MPhil Thesis

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Study of Radicals and Semisimple Classes of Rings
(University of Rajshahi, 2002) Dey, Kalyan Kumar; Majumdar, Subrata
The concept of the radical of a ring was introduced by Artin for rings with the descending chain condition with a view to obtaining a nice structure theorem for the ring. The idea was to single out the Toblerone part, of a ring, called the radical of the ring, and factor out the original ring with respect to the radical. The resulting ring, termed, semi simple has a nice description. Radica1s for rings without chain conditions were proposed by Koethe, Jacobson, Brown, McCoy, Levitzki and others for a similar purpose in an attempt to generalize Artin 's radical. All these attempts were later further generalized by Kurosh and Amitsur to define the concept of a general radical of a ring and the cones ponding semi simple ring and study these in their generality. Andrunakievic advanced these studies further. The class of rings which are radicals of themselves with respect to some radical is called a radical class, or simply, a radical, and the correponding class of the semisimple rings is called a semisimple class. A class rings may be simultaneously a radical ring with respect to some radical and a semisimple ring with respect to another radical. Such a class of rings is called a semisimple radical class. In this thesis we have studied radical classes, semisimple classes and semisimple radical classes of rings………………….
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.
Investigation of the Soliton and Multi-soliton Solutions of nonlinear evolution equations In Mathematical Physics
(University of Rajshahi, Rajshahi, 2020) Hossen, Md. Belal; Ali, M. Zulfikar
Nonlinear evolution equations (NLEEs) play a noteworthy role in various scientific and engineering fields such as applied mathematics, plasma physics, fluid dynamics, optical fibers, biology, solid state physics, chemical physics, mechanics and geochemistry. Various effective procedure have been developed to solve NLEEs. In this work, we have discussed applications of two types methods: first type is modified double sub-equation (MDSE) method which is apply in the (1+1)-dimensional Burger equation, the (1+1)-dimensional Gardner equation and the (1+1)- dimensional Hirota-Ramani equation and secondly, Hirota’s Bilinear method which is apply in (2+1)-dimensional Breaking Soliton, the (2+1)-dimensional asymmetric Nizhnik-Novikov- Veselov equations, and (3+1)-D generalized B-type Kadomtsev-Petviashvili equation. Using Modified double sub-equation method, we have presented some complexiton solutions in terms of trigonometric, hyperbolic functions. Finally, the interaction phenomena of the achieved complexiton solutions between solitary waves and/or periodic waves are presented with in depth derivation. Based on the bilinear formalism and with the aid of symbolic computation, we determine multisolitons, breather solutions, rogue wave, lump soliton, lump-kink waves and multi lumps using various ansatze’s function. We notice that multi-lumps in the form of breathers visualize as a straight line. Besides this, the breather wave degenerate into a single lump wave is determined by using parametric limit scheme. Also, we reflect a new interaction solution among lump, kink and periodic waves via ‘rational-cosh-cos’ type test function. To realize dynamics, we commit diverse graphical analysis on the presented solutions. Obtained solutions are reliable in the mathematical physics and engineering.----
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.
A Study of Characterization of Some Regular Gamma Rings
(University of Rajshahi, 2003) Nazneen, Ayesha; Paul, Akhil Chandra
The 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…….
An Analytical and Numerical Study on Fixed Point Theorems
(University of Rajshahi, 2011) Asaduzzaman, Md.; Ali, M. Zulfikar
Fixed 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.
On Supra Fuzzy Topological Spaces
(University of Rajshahi, 2011) Hoque, Md. Fazlul; Ali, Dewan Muslim
The fundamental concept of a fuzzy set was introduced by L. A. Zadeh [111] in 1965 to provide a foundation for the development of many areas of knowledge. Consequently, this provides a natural frame work for generalizing many algebraic and topological concepts in various directions such as fuzzy groups, fuzzy rings, fuzzy vector spaces, fuzzy supra topology, fuzzy infra topology, fuzzy bitopology etc. many other branches of mathematics have been developed all over the world during the last five decades. In 1968, Chang [19] introduced the concepts of a fuzzy topological space by using the fuzzy set. Wong [105], Lowen [60], Hutton [48], katsaras[52], Ali[3], Pu and Liu[72], etc., discussed various aspects of fuzzy topological spaces. Ying [74] introduced fuzzifying topology and developed this in a new direction with the semantic methods of continuous valued logic. In the frame work of fuzzifying topology, Sinha [93] introduced and studied To-, T1-, T2(Hausdorff)-, T3(regular)-, T4(normal)-, separation axioms. A.S. Mashhour et al. [ 64] introduced and studied the concepts of the family of fuzzifying semi open sets, fuzzifying neighborhood structure of a point and fuzzifying semi-closure of a fuzzy set. A.S. Mashhour et al. [64] also introduced and studied the R-0 and R1 separation axioms and studied their relations with the T1 and Ti-separation axioms respectively. Also, in fuzzifying topology they introduced and studied semi-To-, semi-Ro-, semi-Ti-, semi-R1-, semi-T2(semi Hausdorff)-, semi-T3(semi regular)-, semi-T4(semi normal)-, separation axioms. In 1983, A.S. Mashhour et al. [64] introduced supra topological spaces and studies s-continuous function and s • -continuous functions. In 1987, M.E. Abd EL-Monsef et al. [1] introduced the fuzzy supra topological spaces and studied fuzzy supra continuous functions and characterized a number of basic concepts..........
A Study on Oxygen Diffusion Through Living Tissues
(University of Rajshahi, 2009) Malek, Abdul; Hoque, Md. Ashabul
A 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.
On Ro and R1 Properties in Fuzzy Topological Spaces
(University of Rajshahi, 2011) Azam, S. M. Faqruddin Ali; Ali, Dewan Muslim
The goal of this thesis is to find out some new R1 -concepts for fuzzy topological spaces. Besides some concepts of fuzzy R0, R1, T0, T1, T2 and regular topological spaces that are already existing in the literature are recalled. In this work, twelve R1 -axioms of fuzzy R1 -topological spaces are introduced and studied in detail. Interrelations among various R1 concepts of fuzzy topological spaces are discussed. In analogy with the well-known topological properties, a complete answer is given with regard to all possible (R1 ^ T0 = T2) and (R1 ⇒ R0)-type implications for fuzzy topological spaces. It is also shown that, though a regular topological space is also a R1 -topological space, this is not true for fuzzy topological spaces.
Study On Some Problems of Non-Newtonian Fluid Mechanics
(University of Rajshahi, 2010) Hossain, Md. Aslam; Pk, M. Wazed Ali
This is an elaborate discussion of main theme ideas and the condition in "Fluid mechanics", which help us to realized a clear concept of the background and elements of various viscous Newtonian and non-Newtonian flow problems. The basic equation viz, the equation of a visco-elastic fluid of oldroyds model, the equation of continuity, the equation of motion and the equation for isotropic incompressible fluid of Newtonian and non-Newtonian. Here we also have shown the equation the velocity profile in the dimensionless form. Fluid mechanics is one of the engineering science that form the basis for all engineering like meteorology, oceanography and other subject of physical sciences. The subject branches out into various specialiies such as aerodynamics, hydraulic engineering, marine engineering, gas dynamics and rate processes. It deals with the statics, kinematics and dynamics of fluids, since the motion of a fluid is caused by unbalanced forces exerted upon it. Available methods of analysis stem from the application of the following principles concepts and laws, Newton's laws of motion, the first and second laws of thermodynamics, the principle of conservation of mass, equations of state relating fluid properties. Newton's law of viscosity, mixing-lenth concepts and restrictions caused by the presence of boundaries. In fluid flow calculations, viscosity and density are the fluid properties most generally encountered; they play the principal roles in open-and closedchannel flow and in flow around immersed bodies. Surface tension effects are of importance in the formation of droplets, in flow of small jets and in situations where liquid-gas-solid or liquid-liquid-solid interfaces occurs, as well as in the formation of capillary waves. The property of vapor pressure, accounting for changes of phase from liquid to gas, becomes important when reduced pressures are encountered. In this chapter fluid properties are discussed, as well as units and dimensions and concepts of the continuum.

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