MPhil Thesis

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Asymptotic Methods for some third order Nonlinear Differential Equations
(University of Rajshahi, 1995) Alam, Md. Shamsul; Sattar, Muhammad Abdus
In 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.
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………………….
Eulerian Method for Third Order Linear Systems
(University of Rajshahi, 2003) Haque, Md. Abdul; Sattar, M. Abdus
In this thesis, we have studied the solutions of the various types of third order linear systems of ordinary differential equations. In chapter I, we have considered the third order linear homogeneous system with constant coefficients and developed Eulerian methods for all possible cases of the characteristic roots of the variational matrix. In chapter 2, we have considered the third order linear nonhomogeneous system with constant coefficients. The method covers all the cases when the roots of the characteristic equation of the corresponding homogeneous linear system are real and distinct, real and equal and complex. In finding particular solutions for the nonhomogeneous system of equations we have used the method of variation of parameters. Finally, we have obtained solutions of this system with the help of Crammer's rule. In chapter 3, we have considered the generalized form of third order linear nonhomogeneous system with constant coefficients. We have extended the Eulerian method and developed new techniques for obtaining solutions of this system. In chapter 4, we have discussed the third order linear nonhomogeneous system with variable coefficients. This problem is very difficult to solve, so we have examined a special case of this problem. By using a suitabletransformation, we have reduced it to a third order linear nonhomogeneous system with constant coefficients and have found solutions by the method of Chapter 2. We have illustrated all the methods by several suitable examples.
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…….
A Theoretical Study on Some Aspects of Turbulent Flow
(University of Rajshahi, 2003) Sultana, Mst. Shamima; Sarker, M. Shamsul Alam
The 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…………………………..
Some Topics in Fuzzy Topological Spaces
(University of Rajshahi, 2004) Hossain, Md. Sahadat; Ali, Dewan Muslim
The concept of fuzzy set was introduced by the American Mathematician L. A. Zadeh for the first time in 1965. This provides a natural framework for generalizing many algebraic and topological concepts in various directions. Accordingly fuzzy groups, fuzzy ideals, fuzzy rings, fuzzy vector space, fuzzy measure, fuzzy topology, fuzzy topological groups, fuzzy topological vector space and many other branches have been developed all over the world during the last four decades. It is still developing in many directions. While reviewing the literature in fuzzy topology, we have seen the gap in separation axiom of fuzzy topological space, the counter part of which in ordinary topological space drew attention of several eminent mathematicians like Azad, K. K.; Chang, C. L.; Ali, D. M.; Wuyts, P.; Srivastava, A. K. and Lowen, R. etc. We aim to develop of theories of Fuzzy To, Fuzzy T1, Fuzzy T2, Fuzzy Ro, Fuzzy R1, Fuzzy Regular and Fuzzy Normal spaces analogous to its counterpart in ordinary topology. The material of the thesis has been divided into six chapters, a brief scenario of which we present as follows. Chapter one incorporates some of the basic definitions and results of fuzzy set, fuzzy topology and mappings. These results are ready references for the work in subsequent chapters. Results are stated without proof and can be seen in the papers referred to.
On Some Universal Constructions for Fuzzy Topological Spaces
(University of Rajshahi, 2005) Rashid, Mohammed Harunor; Ali, Dewan Muslim
In this work, many known universal objects such as products, equalizers, pullbacks and their duals, namely, coproducts, coequalizers and pushouts for fuzzy topological spaces are studied in detail. Another known concept of inverse and direct limits on inverse and direct system of fuzzy topological spaces are also studied in more detail. Besides this, we explored some properties of the singular homology groups of fuzzy topological spaces by using the concepts of direct limits on fuzzy direct systems of fuzzy topological spaces.
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.
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.
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.
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..........
On Connectedness Concepts in Fuzzy Topological Spaces
(University of Rajshahi, 2013) Shayid, Md.Abu; Ali, Dewan Muslim
The goal of the thesis is to find out some new connectedness concepts in fuzzy topological spaces. Some concepts of connectedness in fuzzy topological spaces that already exist in the literature are recalled here also. In this work, various type of connectedness like Ci - connectedness (i = I, 2, 3, 4), (C3) - connectedness, stronger forms of connectedness are studied in detail. Interrelations between various connectedness concepts in fuzzy topological spaces are discussed. In each chapter of this thesis, we give several possible definitions, both existing and new, of a concept and then compare the resulting concepts and determine thereby the interrelations among them. Some other properties of these concepts have also been discussed.
Stability Analysis of Some Mathematical Models of Epidemics
(University of Rajshahi, 2013) Islam, Md. Saiful; Asaduzzaman, Md.; Mondal, Md. Nazrul Islam
The spread of communicable diseases is often described mathematically by compartmental models. Many epidemiological models have a Disease Free Equilibrium (DFE) at which the population remains in the absence of disease. The classical Susceptible Infected Removed (SIR) models are very essential as conceptual models like as predator-prey and competing species models in ecology. Some Susceptible Infected (SI) and Susceptible Infected Susceptible (SIS) type models have been considered in this study. There are two major types of control strategies available to limit the spread of infectious diseases, viz. pharmaceutical interventions (drugs, vaccines, etc.), and non-pharmaceutical interventions (social distancing, quarantine, etc.). Vaccination is important for the elimination of infectious diseases as an effective preventive strategy. Vaccination of susceptible individual has been introduced through Susceptible Infected Removed Susceptible (SIRS) models. Effective vaccines have been used successfully to control smallpox, polio and measles. Some models have been presented in this study for the transmission dynamics of infectious diseases to analyze the stability of various equilibrium points mathematically. Some Susceptible Vaccinated Infected Susceptible (SVIS) and Susceptible Vaccinated Infected (SVI) models have been introduced in this study by including a new compartment ‘V’ for vaccinated individual in SIS and SI type models respectively. The above models have various kinds of parameters. Mainly the stability is analyzed by bifurcation curves in the SVIS models. The basic reproductive number (R0) can be calculated due to DFE in the SVI model. Some controlling methods have been given by changing the parameters in the SVI model through R0.
New Approximate Solution of Non-Linear Differential Systems
(University of Rajshahi, 2014) Pervin, Mst. Razia; Shanta, Shewli Shamim
Most of the perturbation methods are developed to find periodic solutions of nonlinear systems; transients are not considered. At 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 processes. In this dissertation, we have modified and extended the KBM method to investigate some second order nonlinear systems. Firstly, a second order time dependent nonlinear differential system is considered. Then a new perturbation technique is developed to find an asymptotic solution of nonlinear systems in presence of an external force. Finally, this technique is used to obtain an asymptotic solution of a time dependent nonlinear differential system with slowly varying coefficients using the extended KBM method. These methods are illustrated with several examples.
A Study on Standard n-Ideals of a Lattice
(University of Rajshahi, 2014) Syeed, Ahmed, Abu Sadat; Latif, Md. Abdul
This 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:
Separation Axioms on L-Topological Spaces
(University of Rajshahi, 2015) Islam, Rafiqul; Hossain, Md. Sahadat
The notion of a fuzzy set, as proposed by L.A. Zadeh[90] in 1965 to provide a foundation for the evolution of many areas of knowledge. After then in quick succession, L-fuzzy sets were introduced by Goguen[24] in 1967. As a result, this provides a natural frame work for generalizing many algebraic and topological concepts in various directions such as L-fuzzy sets, fuzzy logics, fuzzy control, fuzzy groups, fuzzy rings, fuzzy vector spaces, fuzzy topology, fuzzy bitopology, L-topology etc. Many other branches of mathematics have been developed all over the world during the last five decades. Chang [12] first introduced and studied the concept of a fuzzy topological space by using the fuzzy set in 1968. Hutton[31-34] ,Reilly[33-34],Wong [83-84], Lowen[47-48], Srivastava[73-79], Dude[16-17], Cutler[14], Ying[88], Ali[2-9], Hossain[26-30], Pu and Liu[53-54], etc., discussed and developed various aspects of fuzzy topological spaces. Ying [88] introduced fuzzifying topology and developed this in a new direction with the semantic methods of continuous valued logic. With the help of fuzzifying topology, Sinha[69-70] introduced and studied T0, T1, T2-(Hausdorff), T3-(regular), T4- (normal), separation axioms. Mashhour et al. [51-52] introduced and studied the concepts of the family of fuzzifying semi-open sets, fuzzifying neighbourhood structure of a point and fuzzifying semi-closure of a fuzzy set.................
Generalizations of Some Properties of Topological and Bitopological Spaces
(University of Rajshahi, 2015) Biswas, Sanjoy Kumar; Akhter, Nasima
The thesis is concerned with generalizations of some important and interesting properties of topological and bitopological spaces in a span of four chapters. The first chapter constitutes an introduction and study of (i) a weak form of strong continuity, (ii) RC- continuity, (iii) perfect continuity, (iv) contra- precontinuity and (v) contra continuity in bitopological spaces . It thus generalizes the corresponding concepts in topology introduced by Donchev, Jafari and Noiri and studied by them. In addition, generalizing the works of Ekici and Noiri, as investigation of relationships between graphs and contra δ-precontinuous functions in bitopological spaces has also been made in this chapter. In the second chapter the problems of δ-compactness of topological spaces has been generalized to the corresponding properties in bitopological spaces. Some important properties of δ-compactness in bitopological spaces have been established, which are generalizations of results of park, Srivastava and Gupta. Also, a characterization of δ-Hausdorff bitopological spaces has been made and some properties of such spaces have been established, generalizing results of Srivastava and Gupta. The third chapter introduces the notions of weakly β-continuous functions in tritopological spaces and investigates several properties of these functions, thus generalizing the corresponding works in topological spaces by Khedr, Al-Areefi and Noiri and in bitopological spaces by Tahiliani. In the fourth chapter the idea of density topology has been introduced for tritopological spaces and has been used to prove certain theorem involving some separation properties. The concept of density of sets in a tritopological spaces and the notion of its trioclosure generalizing topology have been introduced and fruitfully used for study of separation properties.
A Study on Fixed Point Iterative Procedures
(University of Rajshahi, 2016) Khatun, Miss. Saleha; Ali, M. Zulfikar
Fixed 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.
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.----