http://reposit.library.du.ac.bd:8080/xmlui/xmlui/handle/123456789/87 2026-04-07T07:14:58Z http://reposit.library.du.ac.bd:8080/xmlui/xmlui/handle/123456789/4770 Numerical Solutions of Fractional Order Boundary Value Problems by Weighted Approximation Method Ruman, Umme The major objective of this research work is to employ the weighted residual approach to numerically solve fractional order differential equations with homogeneous and non-homogeneous boundary conditions. This method uses linear combinations of several types of functions to find the approximate solutions, which must satisfy the homogeneous boundary conditions. The piecewise polynomials like the Bernstein, modified Bernoulli and modified Legendre polynomials are utilized as basis functions because these kinds of functions are easily differentiated and integrated in this study. The fractional derivatives are used in the hypothesis of Caputo sense. As a result, we provide a detailed and straightforward comprehensible matrix form of the Galerkin, Least Square and Collocation weighted residual formulation for both linear and nonlinear fractional order boundary value problems. In each chapter, few numerical examples are exhibited to illustrate the precision and usefulness of the current approach. We demonstrate that the results appear to be monotonic convergence within the approximate results and the exact solutions. The approximate results are also compared to the exact solutions along with the solutions that are currently available in the literature. Reliable accuracy is obtained in the present work; the absolute errors are presented both graphically and in tabular form. The thesis entitled Numerical Solutions of Fractional Order Boundary Value Problems by Weighted Approximation Method contains six chapters; out of these, the first chapter is confined as Introduction. In this chapter, we mention the objectives and scope of the thesis and the outline of the research work. We discuss some mathematical preliminaries that are important to establish the problems in detail, such as theorems and lemmas that are used in subsequent chapters, some special functions like Gamma and Mittag- Leffler functions and the basic concepts of fractional derivative and integration in both Riemann-Liouville and Caputo sense. The finite element method is introduced here, especially three weighted residual methods: Galerkin, Least Square and Collocation with the Bernstein, modified Legendre and modified Bernoulli polynomials and their properties. Chapter 2 is devoted to linear fractional differential equations using Bernstein, modified Legendre and modified Bernoulli polynomials as basis functions. We derive rigorous matrix formulations of the following: p(x)du dx + s(x)dαu dxα + u(x) = f(x), under the boundary conditions u(a) = a0, u(b) = b0 where α ≥ 1.5. We examine four examples of second-order linear fractional boundary value problems for the numerical solutions using the suggested formulations. It was found that there is a monotonic convergence between the approximate and exact solutions. Three weighted residual methods for solving fractional Bagley-Torvik equations are studied in chapter 3. A fractional-order differential equation arises in various engineering and physical systems, particularly in modeling viscoelastic materials and dynamic fluid systems. This work concentrates on the numerical solution of the Bagley-Torvik equation, represented as: aD2y(t) + bD3/2y(t) + cy(t) = f(t) The WRM transforms the governing equation into an approximate solution by minimizing the residual error over the problem domain, employing basis functions to represent the solution. The fractional derivative terms are discretized using suitable approximations, such as the Caputo approach, which is incorporated into the weighted residual framework. Results indicate that the weighted residual method provides flexible and efficient results for solving fractional differential equations while maintaining stability and convergence properties. The results suggest that the weighted residual method offers a robust tool for solving fractional order differential equations, making it highly applicable to a range of practical problems in engineering and applied physics. In chapter 4, the Galerkin weighted residual approach is used to quantitatively solve the fourth order fractional differential equations with homogeneous and non-homogeneous boundary conditions. The same process is also introduced to generate the approximate solutions for the two-point fourth-order linear and non-linear integro-differential problems in fractional order. Using piecewise polynomials, the matrix formulation of both scenarios is stated directly. To determine the correctness and effectiveness of the proposed method, we experiment with a variety of instances from the literature utilizing modified Bernoulli and modified Legendre polynomials as basis functions. The absolute errors are displayed in tabular form and we find that reliability has been attained in this study. In Chapter 5, the weighted residual method is used to bring out the approximate solutions for nonlinear fractional differential equations with both homogeneous and nonhomogeneous boundary conditions. We use three techniques: Galerkin, Least Square and Collocation to solve nonlinear two-point boundary value problems numerically in an efficient manner. The accuracy and reliability of the current method, which utilized the modified Legendre and modified Bernoulli polynomials as weight functions, are demonstrated by looking at few nonlinear examples to find the maximum absolute errors. The computational techniques and mathematical formulations are easier to comprehend and less difficult to understand in this literature. The last chapter entitled numerical techniques for the system of fractional differential equations is established by the method of weighted residuals such as Galerkin, Least Square and Collocation methods that are used to solve the boundary value problems (BPVs). This approach is then expanded to obtain approximate solutions of fractional order systems that use differentiable polynomials, specifically modified Legendre polynomials as basis functions. It is possible to efficiently code the algorithm for the residual formulations of matrix form. Here, the Caputo fractional derivatives interpretation is used rigorously. We have employed some examples of linear and nonlinear boundary value problems to quantitatively illustrate these techniques. The findings in absolute errors demonstrate how straightforwardly the current approach locates numerical solutions for the systems of fractional order differential equations. All numerical experiments of this thesis and its algorithm have been implemented using the frameworks of Mathematica and MATLAB, which are used to calculate scientific computations and graphical visualizations. Finally, the conclusion and the list of references are appended at the end of the dissertation. This thesis is submitted for the degree of Doctor of Philosophy. 2025-02-19T00:00:00Z http://reposit.library.du.ac.bd:8080/xmlui/xmlui/handle/123456789/4727 A STUDY OF PULSATILE FLOW PHENOMENA WITH COMPUTATIONAL FLUID DYNAMICS APPROACH Uddin, Md. Jashim Cardiovascular disease (CVD) with the occurrence of plaque formation and sinus development related to stenosed and dilated area vasculatures is a chronic disease and has attracted wide attention among researchers because of its significant effect all over the world by leading to heart attack or stroke. Coronary and carotid arteries and their bifurcations are considered an enthusiastic research area for the pulsatile nature of blood flow. Numerical simulation of the considered left coronary artery has been studied in two-dimensional (2D) stenotic and three-dimensional (3D) geometric models of bifurcation for pulsatile blood flow to better understand the physical mechanism assuming fluid as Newtonian and non-Newtonian characteristics. The computational fluid dynamics (CFD) approach is incorporated in COMSOL Multiphysics with a satisfactory validation. This research indicates an extensive recirculation zone in Newtonian fluid as compared to that in the non-Newtonian rheological model, and hemodynamic parameters like shear stress can be considered a cabalistic factor in the commencement of arterial diseases. Computed hemodynamic parameters such as time-averaged wall shear stress (TAWSS), oscillatory shearing index (OSI), and relative residence time (RRT) are also able to make the difference between Newtonian and non-Newtonian fluids by forming atherosclerotic development variations. Elevated shear stress at the fibrous cap is investigated higher for a shear-thinning fluid. The present investigation also concentrates on the evaluations of the lesion of diagnostic concern on the basis of the diagnostic parameter’s critical values and investigates that the results are affected by the stenosis and rheological model. In 3D bifurcation geometry, the backflow region with a reduction of WSS is investigated computationally at the outer wall of the daughter vessel due to the noticeably low shear rate. The non-Newtonian importance factor (IFc) for the 3D left coronary artery bifurcation model decreases with an increase in bifurcation angle, and the smallest bifurcation angle generates the least time-averaged inlet pressure. Results further concentrate that the flow separation length reduces with developing bifurcation angle in bifurcated geometry. Computational simulation significantly furthermore elucidates that the non-Newtonian blood flow model incorporating hemodynamic and diagnostic parameters has great impacts on instantaneous flow systems. The transient numerical computational approach of fluid-structure interaction (FSI) has been modeled for an atherosclerotic fibrous plaque in a 2D carotid vasculature under the pressure action of normal and hypertension to detect the interactive effect of anatomical blood flow dynamics, the v properties of wall mechanics and pressure conditions on hemodynamics. A significant contribution of the present research on von Mises stress is that its magnitude has increased in HBP compared to that in NBP. The investigated results intend to expose that the variety of wall displacement and separation length occur due to the effect of pressure conditions in the flexible wall model. The TAWSS, OSI and RRT indicate the atherosclerotic thrombus deposition in generating potential risk parameters has a sequentially reduced separation length with an increase of mechanical elastic modulus. The results indicate the influence of elastic modulus on the increased value of the time averaged wall pressure gradient (TAWPG) and the time-averaged pressure drop in which maximum pressure drop is identified for NBP. The findings also illustrate that physiological hypertension gives a greater deformation gradient and stress tensor regarding the development of atherosclerosis. Time-dependent FSI computation of 3D patient data-based carotid geometry has been carried out in an idealized symmetric bifurcating vessel to show the effects of various sinus shapes on atherosclerotic plaque development. The significant impact of the variety of sinus shapes is offered in graphs and contours. In contrast to ellipsoidal and triangular sinuses, the current study shows that the trapezoidal sinus, which is prone to atherosclerosis, exhibits significant recirculation at both bifurcation walls. Based on simulation results, the inner wall of the trapezoidal sinus has a TAWSS that is 1.07 and 1.17 times greater than that of the ellipsoidal and triangular sinuses respectively. The present results demonstrate that the trapezoidal sinus has a larger pressure drop at the bifurcation point. The mass flow rate ratio for the ellipsoidal shape increases by 18% and 27% for that of trapezoidal and triangular shapes respectively. This research also indicates that a trapezoidal sinus of a human being is more susceptible to atherosclerotic plaque progression and development, leading to endothelial dysfunction, and its impact can be used in various biomedical sectors. This thesis is submitted for the degree of Doctor of Philosophy. 2025-11-05T00:00:00Z http://reposit.library.du.ac.bd:8080/xmlui/xmlui/handle/123456789/4725 Effect of Magnetic Particles on Biomagnetic Fluid Flow through Stretched Cylindrical Surface Alam, Md. Jahangir Biomagnetic fluid dynamics (BFD) of liquids containing magnetic particles is a promising method for magnetic drug targeting, gene delivery, the development of magnetic devices, electromagnetic hyperthermia in cancer treatment, and magnetic resonance imaging (MRI). Developing a BFD model for fluids with magnetic particles is crucial to provide medical professionals with a second viewpoint. Here, we focus on theoretical and computational investigations of two-dimensional, steady–unsteady, viscous, incompressible, laminar biomagnetic fluid flow and heat transfer involving magnetic particles (Fe3O4, CoFe2O4, Mn ZnFe2O4) over stretching and shrinking cylindrical surfaces under an applied magnetic field. Considering the intricate interactions between intercellular proteins, membranes, and hemoglobin, blood was considered as the base fluid. BFD flow and heat transfer with magnetic particles over a stretching cylinder under the influence of a magnetic dipole are performed throughout the study. The governing mathematical formulation considers the effects of electrical conductivity and magnetization caused by the magnetohydrodynamics (MHD) and ferrohydrodynamics (FHD) principles, respectively. We also treat the blood flow through a stretching cylinder with MHD and FHD interactions, considering both time-dependent and time-independent cases. The effects of varying fluid parameters, such as ferromagnetic interaction parameter, magnetic field parameter, curvature parameter, particle volume fraction, thermal radiation, etc., were also examined for both stretching and shrinking scenarios. Additionally, we numerically examined the two-dimensional BFD flow, in two specific scenarios: pure blood flow and blood that contains particles in cylindrical geometries under various conditions. The research encompasses several critical chapters, each focusing on distinct interactions and behaviors of biomagnetic fluids in the presence of magnetic fields. At first, we examine the mechanisms of blood–Fe3O4 under FHD and MHD interactions generated by a stretched cylinder, revealing significant alterations in flow characteristics and heat transfer efficiency. We then expand by investigating the flow and heat transfer dynamics of a blood–CoFe2O4 mixture around a rotating stretchable cylinder subjected to a strong magnetic field, and find enhanced thermal conductivity and flow stability. Due to their extreme nonlinearity, finding exact solutions to the governing mathematical equations is still challenging. Researchers have suggested various similarity methods to address this issue, and it has been established that similarity methods are the most effective analytical tools for solving nonlinear partial differential equations. Through similarity transformations, the boundary layer equations related to the boundary conditions are iv converted into a system of non-linear ordinary differential equations. Considering this, we also used group theoretical approaches, such as the one-parameter and two-parameter group techniques, to solve boundary value problems. The steady flow of blood–Mn–ZnFe2O4 past a cylinder considering the FHD concept is analyzed using the one-parameter group technique, which sheds light on the parametric flow behavior. Later on, we employ a two-parameter group theoretical technique to discuss unsteady blood flow with differently shaped magnetic particles considering MHD and FHD interactions, which advances to how particle morphology affects thermal characteristics and flow stability. A dual solution along with stability analysis of the blood–Mn–ZnFe2O4 flow under a magnetic dipole across a shrinking cylinder is also explored. Finally, we examine the intricacies of thermal profiles and flow patterns via a non-similar solution for biomagnetic fluid flow with magnetic particles along an inclined stretched cylinder with sinusoidal surface temperature and magnetic dipole. The findings of all of the problems considered provide important new findings about the behavior of biomagnetic fluids and lay groundwork for further studies and possible applications in heat management, material processing, and biomedical engineering. We used two methods by which previous researchers have tackled the above problems numerically: two-point boundary value technique based on a common finite difference method with central differencing, tridiagonal matrix manipulation, and an iterative procedure, and MATLAB-based bvp4c functions. The numerical results were obtained for fluid velocity, temperature, pressure, and physical quantities like skin friction coefficient, wall pressure gradient, and heat transfer rate. Before moving on to numerical solutions, we contrasted our study with previous research. Once we had good accuracy between studies, we moved on to the in-depth results. The numerical results show that the presence of a magnetic dipole, which generates a magnetic field strong enough to saturate the biofluid, substantially impacts the properties of blood-containing magnetic particles. This thesis is submitted for the degree of Doctor of Philosophy. 2025-11-05T00:00:00Z http://reposit.library.du.ac.bd:8080/xmlui/xmlui/handle/123456789/3593 Spectral Relaxation numerical simulation of boundary layer flow with nanoparticles on a moving surface influenced by induced magnetic field Akter, Shahina The concept of boundary-layer flow is of utmost importance in the science of fluid dynamics, playing a significant role in a wide variety of engineering applications and natural phenomena. The proliferation of research on heat and mass transfer in boundary layers over perpetually moving surfaces can be attributed to the diverse range of manufacturing processes in which they are used, including paper production, metal extrusion, material-handling conveyors, and glass fibre production. Additionally, the expansion of nanotechnology has spurred researchers to explore the flow behaviour at the boundary layer in nanofluids. A nanofluid is a fluid that contains nanoparticles, which results in a significant enhancement of its heat transfer properties due to the increased thermal conductivity. The ability to boost heat and mass transfer with a low concentration of nano-sized particles, and to regulate the transport processes, have led to a large variety of applications for nanofluids. Furthermore, the interaction of a magnetic field with a nanofluid has numerous potential uses that rely on the potential variation in the fluid perpendicular to both its motion and the magnetic field. This thesis presents numerical studies of boundary-layer flow and heat transfer from a moving flat plate subject to different boundary conditions under the influence of an applied induced magnetic field. The focus is on flows of two-dimensional, stable, viscous, incompressible, laminar, and electrically conducting water-based nanofluids incorporated with different metallic and magnetic nanoparticles. At the beginning of this thesis, a theoretical model is studied for steady magnetohydrodynamic (MHD) viscous flow resulting from the motion of a semi-infinite flat plate in an electrically conducting nanofluid. Thermal radiation magnetic induction effects and thermal convective boundary conditions are included. Buongiorno’s two-component nanoscale model is deployed, which features Brownian motion and thermophoresis effects. The second study examines the continuous laminar boundary-layer flow of 􀜣􀝃􀈂water and 􀜥􀝑􀈂water nanofluids with convective heat transport from an inclined stationary or moving flat plate, with a convective surface boundary condition, when there is an induced magnetic field. Then, we perform a computational analysis of ii the combined impacts of temperature- and space-dependent internal heat generation/absorption across a moving flat surface in the presence of an induced magnetic field with a momentum slip condition on the boundary-layer flow of a nanofluid. Later, we investigate the effects of viscous dissipation on a convective aligned MHD flow of a nanofluid over a semi-infinite moving flat surface, where the vectors of the magnetic field and the flow velocity are parallel far from the plate. Lastly, the influence of an induced magnetic field on the MHD heat transfer flow of waterbased ferrofluids, under the influence of slip, over a moving plate subject to uniform heat flux, is analysed. A transverse magnetic field is applied to the plate. The aforementioned mathematical problems are solved by applying appropriate similarity transformations to convert the governing boundary-layer equations and related boundary conditions into a system of nonlinear coupled ordinary differential equations. The fluid is assumed to be a water-based nanofluid containing metallic and magnetic nanoparticles with Prandtl number 􀜲􀝎 􀵌 􀍸􀇤􀍴, without a slip condition. The transformed system of differential equations is solved numerically, employing the spectral relaxation method (SRM) via the MATLAB R2018a software. The SRM is a simple iteration scheme for solving a nonlinear system of equations that does not require any evaluation of derivatives, perturbation, or linearization. Throughout this thesis, the profiles of velocity, induced magnetic field, temperature, and nanoparticle concentration are derived numerically and displayed for a range of physical parameter values. The significance of multiple embedded physical parameters, including the sheet velocity parameter, magnetic field parameter, Prandtl number, magnetic Prandtl number, thermal radiation parameter, Lewis number, Brownian motion parameter, thermophoresis parameter, Grashof number, Biot number, angle of inclination, nanoparticle volume function, Eckert number, and slip parameter, on the fluid flow is examined, and the findings are presented graphically. The numerical values of the skin friction coefficient, heat transfer rate, mass transfer rate, and other missing slope characteristics are tabulated. The impacts of different metallic and magnetic nanoparticles on the boundary-layer flow, friction drag, and heat flow rate are also investigated. To determine the validity of the computational results, they are compared with those of earlier studies. Thus, this study yields a conclusion that supports the accuracy and reliability of the SRM outcomes. The significance of the electrical conductivity of nanofluids is that the flow and heat transfer may be controlled by an external magnetic field, which could lead to important applications in areas iii such as electronic packing, mechanical engineering, thermal engineering, aerospace, and bioengineering. Increasing the magnetic body force parameter strongly reduces the flow velocity and suppresses magnetic induction, but increases the temperature due to the extra work expended as heat in dragging the magnetic nanofluid. Temperatures also increase with the nanoscale thermophoresis parameter and radiative parameter, whereas they are reduced by a higher wall velocity, Brownian motion parameter, or Prandtl number. Both the hydrodynamic and magnetic boundary-layer thicknesses are reduced by greater reciprocal values of the magnetic Prandtl number. The nanoparticle (concentration) boundary-layer thickness increases with the thermophoresis parameter and Prandtl number, whereas it decreases with increasing wall velocity, nanoscale Brownian motion parameter, radiative parameter, and Lewis number. The simulations are relevant to electroconductive nanomaterial processing. The nanofluid with 􀜣􀝃 nanoparticles is found to have remarkably high thermal conductivity, whereas the lowest cooling rate is observed in the 􀜶􀝅􀜱􀬶–water nanofluid. Moreover, higher flow resistance and a faster rate of heat transfer are detected in magnetite nanofluids. The magnetic field’s principal effects are to lower the dimensionless velocity and raise the dimensionless surface temperature compared with the hydrodynamic situation, which in turn increases the ferrofluid skin friction and heat transfer rate. In recent times, there has been a growing trend of companies recognizing the potential of nanofluid technology and directing their attention to its specific industrial applications. The utilization of boundary-layer flow over a flat surface coupled with nanofluid has found extensive application for addressing thermal management issues. The interaction of magnetic fields and fluid dynamics is crucial in various applications such as liquid metal cooling, magnetic drug targeting, and MHD power generation. In essence, the investigation of boundary-layer flow involving nanoparticles on an evolving surface subjected to an induced magnetic field is of great importance due to its capacity to enhance heat transfer, diminish friction, and propel advancements in fields such as energy production and medicine. A dissertation submitted for the degree of Doctor of Philosophy. 2025-02-09T00:00:00Z