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Item A Study of Characterization of Some Regular Gamma Rings(University of Rajshahi, 2003) Nazneen, Ayesha; Paul, Akhil ChandraThe present thesis entitled, "A Study of Characterization of Some Regular Gamma Rings " is the outcome of researches carried out by me under the close supervision of Dr. Akhil Chandra Paul, Professor, Department of Mathematics, Rajshahi University. The thesis is of six chapters. In the first chapter we have tried to introduce all types of the conceptions of the complete thesis. In the second chapter we have given the definition of r - ring due to Barnes and of the relevant things. Various types of r- rings and their examples are also presented there. Some kinds of radical and corresponding theorems are also stated and important ones are proved. The definition of k - regular r- ring is given in the third chapter. Kyuno defmed this regular r - ring. We have tried to prove that the class of all k - regular r- rings forms a radical. Some of the characterizations of this r- rings are developed. We have also shown that k - regular r- ring without zero divisors is a skew r- field…….Item A Study on Convergence of Newton's Method in Real and Interval Number(University of Rajshahi, 2003) Rahman, Md. Majedur; Ali, Md. ZulftkarIn order to find the approximate numerical solution to a system of nonlinear equations as well as an integral and a differential operator equations, Newton's algorithm is widely used. L. V. Kantorovich [1948] and Moore [1977] studied the existence and uniqueness of solution to the system of nonlinear equations and their error bounds. M. Urabe [ 1965] also studied the existence and uniqueness of the solution to nonlinear operator equations (mainly differential operator equations). Kantorovich and Urabe's methods are two variants of Newton's method in some sense. We study the existence and uniqueness of solutions to the nonlinear systems and their error bounds. Our results will be stated in a theorem that ensures the best possible generalized error bound that is different from that given by Kantorovich and Moore. We also develop a technique that may be applied to find an approximate numerical solution to an algebraic as well as to a system of nonlinear equations both in real and interval number systems. Finally, we have treated the error estimation for the quasiperiodic solution to the Van der pol type differential operator equation based on Urabe's theorem.Item A Study on Distributive Nearlattices(University of Rajshahi, 1994) Rahman, Md.Bazlar; Noor, A.S.A.This thesis studies the nature of distributive nearlattices. By a nearlattice S we will always mean a (lower) semilattice which· has the property that any two elements possessing a common upper bound,have a supremum, Cornish and Hickman in their paper [14],referred this property as the upper bound property, and a semilattice of this nature as a semilattice with the upper bound property. Cornish and Noor in [15] preferred to call these semilattices as nearlattices as the behaviour of such a semilattice is closer to that of a lattice than an ordiary semilattice, In this thesis we give several results on near lattices which certainly extend and generalize many results in lattice theory, In chapter 1 we discuss ideals, congruences and other results which are basic to this thesis. We include some characterizations of distributive and modular nearlattices, We generalize the separation properties given by M.H.Stone for distributive lattices. We also show that the set of prime ideals of a nearlattice Sis unordered if and only if Sis semiboolean. Chapter 2 discusses the skeletal congruences of a distributive near lattice. Skeletal congruences on distributive lattices have been studied extensively by Cornish in [ 11], Here we extend several results of Cornish for nearlattices. We also introduce the notion of disjunctive nearlattices, A distributive nearlattice S with O is called disjunctive if for O a < b there is an element x ES such that x A a= 0 and O < x b, Then we give several characterizations of disjunctive nearlattices and semiboolean algebras using skeletal congruences, Finally we show that a distributive naerlattice is semiboolean if and only if 8 -----> ker8 is lattice isomorphism of Sc(S) onto KSc(S) whose inverse is the map .J ---> 8(J), In chapter 3, we discuss on normal and n-normal nearlattices, Normal lattices have been studied by several authors including Cornish [8] and Monteiro [34]; while n-normal lattices have been studied by Cornish [9] and Davey [16], In proving some of the results we have used Principle of Localization, which is an extension of lecture note of Dr. Noor on localization. This technique is very interesting and quite different from those of the previous authors,Item A study on Extended Newton-type Methods for Variational Inclusions(University of Rajshahi, Rajshahi, 2021) Khaton, Mst. Zamilla; Rashid, Mohammed HarunorIn this work, we deal with the two types of variational inclusions. Firstly, we consider the variational inclusion problem of the form 0 ∈ ζ(¯s) + g(¯s) + ξ(¯s), (A) where S and T are Banach spaces, ζ : S → T is differentiable in a neighborhood Υ ⊆ S of a solution s∗ of (A), g : S → T is differentiable at s∗ but may not differentiable in Υ and ξ : S ⇒ 2T is a set-valued mapping with closed graph. This work consists three parts and the main works we have done in this dissertation that are organized as follows. In the first part, particularly in Chapter 3, we study the Newton-type method for solving the variational inclusion problem (A) which is introduced in [2]. Under some suitable assumptions on the Fr´echet derivative of the differentiable function and divided difference admissible function, we establish the existence of any sequence generated by the Newtontype method and prove that the sequence generated by the method (3.1.3) converges linearly, quadratically and superlinearly to a solution of the variational inclusion (A). Specifically, when the Fr´echet derivative of the differentiable function is continuous, Lipschitz continuous and H¨older continuous, divided difference admissible function admits first order divided difference and the set-valued mapping is pseudo-Lipschitz continuous, we show the linear, quadratic and superlinear convergence by the method (3.1.3). In Chapter 4, we introduce and study the extended Newton-type-----Item A Study on Finitely Generated N-Ideals of a Lattice(University of Rajshahi, 1998) Ali, Md. Ayub; Noor, A. S. A.This thesis studies extensively the finitely generated n-ideals of a lattice. The idea of n-ideals in a lattice was first introduced by Cornish and Noor in studying the kernels around a particular element n, of a skeletal congruence on a distributive lattice. Then Latif in his thesis "n-ideals of a lattice" studied thoroughly on the n-ideals and established many valuable results. For a fixed element n of a lattice L, a convex sub lattice of L containing n is called an n-ideal. If L has a "O", then replacing n by 0, an n-ideal becomes an ideal and if L has a "1" then it becomes a filter by replacing n by 1. Thus, the idea of n-ideals is a kind of generalization of both ideals and filters of lattices. The n-ideal generated by a finite number of elements of a lattice is called a finitely generated n-ideal, while the n-ideal generated by a single. element is known as a principal n-ideal. Latif in his thesis has given a neat description on finitely generated n-ideals of a lattice a'nd has provided a number of important results on them.Item A Study on Fixed Point Iterative Procedures(University of Rajshahi, 2016) Khatun, Miss. Saleha; Ali, M. ZulfikarFixed point theory has fascinated hundreds of researchers since 1922 with the celebrated Banach’s fixed point theorem. Fixed point iterative procedures are one of the early achievements of fixed point theory for their usefulness to construct the solving technique of different nonlinear problems. Most of the physical problems of applied sciences and engineering are usually formulated as functional equations. Such equations can be written in the form of fixed point equations in an easy manner. It is always desired to develop an iterative procedure which approximates the solution of these equations in fewer numbers of steps. From this point of view, the main objective of our research is to fit a best fixed point iterative procedure whose working ability (rate of convergence) is better than that of the analogous fixed point iterative procedures. There exist a numeral number of fixed point iterative procedures in literature. But there raised a natural question that, “Which is the best fixed point iterative procedure under the equivalent situation?”. To find the answer of that question already many works have been completed by various renowned researchers; see for instance [12, 27, 37, 39, 47, 49] and their references. By the inspiration of these works here we have proposed a new three-step fixed point iterative procedure whose rate of convergence is better than that of analogous fixed point iterative procedures in case of contraction mapping. Using our new fixed point iterative procedure we have also established some weak and strong convergence theorems for non-expansive mapping and we apply these results to find the solutions of constrained minimization problems and feasibility problems. In the last part of our research, we have studied the fixed point iterative procedures with errors and proveda convergence theorem of multi-step Noor fixed point iterative procedure with errors for Zamfirescu operator, which generates the convergence theorems of rest fixed point iterative procedures with errors for the same operator.Item A Study on Interval Solutions of Nonlinear Systems(University of Rajshahi, 2011) Mollah, Md. Shirazul Hoque; Ali, M. ZulfikarSuppose we seek a solution of the nonlinear system f(x={f;(x1,x2, ..., Xn)} = 0 i = 1,2, ..., n where f1, f2, f3, ..., fn are continuous functions on an open set D in R”. There is good many methods for iterative interval solutions of system (I) for any such methods, R. E. Moore developed a technique for finding a safe starting point from which iterates converge, with a particular iterative method in mind, Krawczyk’s operator. Minoru Urabe established an existence and uniqueness theorem (1965) which helps verify the existence and uniqueness of an exact solution and to know the error bound to an approximate solution of a system like (1). His theorem assumes that all the computations are to be carried out in real numbers exactly. Our attempts will be made to combine M. Urabe's theorem and R. E Moore's technique theoretically as well as numerically considering interval version of Newton's method.Item A Study on Oxygen Diffusion Through Living Tissues(University of Rajshahi, 2009) Malek, Abdul; Hoque, Md. AshabulA theoretical study of oxygen diffusion process through living tissues and its various consequences have been presented. In this aspect we have reviewed some physiological terms and fundamental lows of diffusion. A Mathematical model of the partial pressure of oxygen across the alveolarpulmonary capillary membrane has been determined and expressed in term of membrane thickness due to the oxyhemoglobin dissociation curve. On the other hand, the effect of partial pressure of carbon dioxide has been found to be the function of partial pressure of oxygen. The partial pressure of oxygen along the pulmonary capillary has also been discussed when deoxygenated blood converts to the oxygenated blood taking Hill's modified oxyhemoglobin dissociation equation and Bohr effect. It is found that the partial pressure of oxygen increases with the increasing of capillary length. The mathematical equations have been developed based on the partial pressure of oxygen in the capillary blood is reached in equilibrium position. It is found that the molar flux decreases exponentially with the increasing of radial thickness of the capillary. Moreover, It is found that the molar flux of oxygen increases linearly with the increasing of diffusion coefficient of oxygen. A mathematical model of molar flux of oxygen has been developed across the capillary-tissue membrane by neglecting the convective transport across the capillary membrane. We have found that the maximum consumption rate of oxygen increases rapidly at initial stages after that it decreases with the increasing of wall thickness. Moreover, the result indicates that the pressure profile of oxygen along the wall is approximately linear with the increasing of the thickness of capillary wall.Item A Study on Standard n-Ideals of a Lattice(University of Rajshahi, 2014) Syeed, Ahmed, Abu Sadat; Latif, Md. AbdulThis thesis studies the nature of standard n-ideals of a lattice. The idea of n-ideals in a lattice was first introduced by Cornish and Noor. For a fixed element n of a lattice L, a convex sublattice containing n is called an n-ideals. If L has an 'O', then replacing n by 0, an n-ideal becomes an ideal. Moreover, if L has 1, an n-ideal becomes a filter by replacing n by 1. Thus, the idea of n-ideals is a kind of generalization of both ideals and filters of lattices. So, any result involving n-ideals will give a generalization of the results on ideals and filters with 0 and 1 respectively in a lattice. In this thesis we give a series of results on n-ideals of a lattice which certainly extend and generalize many works in lattice theory. Chapter-1, discusses n-ideals, finitely generated n-ideals and other results on n-ideals of a lattice which are basic to this thesis. We have shown that, a lattice L is modular (distributive) if and only if in (L), the lattice of n-ideals is modular (distributive). In chapter-2, we have discussed lattices and elements with special properties. Here we have proved the coincidence of standard and neutral elements in a wide class of lattices including modular lattices, weakly modular lattices as well as relatively complemented lattices. In modular lattices and relatively complemented lattices the proves of the results are trivial but in weakly modular lattices this prove is not so simple. In this chapter, we have proved the following results:Item A Theoretical Study on Some Aspects of Turbulent Flow(University of Rajshahi, 2003) Sultana, Mst. Shamima; Sarker, M. Shamsul AlamThe conception of turbulent flow and the accompanying transition from laminar to turbulent flow is of fundamental importance. In everyday life, we recognized three states of matter: solid, liquid and gas. Although different in many respects, liquid and gases have a common characteristic in which they differ from solids: they are fluids, lacking the ability of solids to offer permanent resistance to a deforming force. Fluids flow under the action of such forces, deforming continuously for as long as the force applied. The fluids may be classified into different types depending upon the presence of viscosity. Osborne Reynolds shows that two entirely different types of fluids flow exist. In general words, turbulent flow is a flow, in which the inertia force is dominating over the viscosity. On the other hand, Laminar flow is a flow, in which the viscosity of the fluid is dominating over the inertia force. Osborne Reynolds demonstrated this in 1883 through an experiment. Reynolds’s apparatus consist of a tank, containing water and a small tank containing dye. 0. Reynolds’s [50) was also the first to investigate in greater detail the circumstances of the transition from laminar to turbulent flow. The previously mentioned dye experiment was used by him in this connection, and he discovered the law of similarity which now bears his name, which states…………………………..Item An Analytical and Numerical Study on Fixed Point Theorems(University of Rajshahi, 2011) Asaduzzaman, Md.; Ali, M. ZulfikarFixed point theory has fascinated hundreds of researchers since 1922 with the celebrated Banach's fixed point theorem. There exists a vast literature on the topic and this is a very active field of research at present. Let X be a set and let T be a mapping from X intoX. A fixed point of Tis an element x EX such thatT(x) = x. In mathematics, a fixed-point theorem is a result saying that a function T will have at least one fixed point, under some conditions on T that can be stated in general terms. In other words, a fixed-point theorem is a theorem that asserts that every function that satisfies some given property must have a fixed point. Fixed point theorems give the conditions under which maps (single or multivalued) have solutions. Fixed point theory is a beautiful mixture, of analysis, topology, and geometry. If we have an equation whose explicit solution is not so easy to find, in that case we rewrite the equation in the form T(x) = x to find its solution by applying any suitable fixed-point theorem. This method can be applied not only to numerical equations but also to equations involving vectors or functions. In particular, fixed-point theorems are often used to prove the existence of solutions to differential equations. Fixed point theorems also play a fundamental role in demonstrating the existence of solutions to a wide variety of problems arising in social sciences, biology, chemistry, economics, engineering, physics and mathematics. For instance, the Banach [ 18, 40], Brouwer [ 18, 49] and Kakutani [ 18] fixed point theorems have been among the most-used tools in economics and game theory. Over the last 50 years the theory of fixed points has been revealed as a very powerful and important tool in the study of nonlinear phenomena.Item An analytical method for finding exact Traveling wave solutions of some nonlinear Evolution equations (nlees) in biological and Mathematical problems(University of Rajshahi, Rajshahi, 2021) Ullah, Mohammad Safi; Ali, M. Zulfikar; Roshid, Harun-Or-Most of the natural happenings can be present by nonlinear modeling. The soliton theory is a highly effective section of nonlinear sciences that includes soliton, multi-soliton, rational, breather line, breather kinky, lump and rogue wave solutions. Such solutions are essential to realizing the internal properties of the nonlinear models. This dissertation presents exact traveling wave solutions of the three nonlinear models such as the (2+1) Bogoyavlenskii’s breaking soliton (BBS) equation, the (2+1)-dimensional Benjamin-Bona-Mahony-Burgers (BBMB) equation and the (3+1)-dimensional Sharma–Tasso–Olver-like (STOL) equation by applying Hirota bilinear method. By this method, we construct the bilinear form and find the interaction solutions of the above three models. We determine the multi-soliton and their interaction solutions of the BBS model and STOL model. Various properties of the obtained solutions are illustrated clearly with a number of 3D plot, 2D plot, density plot, curve plot and contour plot by choosing suitable parametric values via the computational software Maple 18.Item Analysis of Viscous Incompressible Fluid Flows and Heat Transfer(University of Rajshahi, 2010) Uddin, Md. Sharif; Pk, M. Wazcd Ali; Gorachand LayekWe first present a brief ideas and principles in "Fluid Mechanics" which may serve as the background materials of viscous, incompressible laminar Newtonian or non-Newtonian fluid flow problems considered in the present thesis. The basic equations viz, the continuity equation and the momentum equation for the motion of viscous incompressible fluid under the limits of continuum hypothesis are presented. The laws of classical mechanics apply throughout the continuous medium under consideration. The length scale of the flow is always taken to be large compared with the molecular mean-free-path. so that the fluid can be considered as a continuum. It excludes the flow of gases at very tow pressures i.e., rarefied gases. Liquid flows can usually be treated as incompressible fluid. The classification of fluids, say, Newtonian and non-Newtonian, Prandtrs boundary layer concept, boundary layer equations, concept of similarity variable for analyzing the viscous flow problems, group theoretic approach of for finding invariant solution of an incompressible viscous fluid flows, arc discussed systematically. Using the similarity variable for some specific flow problems, a set of nonlinear ordinary differential equations, known as self similar equations arc derived. Analytical or closed-form solution as well as numerical solution of these nonlinear differential equations relating to particular class of flow problems arc obtained and the corresponding flow quantities are shown graphically and discussed physically. In general, matter is found to exist in four phases or states e.g. solid. liquid, gas and plasma (ionized gases). Out of these last three states of matter are termed as fluid. Fluid mechanics is the subject in which wc deal with the flow problems pertaining to one of these phases e.g. liquid. gas and plasma or combination, mainly of the first two or last two phases. Essentially, the fluid flow problems are of widely spread interest in various fields of engineering as well as in meteorology, oceanography and other subjects of physical sciences. We live in a world which is largely a fluid. Air, oceans, rivers and so on arc all fluids whose behaviour is mostly described using the principles of continuum hypothesis. The constitutive equations are framed using some assumptions based on the material behaviors of fluid and the flow conditions. lt's study 1s important to physicists or applied mathematicians whose main interest is in understanding the related physical phenomena. On the other hand fluid dynamical engineers worked out many problems of practical interest using empirical formula. Also an understanding of this subject helps us to explain a variety fascinating natural phenomena around us.Item Analysis on Viscous Incompressible Thermal Flow on a Ventilated Enclosure by Finite Element Method(University of Rajshahi, 2014) Ahammad, Main Uddin; Rahman, Md. Lutfor; Rahman, Md. MustafizurHeat transfer in enclosure in which the influence of free (natural) and forced convection (mixed convection) are of comparable magnitude occurs frequently in engineering situations. The applications include the heat transfer improvement in heat exchanger devices, design of solar collectors, thermal design of building, air conditioning, cooling of electronic circuit boards, lubrication technologies, chemical processing equipment etc. Convection in ventilated enclosures containing obstruction has gained recent research significance as a means of heat transfer enhancement. The mathematical model of the present problem is governed by the couple equations of conservation of mass, momentum and energy. Discretization of the governing equations is achieved using a finite element scheme based on the Galerkin weighted residuals method. Then Newton–Raphson iterative algorithm is used to obtain the solutions of the obtained algebraic equations. Comparisons with previously established on particular cases of the problem are performed and the results show excellent agreement. Firstly, for mixed convection flow the effect of inlet and outlet position of a square ventilated enclosure with a centered heat generating solid body has been investigated. The bottom wall of the enclosure is kept at a uniform constant temperature, while the rest three walls of the enclosure are assumed adiabatic. A transverse uniform magnetic field is imposed in the horizontal direction normal to the right vertical wall. An external flow enters the cavity through an inlet opening whereas it exits via another outlet opening. After that, the effect of pertinent parameters in the considered flow problem in this thesis was analyzed for three different types of internal cavity solid body (heat generating, heat conducting, adiabatic) for a selected BT (bottom inlet and top outlet) configuration. Obtained results from the present study are presented in the form of streamlines, isotherms, average Nusselt number along the bottom heated surface and average fluid temperature in the cavity for each of four configurations as well as three different confined blocks for the pertinent parameters namely Reynolds number, Prandtl number, Hartmann number, solid-fluid thermal conductivity ratio, solid block diameter at the three values of Richardson number, varying from 0.1 to 10. The computational findings of this thesis reveals that both the flow and the thermal fields strongly depend on the parameters Reynolds number Re, Prandtl number Pr, Hartmann number Ha at the three convective regimes (Ri = 0.1, 1, 10). The centered solid body of the enclosure influences the steamlines pattern slightly for the smaller dimension of the block D whereas it has a considerable disparity in temperature distribution inside the enclosure. It is also observed that the solid-fluid thermal conductivity ratio K have insignificant effect on the flow fields and have significant effect on the thermal fields at the three convective regimes.Item Analytical Invcstlgations in Tubulent and IVIlID Tttdmlent Flow(198) Rahman, Md. Lutfor; Sarker, M.Shamsul AlamThe thesis entitled "Analytical Investigations in Turbulent and MHD Turbulent Flow" is being presented for the award of the degree of Doctor of Philosophy in Mathematics. It is the outcome of my research conducted in the Department of Mathematics, Rajshahi University during the year 1994-1998 under the guidance of Dr. M. Shamsul Alam Sarker, Department of Mathematics, Rajshahi University, Rajshahi-6205, Bangladesh. The whole thesis has been divided into six chapters. The first is an introductory chapter and gives the general idea of turbulence, magnetohydrodynamic turbulence and its principal concepts. Throughout the work we have considered the flow of fluids to be isotropic and homogeneous. The notions generally adopted are those used by Batchelor, Chandrasekhar and Deissler in their research papers. Number inside brackets [ ] refer to the references which are arranged alphabetical at the end of the thesis. In the second chapter, we have derived the equation for tl1e rate of change of magnetic field covariance in MHD turbulent flow. 111e result shows that the defming scalars of the magnetic field covariance depend on the defining scalar H of two point magnetic field correlation. In tl1e third chapter, the decay of turbulence at ti.mes before the final period in presence of dust particles is studied. Two and three point correlation equation is used to obtain a relation for tl1e triple correlations and the equation is made detenninate by neglecting the quadruple correlations. Finally, we obtained tl1e energy decay law of dusty fluid turbulence before tl1e final period. In the fourth chapter, we have studied the decay of dusty fluid MHD turbulence before the final period. Tirree point correlation equation is used to obtain a relation for the triple correlations applicable at times before the final period. In this case the equation is made determinate by neglecting tl1e quadrnple correlations. Finally, we obtain the energy decay law of dusty fluid MHD turbulence at times before the final period.-------Item Analytical Methods to Investigate Exact Solutions for Space-Time Fractional Differential Equations Arising in the Real Physical Phenomena of Mathematical Physics and Biology(University of Rajshahi, Rajshahi, 2021) Rahman, Zillur; Ali, M. Zulfikar; Roshid, Harun-or-Fractional derivatives are most important to accurate nonlinear modeling of various real-world difficulties in applied nonlinear science and engineering incidents especially in the fields of crystal, optics and quantum mechanics even in biological phenomena. The investigation of exact solutions of such nonlinear models has great important to visualize the nonlinear dynamics. We consider the space-time fractional nonlinear differential equations for pulse narrowing transmission lines model, the space-time fractional Equal-width (s-tfEW) and the space-time fractional Wazwaz-Benjamin-Bona-Mahony (s-tfWBBM), complex Schrodinger and biological population models, the complex time fractional Schrodinger equation (FSE) and low-pass electrical transmission lines equation (ETLE) are studied with the effective unified method, Jacobi elliptic expansion function integral technique, generalized Kudryshov technique, modified simple equation (MSE) method respectively. As a result, we get some solitary wave solutions in the form of hyperbolic and combo hyperbolic-trigonometric :functions including both stable and unstable cases. We obtain kink wave, bright bell wave, dark bell wave, combo periodic-rogue waves, combo M-W shaped periodic-rogue waves in stable cases, and singular kink type in unstable solitonic natures. Lastly, we proposed an Improved Kudryashov method for solving any nonlinear fractional differential models. We apply the proposed approach to the nonlinear spacetime fractional model leading wave spread in electrical transmission lines (s-tfETL), the spacetime M-fractional Schrodinger-Hirota (s-tM-fSH) and the time fractional complex Schrodinger (tfcS) models to verify the effectiveness of the propose approach. The implementations of the introduced new technique on the models provide us periodic envelope, exponentially changeable soliton envelope, rational, combo periodic-soliton and combo rational-soliton solutions, which are much interesting phenomena in the nonlinear sciences. Beside the scientific derivation of the analytical findings, we represent the results graphically for clear visualization of the dynamical properties.Item Asymptotic Method for Time Dependent Nonlinear Differential Systems with Slowly Varying Coefficients(University of Rajshahi, 2013) Roshid, Harun-Or-; Ali, M. Zulfikar; Dey, PinakeeAlmost all perturbation methods are developed to find periodic solutions of nonlinear system where transients are not considered. First Krylov and Bogoliubov introduced a perturbation method which is well known as “asymptotic averaging method” to discuss the transients in the second order autonomous systems with small nonlinearities. Later, this method has been amplified and justified by Bogoliubov and Mitropolskii. Mitropolskii has extended the method for slowly varying coefficients to determine the steady state periodic motions and transient process. In this dissertation, we have modified and extended the KBM method to investigate some fifth order and second order nonlinear systems in both cases with constant and slowly varying coefficients. At first, a fifth order damped nonlinear autonomous differential system is considered and a perturbation solution is developed. Then a procedure is developed for the same system with damped taking three of eigenvalues are real. After then we considered fifth order systems for over damped with small nonlinearity to obtain the transient response. We also developed a formula for fifth order critically damped nonlinear systems to control micro vibration, in micro and nano-technological industries that bring the system to equilibrium as quickly as possible without oscillating. After then we presented an analytical technique based on the extended Krylov-Bogoliubov-Mitropolskii method (by Popov) to determine approximate solutions of nonlinear differential systems whose coefficients change slowly and periodically with time. Furthermore, a non-autonomous case also investigated in which an external force acts in this system. At last, Krylov-Bogoliubov-Mitropolskii (KBM) method has been extended to certain damped-oscillatory nonlinear systems with varying coefficients. The implementations of the methods are illustrated by several examples.Item Asymptotic Methods for some third order Nonlinear Differential Equations(University of Rajshahi, 1995) Alam, Md. Shamsul; Sattar, Muhammad AbdusIn this thesis, we investigate the oscillations of third order nonlinear systems by the asymptotic method. The asymptotic method of Krylov-Bogoliubov-Mitropolskii (KBM) is a popular technigue for obtaining analytic solution of a second order nonlinear oscillatory system. First a third order nonlinear differential system modeling nonoscillatory process and characterized by critical damping is considered and a new perturbation technigue is developed, based on the work of Krylov-Bogoliubov-Mitropolskii, to £ind the solution of the system. Then a method is presented unifying both third order damped and overdamped systems. This method is a generalization of Bogoliubov·s asymptotic method and covers all the cases when the roots of the corresponding linear equation are real, real and complex, and real and purely imaginary.Later a third order forced nonlinear differential system modeling oscillatory process is considered and a new perturbation technique is developed to find the solution of the system. The methods are illustrated by several examples.Item Bekenstein-Hawking Entropy by Energy Quantization from Different Black Holes(University of Rajshahi, 2016) Hossain, Md. Jakir; Rahman, Md. Atiqur; Hossain, Md. IliasWe investigate the Bekenstein-Hawking entropy from different types of non-rotating black holes in de Sitter and Anti-de Sitter spaces by using the energy quantization mechanism in analogy with Bohr’s atomic model. We quantize the energy of the particle from the quantization of angular momentum. We also investigate the change of entropy between two nearby states as well as the thermal emission rate. In the limiting case all the results coincide with one another for the black holes in de Sitter and Anti- de Sitter spaces. The thesis is organized as follows: In chapter 1 we provide a brief discussion about our work of studying Bekenstein-Hawking entropy from black hole spacetime. In chapter 2 to 7 by using the quantization method we investigate the Lagrangian and canonical momenta of test particle, Radial motion and Effective Potential, Quantization of Circular Orbit, Energy quantization and Hawking Radiation for Schwarzschild, Schwarzschild-de Sitter (SdS), Schwarzschild Anti-de Sitter (SAdS), Reissner-Nordström (RN), Reissner-Nordström-de Sitter (RNdS), Reissner-Nordström Anti-de Sitter (RNAdS), black holes. Our new process is universally robust and the entropy framework given in this work indeed support the new perspective on quantum properties of gravity beyond classical physics, however, suggests a new idea to unify gravity with quantum theory and in the limiting case, the results are in line with that obtained by Sakalli et al. and He et al.’s method of the black hole.Item Characterizations of K-Derivations and Bi-Derivations on Lie Ideals of Gamma Rings(University of Rajshahi, 2016) Nazneen, Ayesha; Paul, Akhil ChandraThe present thesis entitled, "CHARACTERIZATIONS OF K-DERIVATIONS AND BI-DERIVATIONS ON LIE IDEALS OF GAMMA RINGS" is the outcome of researches carried out by me under the close supervisions of Dr. Akhil Chandra Paul, Professor, Department of Mathematics, Rajshahi University, Rajshahi.The main goal of this thesis is to characterize kderivations and then to generalize it. At the begining we introduce the concept of gamma rings and then we have mentioned the extention works on gamma rings, that means kderivations, jordan k-derivations, Jordan generalized k-derivations, biderivations e.t.c. We have also mentioned the mathemeticians who have worked on these fields. In the first chapter we mainly described our works on Jordan kderivations. Here we have given the definition of k-derivation, Lie ideal, Jordan k-derivation etc. Some examples are also given. Here we define cpa (u, v) for u, v E U, a E r; where U is a Lie ideal of a r-ring M.When M is a 2-torsion free prime r-ring , then we have proved that every Jordan k-derivation is a k-derivation also. In the second chapter, we have discussed about Jordan generalized kderivation. Here we have define a new additive mapping, which is a generalized form of k-derivation.We have defined it and then defined Jordan generalized k-derivation. For a generalized k-derivation we have defined it with k-derivation. We added some examples. We have defined Jordan generalized k-derivation on Lie ideals. Here we have also defined \Va (u, v), for u, v E U, a E r; where U is a Lie ideal of a r-ring M. At the end of this chapter we have proved that every Jordan generalized k- derivation is a Jordan k-derivation and so is a k-derivation on a Lie ideal of a 2-torsion free prime r-ring M also. We have worked on semi prime r -rings in the third chapter. In this chapter we have characterized semi prime r -rings and proved that every Jordan k-derivation on a Lie ideal U of a 2-torsion free r-ring Mis a kderivation on U of M if Mis semi prime. The conception of Left centralizer is presented in the forth chapter. We have denoted it by T. Here we have mentioned an additive mapping Ba( u, v) for all u, v E U; a E r.we have also defined Jordan left centralizers. Here we have proved that every Jordan Left centralizer T is a Left centralizer ifM is a 2- torsion free semiprime r-ring. In the fifth chapter, we have discussed Jordan generalized k-derivations on Lie ideals of semiprime r-rings. We have studied Jordan generalized k-derivations earlier. We have worked those on semiprime r-rings. With a special condition we have proved first that every Jordan generalized kderivation on a Lie ideal U of a 2-torsion free semiprime r-ring M is also a generalized k-derivation on U of M. Then we have proved the same result without any special condition by using the left centralizer. In the sixth chapter, we have studied bi-derivations. We have defined symmetric mapping, bi-derivation, symmetric bi-derivation etc. Here we have also defined trace, which is associated with a bi-derivation. Using different types of conditions, we have proved that either the Lie ideal U is contained in Z(M) or the trace d is zero, if M is a prime r -ring. We have developed these results for a semiprime r-ring also. We have worked on bi-additive mappings on semiprime r-rings in chapter seven. In this chapter we have studied the commutativity of a Lie ideal of a 2-torsion free semiprime r-ring. In the eighth chapter, we have discussed symmetric bi-derivations with symmetric generalized bi-derivations. Here we have worked with two symmetric bi-derivations associated with their respective traces and have found some important results. We have worked on commutativity with symmetric bi-derivations in the nineth chapter. K. K. Dey and A. C. Paul have worked on symmetric biderivations. Some of their results are extended here on Lie ideals of prime and semiprime r-rings. At the first stage of this thesis we have discussed about Nobusawa rrings. In the tenth chapter we have tried to characterize k-derivations on Lie ideals of Nobusawa r-rings. When M is a r-ring, then it is clear that r is also an M-ring. In the basis of this idea we have found ad-derivation on a Lie ideal Q of an M-ring r. In this chapter, we have tried to find out the same types of results on these two categories of derivations on respective Lie ideals of those rings. Also we have worked on d2 and d3. In the eleveth chapter, we have described Left k-derivation. We have defined Jordan left k-derivation also. In this chapter we have used Mas a completely prime r-ring. We have proved that if M is a 2-torsion free completely prime r-ring and U is a Lie ideal of M, then every Jordan left k-derivation on U ofM is also a left k-derivation on U ofM. A complete bibliography which have been helped us to finish my total research is also added at the end of this thesis.