The objective of this thesis is to study first the theory of fractional calculus and some of well known methods for evaluating derivatives of fractional orders for certain functions.The second objective is to study the G-spline interpolation functions and its construction using a new approach in formulating the Heremite-Birkhoff problem using fractional derivatives instead of integer order derivatives.
The NTRU [Number theory research unit] cryptosystem is a relatively new public key cryptographic algorithm that was first introduced in 1998, and that key runs are much faster than conventional public key algorithms such as RSA, ECC. The main advantage of this cryptosystem is its high speed generation keys, which is often the most important part of public key cryptography. The security of NTRU cryptosystem comes from the interaction of the polynomial mixing system with the independence of reduction modulo two relatively prime integers' p and q.
The main objective of this work is to study the numerical solution of fractional ordinary differential equations using G-spline interpolation functions. Two numerical approaches are used, the first approach utilize the explicit linear multistep methods which can be applied easily for linear and nonlinear problems while the second approach is a modified approach by using the implicit linear multistep methods for solving nonlinear fractional ordinary differential equations which has so many difficulties in their solution.
The Fractional Order Bounded Delay Differential Equations (FOBDDE’s) has been studied in this work. The Existence and Uniqueness theorems of such type of differential equation have been proved, by using the successive approximation techniques. Also, the analytic solution of (FOBDDE’s) are presented, using Laplace Transformation, and the numerical solutions are discussed, using general one-step methods and linear multi-step methods. The comparison, among these methods and the exact solutions are presented.
In this thesis, some properties and basic definitions of fractional integral and derivatives of Riemann-Liouvill are presented , to construct the optimality conditions of mixed order unconstrained and constrained variational problems with continuous and discontinuous functional, on fixed and moving boundaries ,based on the classical product rule for Riemann-Liouvill , Several tested example are presented to demonstrate the implementation of the optimality necessary conditions.
Fractional (or non-integer) differentiation is an important concept both from theoretical and applicational points of view. The study of problems of the calculus of variations with fractional derivatives is a rather recent subject. In this work, some properties and basic definitions of fractional integral and derivatives of Riemann-Liouvill are presented.
This work is oriented towards two objectives:
The first objective is to classify and study the generalized one-dimensional integral equations that contain n one-dimensional integral operators. This study includes the existence of a unique solution for special types of these integral equations and their solutions by using some quadrature methods, namely the trapezoidal rule, the modified trapezoidal rule and Simpson's rule.
This thesis consider the normal distribution with its important appearance in many statistical fields of applications. Some mathematical and statistical properties of the distribution have been collected and illustrated with moments and higher moments. Six related theorems have been studied in the applications of this type of distribution. The estimation manner and its properties have been illustrated throughout two methods (Moment and Maximum Likelihood methods) which are used to estimate the distribution parameters theoretically.
This work has three objectives:
The first objective is to study fuzzy set theory including definitions, notations and examples. The second objective is to study and proof the existence and uniqueness theorem of fuzzy boundary value problems directly without transforming the problem into fuzzy initial value problem. The third objective is to study the numerical solution of fuzzy boundary value problems.
The main objective of thesis is oriented toward two objectives.
