Theses supervised by Nazim Mahmudov
11 theses · Eastern Mediterranean University
Adomian’s Decomposition of Multi-Order Fractional Differential Equations
Adomian's Decomposition Method (ADM) was introduced about three decades ago, it has proven to be efficient, reliable and easy to compute the solutions of non-linear and linear differential equations. It can also be used to compute various types of equations such as Boundary value problems, Integral equations, Equations arising in fluid flow e.t.c. This thesis work presents the derivation of Adomian's decomposition algorithms and the possible solution of fractional differential equations of the multi-order type in the Caputo sense. It consist of four chapters, Chapter 1 contains a brief introduction of Adomian's Decomposition Method(ADM) and definitions, while the second chapter deals with basis proofs and methodology with respect to Adomian's Decomposition Method(ADM). In Chapter 3, we applied the method of solution to multi-order fractional differential equations. We then discuss the results and make conclusion in Chapter 4.
Numerical Solutions of Fractional Differential Equations
In this thesis, we focus on numerical solutions of general linear multi-term fractional differential equations (FDEs) with fractional derivatives defined in the Caputo sense. Multi-term fractional order differential equations are involving both ordinary and fractional derivative operators. Numerical methods plays very crucial role for solving fractional differential equations, since analytical solutions are not always possible for solving them. Memory trait of fractional calculus is one of the main reason for difficulty of developing analytical techniques for such a equations. Therefore, there has been considerable interest in solving FDEs numerically in recent years and many powerful schemes have been developed. Essentially, most of the developed methods are modified from original versions for classical differential equations and applied to FDEs. In this study, we introduce a numerical technique based on the fractional Taylor vector and we construct fractional Taylor operational matrix of fractional integration to solve multi-term FDEs. The main characteristic of this technique is to reduce the given IVP of fractional order to a system of algebraic equations by employing the fractional Taylor operational matrix of fractional integration. Finally, this set of algebraic equations can be solved easily and efficiently for unknown coefficients by using computer programming. Consequently, by using these coefficients, the approximate solution of the given problem can be obtained. Some numerical examples are presented to demonstrate the accuracy and applicability of given method. The approximate solutions obtained by use of given technique are compared with numerical results of some other methods in literature and exact solutions of given problems. From these results, we can conclude that the presented technique is efficient and applicable for solving high order multi-term fractional order differential equations numerically. Keywords: numerical solutions, fractional Taylor vector,fractional differential equations, spectral method, Caputo fractional derivative, Riemann-Liouville fractional integral, operational matrices.
Series Approximate Analytical Solution of Fractional Partial Differential Equations
The primary objective of this thesis work is the presentation of a new iterative procedure to achieve both series approximate solution and analytical solution of positive non-integer order partial differential equations. The order of the derivative is considered according to Caputo’s assumption. This iterative procedure is called Aboodh transform iterative method. The Aboodh transform iterative method is a combination of the new iterative method with the Aboodh transform. The new iterative method was introduce as important tool to linearize all the associated nonlinear terms since the Aboodh transform cannot handle the nonlinear terms. Several examples and cases are examined. The solutions obtained were compared with solutions obtained by other existing methods in literature. Also, the solutions reveals that the Aboodh transform iterative procedure is less computational involving and requires no restrictive assumption, Lagrange multipliers and Adomian polynomial. The software used to implement the Aboodh transform iterative procedure are LaTex, MATHEMATICA 10.0 and MATLAB R2021.
Integral Type Fractional Gronwall Inequalities
The current research involves the ideas and principles about integral inequalities of Gronwall type. It deals with the possibilities that we mathematicians use in order to solve equations in various ways. The first case we adopted to solve equations is Linear Generalization. The latter deals with equations that are different from those treated with Non-Linear Generalization. The research we conduct overlaps to study the relation between fractional and Gronwall inequalities by analyzing how Gronwall inequalities are included and used in fractional inequalities. Keywords: Gronwall inequalities, Fractional inequalities, Linear generalizations and Non-Linear generalizations.
Approximation by kantorovich type operators
In this thesis, new type q-Bernstein - Kantorovich polynomials and complex q-Szász-Kantorovich operators are introduced. In additon, The exact order of approximation, quantitative Voronovskaja-type theorems, simultaneous approximation properties for complex q-Bernstein - Kantorovich polynomials , complex Szász-Kantorovich and complex q-Szász- Kantorovich operators are studied.
Computational Numerical Solution Algorithm for Fractional Differential Equations
This study focused on three main problems, firstly, a study on the existence of the solution for a coupled system of fractional differential equations with integral boundary conditions. The solution process for the existence and uniqueness of solutions for the proposed problem was obtained by using the contraction mapping principle, and then by using Leray–Schauder’s alternative method. Secondly, investigation and approximation of solutions of Caputo type fractional differential equation with nonlinear boundary conditions has been solved by using an appropriate parameterization technique, where nonlinear boundary conditions were transformed to linear boundary conditions by using vector parameters. To solve the transformed problem, a numerical-analytic scheme was constructed to find the relation between different type’s two-point and multipoint linear boundary condition and nonlinear boundary conditions. Finally, efficient numerical - analytical computational algorithm for solving systems of fractional differential equations (SFDEs) Nonlinear Point Boundary-Value Problem with Nonlinear Boundary Conditions were considered. The fractional derivative was described in the Caputo sense. The method is based on numerical approximations of systems of fractional differential equations, where the properties of this method were utilized to reduce SFDEs to the system of algebraic equations. Special attention is given to study the convergence and estimate the error of the presented method. The methods introduce a promising tool for solving many systems of non-linear fractional differential equations. Numerical examples were presented to illustrate the validity and the great potential of both proposed techniques.
Fractional Differential Equations with Fractional Boundary Conditions
This work is dedicated to investigate the existence and uniqueness of solutions for nonlinear fractional differential equations with boundary conditions involving the Caputo fractional derivative in a Banach space. After introducing some basic preliminaries and the important concepts of fractional calculus, we considered two models of boundary value problems of Caputo fractional derivative. The first one is nonlinear fractional differential equation with nonlocal four-point fractional boundary conditions. The second equation is nonlinear impulsive boundary value problem of multi-orders fractional supplemented with nonlocal four-point fractional boundary conditions. The existence and uniqueness of solution are obtained via Banach’s fixed point theorem and Schauder’s fixed point theorem for the two models. In addition, both results are provided by the illustrative examples to support them.
Nonlinear Sequential and Non Sequential Fractional Differential Equations with Integral Boundary Conditions
This thesis relies on various fractional differential equations. Based on the classical fixed point theorem summarized by what known as the Banach contraction mapping theorem, nonlinear alternative of Leray-Schauder type and Krasnoselskii’s fixed point theorem, a three different nonlinear fractional differential equations are considered. In chapter four we study the existence and uniqueness for the solution of the nonlinear sequential fractional differential equation involving Caputo fractional derivative and associated with nonlocal integral boundary conditions. In chapter five with a little modifications on the same problem mentioned in the previous chapter lead us to define a new function space with different norm, the boundary condition for this problem can be considered as a generalization of the boundary conditions associated with the problem in chapter four. For these two chapters we illustrate our results by examples given at the end of each one. Whereas, in chapter six which can be considered as two parts, we investigate the existence and uniqueness for the solution of the nonlinear fractional differential equations involving Hadamard and Caputo-Hadamard fractional derivative associated with three points integral boundary conditions, for the applicability of our results we give some examples at the end of this chapter as well. Keywords: fractional differential equation, sequential, Caputo, Hadamard, nonlocal integral boundary conditions
Stability, Existence and Uniqueness of Boundary Value Problems for a Coupled System of Fractional Differential Equations
The current thesis investigates four different nonlinear systems of fractional differential equations and deals with the existence, uniqueness, and stability of their solutions. The first studied problem is a coupled system of fractional differential equations with four-point integral boundary conditions. Existence and uniqueness of solutions are established by applying the contraction mapping principle and Leray–Schauder’s alternative theorem. Finding and results are demonstrated and supported with numerical examples. The second studied case is a boundary value problem for a coupled system of nonlinear fractional differential equations, where the existence and uniqueness of solutions is proven by using the Banach’s fixed point theorem and Schauder’s alternative. Furthermore, the Hyers-Ulam stability of solutions is discussed, sufficient stability conditions are drawn, and supporting numerical results are presented. In the third problem, a coupled system of Caputo type sequential fractional differential equations with integral boundary conditions is studied. Similarly, existence and uniqueness of solutions are discussed and established by employing contraction mapping principle and Leray–Schauder’s alternative theorem, and Hyers-Ulam stability of the boundary value problem is investigated. The last problem is a nonlinear Caputo type sequential fractional differential equation with non-separated non-local integral fractional boundary conditions. Existence, uniqueness, and Hyers-Ulam stability of solutions are discussed and established, and theoretical findings are presented and supported by numerical examples. Keywords: fractional differential equation, sequential, Caputo, integral boundary conditions, stability, Hyers-Ulam stability, existence and uniqueness of solutions.
On Impulsive Sequential Fractional Differential Equations (ISFDE’s)
In this thesis, we study the existence and uniqueness of a nonlinear impulsive sequential fractional differential equations of order 1,2 involving Liouville-Caputo fractional derivative supplemented with the separate boundary value conditions. The subject of boundary value problem and fractional differential equations are very important in many fields of science and engineering. In fact, both sequential fractional differential equations and Impulsive fractional differential equations are studied individually from various perspectives. However, this topic combining both of them to produce a wider case namely, impulsive sequential fractional differential equations. By doing so, a new existence and uniqueness results of solutions are provided for the problems. Keywords: Nonlinear impulsive sequential fractional differential equations, Caputo fractional derivative, Banach fixed point theorem.
Kalman Filtering under Wide Band Noises
Kalman filtering is a powerful estimation method. One of its weaknesses is related to the white or colored nature of the disturbing noises in the Kalman filtering model. At the same time, real noises are rarely white or colored. They are mostly wide band. In this regard, white or colored noise Kalman filtering makes concessions on adequacy. This pushes system scientists to develop mathematical methods of estimation for systems corrupted by wide band noises. In applications, wide band noises are detected by their autocovariance and cross-covariance functions which do not allow modeling them uniquely. Therefore, it becomes important to develop estimation methods which are independent of a class of wide band noises, but dependent on the unique autocovariance and cross-covariance functions. Such results are called invariant results. In this paper, we prove a complete set of invariant equations for Kalman type filter for a linear signal-observation system corrupted by correlated wide band noises. This filter has a ready form to be used in applications, just respective numerical methods must be developed. We also discuss an application scenario for the proposed filter. Keywords: Wiener process, white noise, wide band noise, Kalman filter.