The aim is to use some probabilistic properties to improve and sharp to operator defined above. Also, the rates of convergence as well as the continuity of the operator are studied. Various methods of approaching the problem are evaluated in this study. Keywords: Bernstein type operator, probabilistic approach, binomial distribution, rates of convergence.
Fixed Point Theorems and Applications
ABSTRACT: Fixed point theory be one of the advanced topics in both pure and applied mathematics, it also has seen great interest since recent decades, because it is considered an essential tool for nonlinear analysis and many other branches of modern mathematics. In particular, when we deal with the solvability of a certain functional equation (differential equation, fractional differential equation, integral equation, matrix equation, etc), we are reformulating the problem in terms of investigating the existence and uniqueness of a fixed point of a mapping. In addition, this theory has several applications in many different fields such as biology, chemistry, economics, game theory, optimization theory, physics, etc. The basic purpose of this thesis is to present some recent advances in this theory with some applications that is an important for our life. For example, first and second order of ordinary differential equations in Banach space and fractional differential equations involving Riemmann-Liouville and Caputo differential operators. Keywords: Fixed points, Banach’s contraction theorem, Contraction, Schauder’s fixed point theorem, Brouwer’s fixed point theorem, Uniqueness, Existence, Fractional differential equations, Boundary value problems.
On Caputo Type Sequential Fractional Differential Equations
The goal of this thesis is to give basic information about fractional calculus, and fractional differential equations of different types and study the existence and uniqueness of certain type of fractional differential equation, namely the Caputo type sequential fractional differential equations. Fixed point theorems due to Banach, Krasnoselskii, and Leray-Schauder alternative criterion is applied to obtain the desired results. The results are well illustrated with the aid of examples. Keywords: sequential fractional derivative, integral boundary conditions, fractional differential equation, fixed point theorems
On the w−Multiple Meixner Polynomials
In this thesis, a new family of discrete MOPs, namely ω-multiple Meixner polynomials, where ω is a positive real number is introduced. For ω-MOPs, orthogonality conditions w.r.t r (with r > 1) different Pascal distributions (Negative Binomial distributions) are used. Depending on the selection of the parameters in the Negative Binomial distribution, two kinds of ω-MMPs, namely 1st and 2nd kinds are considered. Some structural properties of ω-MMPs, such as raising operator, Rodrigue’s type formula and explicit representation are derived. The generating function for ω-MMPs is obtained and by use of this generating function several consequences for these polynomials are reached. A lowering operator for ω-MMPs which will be helpful for obtaining difference equation is also derived. By combining the lowering operator with the raising operator the difference equation which has the ω-MMPs as a solution are obtained. A third order difference equation for ω-MMPs is given . Also it is shown that for the special case ω = 1, the obtained results coincide with the existing results for MMPs of both kinds. In the last part as an illustrated example for the ω-MMPs of the first kind the special case when ω = 1/2 is considered and for the 1/2-MMPs of the first kind,the results obtained for the main theorems are stated. For the ω-MMPs of the second kind the special case when ω = 5/3 is studied and for the 5/3-MMPs of the second kind, the corresponding result obtained for the main theorems are examined.
Quantum Calculus on Finite Intervals and Applications to Impulsive Difference Equations
In Mathematics, quantum calculus is a version of calculus in which limits are not taken. This type of calculus plays important role both in theoretical and practical areas of mathematics. In quantum calculus, derivatives are differences and anti derivatives are sums. Quantum calculus is a theory where smoothness is no more needed. In this work, we study finite intervals in quantum calculus. We review and study the -derivative and -integral of a function and demonstrate their properties. We apply this concept to provide existence and uniqueness results for the initial value problems, namely for first and second order impulsive -difference equations. Keywords: -derivative, - integral, impulsive -difference equation, existence, uniqueness.
Classical Orthogonal Polynomials and Differential Operators
In this thesis we introduce the concept of classical orthogonal polynomials which are Hermite, Laguerre and Jacobi polynomials. We first provide the necessary overview on special functions. Then we give several properties of orthogonal polynomials in Chapter 2. In Chapter 3, we start to classical orthogonal polynomials firstly we obtain the orthogonality relation, Rodrigues formulas and we give the norm of the classical orthogonal polynomials. Finally we divide the examples of classical orthogonal polynomials into three chapters and for each of them we give the weight function, interval of the orthogonality, second order differential equation, hypergeometric representation. Keywords: classical orthogonal polynomials, hypergeometric function, second order differential equation, Rodrigues formula.