B-Spline Collocation Methods For Coupled Nonlinear Schrödinger Equation

In this study, the Coupled Nonlinear Schrödinger Equation (CNLSE) which models the propagation of light waves in optical fiber is solved using numerical methods namely Finite Difference Method (FDM) and B-Spline collocation methods. The equation was discretized in space and time. We propose the discretization of the nonlinear terms in the CNLSE following the Taylor approach and a newly developed approach called Besse. The theta-weighted method is used to generalize the scheme whereby the Crank-Nicolson scheme (i.e θ = 0.5) is chosen. The time derivatives are discretized by forward difference approximation. For each approach, the space dimension is then discretized by five different collocation methods independently. The first method for Taylor approach is based on FDM whereby the space derivatives are replaced by central difference approximation

Fourth-Order Spline Methods For Solving Nonlinear Schrödinger Equation

The Nonlinear Schrödinger (NLS) equation is an important and fundamental equation in Mathematical Physics. In this thesis, fourth-order cubic B-spline collocation method and fourth-order cubic Exponential B-spline collocation method are developed in order to solve problems involving the NLS equation. The established Cubic B-spline Collocation Method and Cubic Exponential B-spline Collocation Method are of second-order accuracy. The methods developed in this thesis are of fourth-order accuracy. The time dimension of the NLS equation is discretized using the Finite Difference Method and the space dimension is discretized based on the particular B-spline methods used. The Taylor series approach and Besse approaches are used to handle the nonlinear term of the NLS equation. Since the methods result in an underdetermined system, the supplementary initial and boundary conditions are used to solve the system. The developed methods are tested for stability and are found to be unconditionally stable. Error analysis and convergence analysis are also carried out. The efficiency of the methods are assessed on three test problems involving solitons and the approximations are found to be very accurate. Besides that, the numerical order of convergence is calculated and associated theoretical statements are proved. In conclusion, the proposed methods in this study worked well and give accurate numerical results for the NLS equation

Mathematical model of interactions immune system with Micobacterium tuberculosis

Tuberculosis (TB) remains a public health problem in the world, because of the increasing prevalence and treatment outcomes are less satisfactory. About 3 million people die each year and an estimated one third of the world's population infected with Mycobacterium Tuberculosis (M.tb) is latent. This is apparently related to incomplete understanding of the immune system in infection M.tb. When this has been known that immune responses that play a role in controlling the development of M.tb is Macrophages, T Lymphocytes and Cytokines as mediators. However, how the interaction between the two populations and a variety of cytokines in suppressing the growth of Mycobacterium tuberculosis germ is still unclear. To be able to better understand the dynamics of infection with M tuberculosis host immune response is required of a model.One interesting study on the interaction of the immune system with M.tb mulalui mathematical model approach. Mathematical model is a good tool in understanding the dynamic behavior of a system. With the mediation of mathematical models are expected to know what variables are most responsible for suppressing the growth of Mycobacterium tuberculosis germ that can be a more appropriate approach to treatment and prevention target is to develop a vaccine. This research aims to create dynamic models of interaction between macrophages (Macrophages resting, macrophages activated and macrophages infected), T lymphocytes (CD4 + T cells and T cells CD8 +) and cytokine (IL-2, IL-4, IL-10,IL-12,IFN-dan TNF-) on TB infection in the lung. To see the changes in each variable used parameter values derived from experimental literature. With the understanding that the variable most responsible for defense against Mycobacterium tuberculosis germs, it can be used as the basis for the development of a vaccine or drug delivery targeted so hopefully will improve the management of patients with tuberculosis. Mathematical models used in building Ordinary Differential Equations (ODE) in the form of differential equation systems Non-linear first order, the equation contains the functions used in biological systems such as the Hill function, Monod function, Menten- Kinetic Function. To validate the system used 4th order Runge Kutta method with the help of software in making the program Matlab or Maple to view the behavior and the quantity of cells of each population

Exploring Written Artefacts

This collection, presented to Michael Friedrich in honour of his academic career at of the Centre for the Study of Manuscript Cultures, traces key concepts that scholars associated with the Centre have developed and refined for the systematic study of manuscript cultures. At the same time, the contributions showcase the possibilities of expanding the traditional subject of ‘manuscripts’ to the larger perspective of ‘written artefacts’

Exploring Written Artefacts

This collection, presented to Michael Friedrich in honour of his academic career at of the Centre for the Study of Manuscript Cultures, traces key concepts that scholars associated with the Centre have developed and refined for the systematic study of manuscript cultures. At the same time, the contributions showcase the possibilities of expanding the traditional subject of ‘manuscripts’ to the larger perspective of ‘written artefacts’