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ABSTRACT: This thesis consists of five chapters. The first Chapter gives general information about the thesis. In the second Chapter, some preliminaries and auxilary results that are used throughout the thesis are given. The original parts of the thesis are Chapters 3, 4 and 5 which are established from [35], [46] and [48]. In Chapter three, extended 2D Bernoulli and 2D Euler polynomials are introduced. Moreover, some recurrence relations are given. Differential, integrodifferential and partial differential equations of the extended 2D Bernoulli and the extended 2D Euler polynomials are obtained by using the factorization method. The special cases reduces to differential equation of the usual Bernoulli and Euler polynomials. Note that the results for the usual 2D Euler polynomials are new. In Chapter four, we consider Hermite-based Appell polynomials and give partial differential equations of them. In the special cases, we present the recurrence relation, differential, integro-differential and partial differential equations of the Hermite-based Bernoulli and Hermite-based Euler polynomials. In Chapter five, introducing k-times shift operators, factorization method is generalized. The differential equations of the Appell polynomials are obtained. For the special case k = 2, differential equation of Bernoulli and Hermite polynomials are exhibited. Keywords: 2D Bernoulli polynomial, 2D Euler polynomial, extended 2D Bernoulli polynomial, extended 2D Euler polynomial, Hermite-based Appell polynomials, factorization method. …………………………………………………………………………………………………………………………………………………………………………………………………………
The aim of my thesis is to examine the Bernstein Operators, which are linear positive operators and the properties of the New Generalized Operators. My thesis consists of four parts. The first part is an introduction and gives information about the parts that we will examine in the following chapters. The second chapter is to give more information about all the fundamental theorems and properties. In this section, the basic theorems used in the thesis are proved and explained with examples. In the third part, Korovkin Theorem’s proof and Bernstein Operators and approximation propeties and converges uniformly of Bernstein operators are given. In the last, new family of generalized Bernstein operators' definition and some important theory of convergence approximation of functions are given. After that we will examine some important results regarding the rate of converges and predictions of new generelized operators, that are appliactions of the properties and formulas which are foretold. Lastly, we examine the shape preservation properties and complete the thesis.
This thesis includes five chapters. In the first chapter, general information and some preliminaries that used throughout the thesis are given. In Chapter 2, the incomplete Pochhammer ratios are defined in terms of the incomplete beta function ( ) With the help of these incomplete Pochhammer ratios, we introduce new incomplete Gauss hypergeometric, confluent hypergeometric and Appell’s functions and investigate several properties of them such as integral representations, derivative formulas, transformation formulas and recurrence relation. Furthermore, an incomplete Riemann-Liouville fractional derivative operators are introduced. This definition plays a key role for our understanding of the linear and bilinear generating relations for the new incomplete Gauss hypergeometric functions. In Chapter 3, we give the definitions of Caputo fractional derivative operators and show their use in the special function theory. For this purpose, new types of incomplete hypergeometric functions are introduced and their integral representations are obtained. Furthermore, we define incomplete Caputo fractional derivative operators and show that the images of some elementary functions under the action of incomplete Caputo fractional derivative operators give a new type of incomplete hypergeometric functions. For the new type incomplete hypergeometric functions linear and bilinear generating relations are obtained. In Chapter 4, generalizations of incomplete gamma, beta, Gauss, confluent and Appell’s hypergeometric functions are introduced. Also, Mellin transforms, transformation formulas, differentiation and difference formulas and fractional calculus formulas are obtained for these functions. In Chapter 5, the extended incomplete Mittag-Leffler functions are introduced by using the extended incomplete beta functions and we investigate several properties of these functions. The Mellin transform of these functions is presented with regards to the incomplete Wright hypergeometric functions. Furthermore, we obtain the images of the extended incomplete Mittag-Leffler functions under the actions of Riemann-Liouville fractional integral and derivative operator. Some miscellaneous properties of these funtions are also given. Keywords: incomplete Pochhammer ratios, incomplete hypergeometric functions, incomplete Riemann-Liouville fractional derivative operators, Generating functions, incomplete Caputo fractional derivative operators, generalized incomplete gamma and beta functions, generalized incomplete hypergeometric functions, incomplete Mittag-Leffler functions, extended incomplete Mittag-Leffler functions
Engineering and physics demand a through knowledge of applied mathematics and a good understanding of special functions. These functions commonly arise in such areas of applications as heat conduction, communication systems, electro-optics, approximation theory, probability theory, and electric circuit theory, among others. The subject of special functions is quite rich and expanding continuously with the emergence of new problems in the areas of applications in engineering and applied sciences. We investigate generalized gamma function, digamma function, the generalized incomplete gamma function, extended beta function. Also, some properties of these functions are taken into hand. Keywords: Approximation, Circuit, Gamma, Beta, Digamma
Modern society faces many nonlinear problems, which nonlinear equations are better suited to understand and address them. Despite having access to high-performance digital computers, we still struggle to find precise and superior solutions to nonlinear problems, particularly the analytical approximation than its numerical consequence. The Aboodh transform iterative method, based on a new iterative method and the Aboodh transform, is the methodology we suggest in this thesis work to solve fractional differential equations, and the fractional order is taken into account by the Caputo operator. The technique combines the Aboodh transform with a fresh iterative approach to produce a series-form solution with easily calculatable components. The decomposition method is suited to visible issues, solve nonlinear problems without linearization, perturbation, or discretization methods, yet requiring less computational work than the traditional conventional methods, the numerical. The nonlinearity and linearity terms are decomposed in a series form. A few examples that are sufficiently backed up by numerical evidences have been provided to show the effectiveness of the scheme. The graphical solution showed how the fractional order of the scheme affected the results. In general, the solution profiles have demonstrated or shown that the scheme is almost exact and strength forward, simple to apply, and computationally less expensive. The solutions to the fractional differential equation’s exact problems are completely consistent with the findings, and the proposed scheme effectively and fully captures its true behavior and fractional effects.
ABSTRACT: In the study called non-Newtonian Calculus, Grossman and Katz introduced a new type of calculus that includes branches such as geometric and bi-geometric calculus. The aim of this thesis is to examine the basic features of the geometric and bigeometric calculus. This thesis is divided into five parts. In the first part, the literature on non-Newtonian calculus is summarized. In the second part, a sub-branch of non-Newtonian calculus called geometric arithmetic is introduced, along with the properties of geometric real numbers which is called 𝛼-arithmetic. In the third part, the applications of the 𝛼- arithmetic of non-Newtonian calculus is studied. In chapter four the arithmetic, differentiation and integration are applied to a specific example which is hyperbolic tangent. In chapter 5 the application of arithmetic to the cubic calculi is given and according to this geometric and bi-geometric differentiation and integrations are defined. Apart from this some other concepts are given such as absolute value, commutativity, associativity and distributivity properties and q-limit. In conclusion part a discussion about quadratic Calculi is given. Keywords: Non-Newtonian Calculus, 𝛼-Arithmetic, Geometric Calculus, BiGeometric Calculus, Cubic Calculi and Quadratic Calculi.
ABSTRACT: In this thesis, q-Szász-Durrmeyer (0 < q < 1) and q-Phillips (q > 0) operators are defined and some properties of these operators are studied. More precisely, local approximation results for continuous functions in terms of modulus of continuity are proved and Voronovskaja type asymptotic results are investigated. Keywords: q-Szász-Durrmeyer operators, k-functional, modulus of continuity, qcalculus, q-Phillips operators, q-integers, q-gamma functions, rate of convergence, qintegral. ……………………………………………………………………………………………………………………………………………………………………………………………………………………
ABSTRACT: One of the main starting point for the theory of calculus is the differentiation operation, which is defined as follows. Firstly, divide the difference of two function values by the difference of the corresponding two arguments, and then take the limit as the two arguments converge to each other. The result of this limit is called the derivative of the original function. Many variants of this basic operation have been proposed, giving rise to different theories and types of calculus. In this thesis, I will study some particular variants in which the limiting process is omitted but the two arguments in the quotient expression are linear functions of each other. The most basic one is the q-calculus (or quantum calculus), which is a particular case of both the (q,ω)-calculus (or Hahn calculus) and the (p,q)-calculus, which are the both special cases of the new type called (p,q)-Hahn calculus. These approaches give more discrete theories than the original calculus, more applicable to quantum physics. But a lot of the structure remains the same: in all cases there are derivatives, integrals, product and chain rules, exponential and Appell functions. In this thesis, I will study important properties and special functions associated with each of these three known types of calculus, and finally, I introduce the new (p,q)-Hahn Calculus. Keywords: q-calculus or quantum calculus, q,ω-calculus or Hahn Calculus, (p,q)-calculus, (p,q)-Hahn Calculus, Exponential Functions, Appell Polynomials.
ABSTRACT: This thesis consisting of three chapters is concerned with Bernstein polynomials. In the first chapter, an introduction to Bernstein polynomials is given. Then, basic properties of Bernstein polynomials are studied in the second chapter. Last chapter studies the generalized Bernstein polynomials and since it is known that generalized Bernstein polynomials are related to q-integers, we gave basic properties of q-integers. In this chapter, convergence properties of Bernstein polynomials are also given. In addition, we introduced some probabilistic considerations of generalized Bernstein polynomials. Keywords: Bernstein polynomials; generalized Bernstein polynomials; q-integers; Convergence. …………………………………………………………………………………………………………………………
In data mining, many algorithms were suggested to define the frequent rules within the data set. One of the problems is to choose a correct algorithm for the problem and the determination of the efficiency of the algorithm has important role during the investigation of hidden knowledge. The thesis describes how to handle data set with Association Rules Analysis/ Market Basket Analysis with the popular Apriori algorithm and k -Map algorithm of data mining. The goal of this thesis is to find the most frequent patterns within the data set and then using different measurements to do further investigation on the obtained frequent patterns. Keywords: Data Mining, Association Rules Analysis, Market-Basket Analysis