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Item Analytical Technique for Solving Second Order Generalized Strongly Nonlinear Duffing Equation with Varying Coefficients in Presence of Small Damping(Khulna University of Engineering & Technology (KUET), Khulna, Bangladesh, 2019-08) Kundu, Dipa; Uddin, Prof. Dr. Md. AlhazIn this thesis, we have extended He’s homotopy perturbation method for obtaining the approximate analytical solution of second order generalized strongly nonlinear Duffing equation with varying coefficients in presence of significant small damping based on the extended form of the Krylov-Bogoliubov-Mitropolskii (KBM) method. Accuracy and validity of the solutions obtained by the present method are compared with the corresponding numerical solutions obtained by the well-known fourth order Runge- Kutta method in graphically. The method has been illustrated by examples. In this study, the present technique gives acceptable results avoiding any numerical complexity. The results presented through figures show that the approximations are of extreme accuracy with significant damping. The proposed method is simple and suitable for solving the above mentioned nonlinear damped systems.Item A study on standard n-ideal of a lattice(Khulna University of Engineering & Technology (KUET), Khulna, Bangladesh, 2019-04) Hossen, Md Imran; Azad, Prof. Dr. Md Abul kalamLattice theory is an important part of mathematics . Ideal lattice and n-ideal of a lattice have played many roles in development of lattice theory. Historically, lattice theory started with Boolean distributive lattices: as a result, the theory of ideal lattice and n-ideal of a lattice is the most extensive and most satisfying chapter in the history of lattice theory. Ideal lattice have provided the motivation for many results, in general lattice theory. Many conditions on lattices and on element and ideals of lattices are weakened forms of distributivity is imposed on lattices arising in various areas of mathematics, especially algebra. In lattice theory there are different classes of lattices known as variety of lattices. Class of Boolean lattice is of course most powerful variety. Throughout this thesis we will be concerned with another large variety known as the class of ideal lattice and n-ideal of a lattice have been studied by several authors. The realization of special role of ideal lattices moved to break with the traditional approach to lattice theory, which proceeds from partially ordered sets to general lattices, semi modular lattices, modular lattices and finally ideal lattices. In this thesis we give several result on ideal and n-ideal which will certainly extend and generalize many results in lattice theory. In order to review, we include definations, examples, solved problems and proof of some theorems. The thesis contains four chapter. Chapter 1 we have discussed the basic defination of relation, poset, lattice, complete lattice, convex sub lattice, complemented and relatively complemented lattice. We also proved that, Dual of a complete lattice is complete . Chapter 2 have discussed basic concept of ideal and n-ideal of lattice. Here we study the defination and examples of ideal and n-ideal. Some imprtant theorem like “If n is a neutral element of a lattice, then I (L) n is modular if and only if L is modular”. Chapter 3 we have discussed Standard element and n-ideals. We also discussed in this chapter Congruence relation. Chapter 4 deals with standard n-ideal and Principal n-ideal. This is the main part of this thesis work. In this chapter we have discussed some defination and some important theorems like “For a neutral element n and a standard n-ideal S and an n-ideal I, S I is also a standard n-ideal” .Item Steady MHD Natural Convection Heat and Mass Transfer Flow above a Vertical Porous Surface with Thermal Diffusion and Inclined Magnetic Field(Khulna University of Engineering & Technology (KUET), Khulna, Bangladesh, 2018-08) Rana, Md. Sohel; Hossain, Prof. Dr. M. M. TouhidIn this study the effect thermal diffusion and inclined magnetic field on the steady laminar natural convection heat and mass transfer flow of viscous incompressible MHD electrically conducting fluid past a vertical porous surface is considered under the influence of induced magnetic field. The governing non-dimensional equations relevant to the problem, containing the partial differential equations, are transformed by usual similarity transformations into a system of coupled non-linear ordinary differential equations and have been solved by using the perturbation technique. On introducing the non-dimensional concept and applying usual Boussinesq’s approximation, the solutions for velocity fields, temperature distribution, induced magnetic fields and mass concentration are obtained up to the second order approximations for different selected values of the established dimensionless parameters and numbers. The influences of these various establish parameters and numbers on the velocity and temperature fields, induced magnetic field and mass concentration are exhibited under certain assumptions and are studied graphically. Further, the effects of these dimensionless parameters on the coefficients of skin friction and rate of heat and mass transfers are also studied in tabular form in the present analysis. The effect of different angle of inclinations of the externally applied uniform magnetic field on the field variables have been investigated for the present problem. It is observed that with other useful associated parameters, the thermal diffusion and the angle of inclination of the applied magnetic field have a retarding influence on the fluid velocity, induced magnetic field and mass concentration as well. It is also found that the dimensionless Prandtl number, Grashof number, Modified Grashof number, magnetic parameter and suction parameter have their own dependency on the concerned independent field variables like the velocity, temperature, concentration and induced magnetic fields as well as on other physical parameters of interests like local skin-friction coefficient (Cf), Nusselt number (Nu) and Sherwood number (Sh).Item Similarity Solution of Unsteady Convective Heat and Mass Transfer Flow along an Isothermal Vertical Plate with Suction(Khulna University of Engineering & Technology (KUET), Khulna, Bangladesh, 2018-07) Ghosh, Tumpa Rani; Hossain, Prof. Dr. M. M. TouhidIn this study the similarity solutions of unsteady convective boundary layer heat and mass transfer flow of viscous incompressible fluid along an isothermal plate with suction is investigated. The similarity equations of the specified problem are obtained first by employing the usual similarity technique and Bousinesq approximation. Then the set of transformed ordinary differential equations has been solved numerically adopting the 6th order Range-Kutta method and to enumerate the unspecified initial conditions shooting method has been adopted. The non-dimensional solutions regarding the velocity, temperature and mass concentration have been found for different selected values of the established dimensionless numbers and parameters entering into the problem. The effect of suction on fluid flow and temperature fields as well as concentration distributions and other topics of interest like skin friction and heat transfer coefficients are extensively investigated. It is observed that the velocity at any point within the boundary layer increases with the increase of suction parameter (Fw). But both temperature and mass concentration decrease rapidly with the increasing values of Fw. The effects of some other involved dimensionless numbers and parameters like Prandtl number (Pr), local concentration Grashof number (Gc) and unsteadiness parameters (A1, A2, A3) on the velocity and temperature fields and on the concentration distributions have been investigated. The results show that fluid velocity decreases slowly but temperature decreases significantly with the increase of Pr whereas no appreciable change is found on concentration for increasing values of Pr. It is also found that velocity increases highly with the increase of Gc while the impact of Gc on the temperature and concentration are very diminutive. All the unsteadiness parameters also apparently affect the velocity, temperature and concentration distributions. The obtained numerical results involving the effects of these non-dimensional numbers and parameters on the velocity, temperature and concentration profiles are displayed with the help of various graphs. Finally, the dependency of wall shear stress (in terms of local Skin-friction coefficients) and wall heat flux (in terms of Nusselt number), which are of physical interests, on the concerned non-dimensional parameters and numbers are also illustrated and discussed through the tables. The obtained results might be of great interest for the creative learners regarding aerodynamics and hydrodynamics including industrial applications.Item A Weighted Least Cost Matrix Approach in Transportation Problem(Khulna University of Engineering & Technology (KUET), Khulna, Bangladesh, 2018-07) Jannat, Fatima; Jamali, Prof. Dr. A. R. M. Jalal UddinTransportation models are of multidisciplinary fields of interest. Mainly in the business arena, it is common to encounter the problem of transportation of some products/goods from source/origin to sink/destination so that cost of transportation is minimal and also satisfy the constraints related to demand and the supply. A huge number of physical problems are modeled as Transportation problem (TP) which includes inventory problem, assignment problem, traffic problem and so on. TPs are required for analyzing and formulating such models. In order to minimize the transportation cost satisfying all constraint, transportation model first provides the Initial Basic Feasible Solution (IBFS) and then IBFS be optimized by some related optimization algorithm if IBFS is not optimized. So the primary objective of transportation model is to find out a good IBFS of TPs. In classical transportation approaches, the flow of allocation is controlled by the cost entries such as West Corner Method (WCM), Least Cost Method (LCM) etc. and/or manipulation of cost entries, so called Distribution Indicator (DI) or Total Opportunity Cost (TOC) like Vogel’s Approximation Method (VAM) and its variations. In LCM, the flow of allocation is directly controlled by the cost entries i.e. lowest cost prefers first. On the other hand, for examples, on VAM, its variants and some other methods, the flow of allocations is controlled by the DI or TOC tables. But these DI or TOC tables are formulated by the manipulation of cost entries only. None of them considers demand and/or supply entry to formulate the DI/ TOC table. In this thesis, we have first developed a new procedure of control of allocation named Weighted Opportunity Cost (WOC) matrix by incorporating supply/demand entries. At first, weight factors are formulated by using demand and supply entries which is off-course statistically valid. Then virtual weighted cost entries are formulated by manipulation of cost entries along with weight factors. Finally WOC matrix is formulated in which supply/demand entries acts as weight factor upon corresponding cost entries. Several examples are provided to demonstrate the concept of WOC matrix. After successfully development of WOC, our intension is go upon the development of an algorithm to find out IBFS of TPs. It is known that, in Least Cost Matrix method, the flows of allocations are controlled by the cost entries only. The flows of allocations are predefined according to the cost entries i.e the ascending order of cell cost and the whenever identical costs are encountered whatever be the structure of demand/supply entries. So, the algorithm does not need to update allocation direction in subsequent steps. On the other hand in VAM, the flow of allocation is controlled by the DI table rather than directly cost matrix. By incorporating these two ideas, we have proposed a Weighted Opportunity Cost based on LCM (WOC-LCM) approach. In this proposed approach, the flow of allocation is controlled by the WOC matrix rather than cost matrix as in LCM approach. But WOC matrix is invariant through all over the allocation procedures like cost matrix in LCM method whereas DI table is updated after each step of allocations. Some experiments have been carried out to justify the validity and the effectiveness of the proposed WOC-LCM approach. Experimental results have shown that the WOC-LCM approach outperforms LCM. Moreover, sometime this approach is able to find out optimal solution too.Item A Case Study of the Solution of Laminar Convective Boundary Layer Flow around a Vertical Slender Body with Transpirations(Khulna University of Engineering & Technology (KUET), Khulna, Bangladesh, 2016-01) Razzak, S. Md. Abdur; Hossain, Prof. Dr. M. M. TouhidIn this thesis, a case of the similarity solution of laminar convective boundary layer flow around a vertical slender body with transpirations has been investigated. Firstly, the governing boundary layer partial differential equations have been made dimensionless and then simplified by using Boussinesq approximation Secondly, similarity transformations are introduced on the basis of detailed analysis in order to transform the simplified coupled partial differential equations into a set of ordinary differential equations. In the present thesis one of the important similarity case, out of four cases has been studied. Under the considered case, the transformed complete similarity equations are solved numerically by using computer software MATLAB. Further the flow phenomena have been characterized with the help of obtained flow controlling parameters such as suction/blowing parameter, buoyancy parameter, Prandtl number, body-radius parameter and other driving parameter. Finally, the effects of these involved parameters on the velocity and temperature fields are presented graphically. It is observed that a small suction or blowing played a significant role on the patterns of' flow and temperature fields.Item Approximate Solution of the Quadratic Nonlinear Oscillator by Iteration Procedure(Khulna University of Engineering & Technology (KUET), Khulna, Bangladesh, 2015-12) Hossain, Md. Rakib; Haque, Dr. B. M. IkramulAn analytical technique has been developed based on an iteration method to determine higher order approximate periodic solutions for a nonlinear oscillator with discontinuity for which the elastic force term anti-symmetric and quadratic. Usually, a set of nonlinear algebraic equations is solved with this method. However, analytical solutions of these algebraic equations are not always possible, especially in the case of large oscillations. A modified approximate analytic solution of the quadratic nonlinear oscillator "ẍ + x2 sgn(x) = 0" has been obtained based on an iteration procedure. Here we have used the truncated Fourier series in each iterative step. The approximate frequencies obtained by the technique shows a good agreement with the exact frequency and periodic solutions with the exact ones. The percentage of error between exact frequency and fourth approximate frequency of the method is as low as 0.00003%.The method is mainly illustrated by the quadratic nonlinear oscillator but it is also useful for many other nonlinear problems.Item Approximate Analytic Solutions of the Nonlinear Cubic Oscillator by Iterative Method(Khulna University of Engineering & Technology (KUET), Khulna, Bangladesh, 2016-11) Rahman, Md. Mominur; Haque, Dr. B. M. IkramulA modified approximate analytic solution of the cubic nonlinear oscillator “ẍ + x3 = 0" has been obtained based on an iteration method. Here we have used the truncated Fourier series in each iterative step. The approximate frequencies obtained by this technique show a good agreement with the exact frequency. The percentage of error between exact frequency and our fifth approximate frequency is as low as 0.009%. The calculation with this technique is very easy. The modified technique accelerates the rapid convergence of the solution, reduces the error solution and increases the validity range.Item Analytical Solutions of Second Order Nonlinear Ordinary Differential Systems with high Order Nonlinearity(Khulna University of Engineering & Technology (KUET), Khulna, Bangladesh, 2016-01) Ghosh, Deepa Rani; Uddin, Prof. Dr. Md. AlhazNonlinear oscillator models have been widely used in many areas of physics and engineering and are of significant importance in mechanical and structural dynamics for the comprehensive understanding and accurate prediction of motion. The aim of the present study is to solve the second order autonomous Nonlinear Differential Systems with strong and high ((9th)) order nonlinearity in presence of small damping by combing He’s homotopy perturbation and the extended form of the KBM methods. The results obtained by the presented method are compared with those solutions obtained by the fourth order Runge-Kutta method in graphically.Item A study on Weakly Complemented Nearlattice(Khulna University of Engineering & Technology (KUET), Khulna, Bangladesh, 2016-07) Rahman, Mahfuza; Rahrnan, Prof. Dr. Md. ZaidurIn this thesis study of the nature of the weakly complemented nearlattice is presented. By a nearlattice S we will always mean a meet semilattice together with the property that any two elements possessing a common upper bound, have a supremum. Cornish and referred this property as the upper bound property, and a semilattice of this nature as a semilattice with the upperbound property. Cornish and Noor [8] preferred to Hickman [7] call these semilattices as nearlattices, as the behaviour of such a semilattice is close to that of a lattice than an ordinary semilattice. Of course a nearlattice with a largest element is a lattice. Since any semilattice satisfying the descending chain condition has the upper bound property, so all finite semilattices are nearlattices. In lattice theory, it is always very difficult to study the non-distributive and non-modular lattices. Gratzer [12] studied the non-distributive lattices by introducing the concept of distributive, standard and neutral elements in lattices. Cornish and Noor [8] extended those concepts for nearlattices to study non-distributive nearlattices. On the other hand, J.C Varlet [33] studied another class of non-distributive lattices with 0 by introducing the concept of 0-distributivity. In fact this concept also generalizes the idea of pseudocomplement in a general lattice. In this thesis we have extended the concept of weakly complemented nearlattice in terms of homomorphism theorem.
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