PhD Thesis http://repository.library.du.ac.bd:8080/xmlui/xmlui/handle/123456789/283 2020-12-03T01:54:49Z 2020-12-03T01:54:49Z Dynamics of fractals in euclidean and measure spaces Islam, Md. Jahurul http://repository.library.du.ac.bd:8080/xmlui/xmlui/handle/123456789/1180 2019-11-24T05:09:15Z 2019-01-13T00:00:00Z Dynamics of fractals in euclidean and measure spaces Islam, Md. Jahurul In this thesis, we discuss the constructions of the generalized Cantor sets which are the prototypical fractals and also discuss the Markov operators defined on separable complete metric space. We show that these special types of sets are Borel set as well as Borel measurable whose Lebesgue measures are zero. We formulate Iterated Function System of the Generalized Cantor Sets (IFSGCS) using affine transformation and fixed points method. We discuss the Hausdorff dimension of the invariant set for iterated function system of generalized Cantor sets. We also formulate Iterated Function System with probabilities of the Generalized Cantor Sets (IFSPGCS). We show their invariant measures using Markov operators and Barnsley-Hutchinson multifunction. We observe that these functions satisfy the sweeping properties of Markov operator. In addition, we show that these iterated function systems with probabilities are non-expansive and asymptotically stable if the Markov operator has the corresponding property. Further we study two dimensional fractals such as the Koch snowflake, the Koch curve, the Sierpiński triangles, the Sierpiński carpet, the box fractal and also three dimensional fractals such as the Menger sponge and the Sierpinski tetrahedron. We show fractal and topological dimensions and Lebesgue measures of those fractals. We formulate iterated function system of higher dimensional fractals such as the square fractals, the Menger sponge, the Sierpinski tetrahedron and the octahedron fractal. We also discuss the Hausdorff dimension of the invariant set for iterated function system of those fractals. This dissertation submitted in fulfillment of requirement for the degree of Doctor of Philosophy (Ph.D) in Mathematics. 2019-01-13T00:00:00Z Numerical solutions of higher order boundary value problems using piecewise polynomial bases Hossain, Md. Bellal http://repository.library.du.ac.bd:8080/xmlui/xmlui/handle/123456789/1178 2019-11-24T05:07:43Z 2015-04-21T00:00:00Z Numerical solutions of higher order boundary value problems using piecewise polynomial bases Hossain, Md. Bellal The Boundary Value Problems (BVPs), either the linear or nonlinear, arise in some branches of applied mathematics, engineering and many other fields of advanced physical sciences. Many studies concerned with solving second order boundary value problems using several numerical methods. But few studies concerned with especial cases of higher order BVPs have been solved applying several numerical techniques. In our thesis, we have used the Galerkin technique for solving higher order linear and nonlinear BVPs (from order four up to order twelve). The well known Bernstein and Legendre polynomials are exploited as basis functions in the technique. The main steps, in this thesis, depend on: 1. To use the Bernstein and Legendre polynomials we need to satisfy the corresponding homogeneous form of the boundary conditions and modification is thus needed. 2. A rigorous matrix formulation is developed by the Galerkin method for linear and nonlinear systems and solved it using Bernstein and Legendre polynomials. 3. Using the Newton's iterative method for nonlinear problems to obtain more accurate results. This thesis submitted to the University of Dhaka in partial fulfillment of the requirement for the award of the degree of Doctor of Philosophy in Mathematics. 2015-04-21T00:00:00Z Conjugate effect of fluctuating thermal and mass diffusion on the flow of viscous fluid along flat and cylindrical surfaces Hussain, Sharmina http://repository.library.du.ac.bd:8080/xmlui/xmlui/handle/123456789/1176 2019-11-24T05:10:04Z 2018-12-17T00:00:00Z Conjugate effect of fluctuating thermal and mass diffusion on the flow of viscous fluid along flat and cylindrical surfaces Hussain, Sharmina Numerical investigations have been carried out to look into the e ect of double-di usion on laminar ow along vertical at and cylindrical surfaces. Conventional convective processes i.e. natural convection, mixed convection and magnetohydrodynamic (MHD) natural convection have been considered and six di erent models are studied deeming oscillating surface temperature and species concentration boundary conditions. In each case, extensive parametric simulations are performed in order to elucidate the e ects of some important parameters i.e. Prandtl number, Schmidt number, buoyancy ratio parameter, Straouhal number, and magnetic parameter on ow eld in conjunction with heat and mass transfer. Di erent numerical techniques are applied to solve the governing equations. Asymptotic solutions for low and high frequencies are obtained for the conveniently transformed governing coupled equations. Solutions are also obtained for wide ranged values of the frequency parameter. At the very outset of this dissertation, natural convection ow along vertical at plate is considered. The surface temperature and species concentrations are assumed to be of small amplitude oscillation. In this study, the similar boundary conditions are imposed for surface temperature and concentration as well as surface heat and mass ux. Finally, comparative study has been made to nd a correlation between these two cases. Comparison between the perturbation solutions and the solutions for the wide ranged values are made in terms of the amplitude and phase of the shear stress, surface heat transfer and surface mass transfer coe cient. It has been found that the amplitudes and phase angles obtained from asymptotic solutions are in good agreement with the nite di erence solutions obtained for wide ranged values of the frequency parameter. This simulated results are validated against some published results. Flow along a vertical wedge is taken into account and the convective process is deemed as mixed convection in the subsequent model study. In this study also, the surface temperature and velocity boundary conditions are assumed to be sinusoidal and amplitude of the oscillation is considered very small. Implicit nite di erence method is used to solve the governing equations of the entire ow eld and results obtained from this method are regions of the ow eld, respectively. To study the parametric e ects on the heat transfer rate, results are calculated in terms of amplitude and phase angles of shear stress and heat transfer coe cient. In the next investigation, the preceding model is extended to study heat and mass transfer simultaneously. In order to take into account the concentration eld, another convection-di usion equation is added with the governing set of equations and corresponding boundary condition is assumed to be similar to surface temperature boundary condition. Wide-ranging parametric studies have been carried out and results are presented in both tabular and graphical forms. Magnetohydrodynamic (MHD) natural convection ow is investigated considering uctuating surface temperature and concentration boundary condition. Results obtained in the present investigation are compared with some other published results and further numerical investigations are accomplished to study the both heat and mass transfer by using implicit nite di erence method. It has been assumed in this study that, in some undisturbed ow region, there is an uniform magnetic eld making a non-zero angle with it. Mixed convection ow along horizontal heated cylinder have been considered in the last two models and full set of Navier-Stokes equations in terms of vorticity and stream function is solved. Parametric studies have been carried out in these cases and heat and mass transfer rates are observed by calculating the Nusselt number and Sherwood number. In the rst investigation of these type of ow, non-uniform surface temperature and concentration are considered with no oscillations. After that, the same problem is studied by allowing small amplitude oscillations in surface velocity, temperature and species concentration. In both the cases, isothermal lines and isoconcentration contours are drawn to visualize the temperature and concentration distribution by varying the relevant parameters. Important ndings are listed after each investigation and nally concluding remarks are brought out based on the overall investigation and understanding. Almost all the cases, the present simulations are validated either qualitatively or quantitatively by comparing with some published results. This thesis submitted in partial ful llment of the requirementsfor the Degree of Doctor of Philosophy in the subject of Mathematics. 2018-12-17T00:00:00Z