M.Phil / MS

Now showing 1 - 4 of 4

Imex Methods For Ordinary Differential Equations
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2019) Nida Khalid; FA17-RMT-053; Dr. Yousaf Habib; LHR TP 5674
The aim of this thesis is to construct partitioned Runge-Kutta methods (PRK) for the numerical solution of non-separable system of ordinary differential equations y0 = f(y; z), z0 = g(y; z) as IMEX scheme (implicit explicit). The partitioned Runge-Kutta methods consist of two RK methods such that one method solve y0 = f(y; z), the non stiff differential equation, the other method solve z0 = g(y; z), the stiff differential equation. Explicit RK method is used for the solution of non stiff part and implicit RK method is used for the solution of stiff part. In order to construct PRK method, we have used the idea of effective order methods. The development of these methods require the solution of large number of order conditions, however, there exist simplifying assumptions which reduce the number of order conditions. In this thesis, simplifying assumptions for PRK method for non-separable differential equations are derived for the first time and applied successfully to reduce the effective order conditions, thus allowing us to construct effective order 3 PRK method with just two stages. This result in huge savings in terms of computational cost.
General Linear Methods With Projection
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2018) Lubna Mustafa; FA16-RMT-031; Dr. Yousaf Habib; LHR TP 5194
The present work deals with the structure-preserving numerical methods for the solution of ordinary differential equations representing dynamic systems from the Hamiltonian perspective of classical mechanics. These differential systems involve quantities that should remain constant over the course of enquiry. Whereas the usual numerical methods do not account for the preservation of these invariants, structure-preserving methods such as the symplectic Runge-Kutta methods and the G-symplectic general linear methods faithfully maintain these characteristic properties of a Hamiltonian system during the numerical discretization of the problem. Physically significant invariant properties of a Hamiltonian system include the conservation of total energy and the symplecticity of flow. In Chapter 1, basic theory of ordinary differential equations is introduced. The exposition combines definitions with the construction and implementation of numerical methods for ordinary differential equations and is central to all the subsequent investigations carried out in this work. In Chapter 2, long - term integrators for a Hamiltonian system are treated in detail. The general linear methods subsume the Runge-Kutta methods as well as the linear multi-step methods introduced previously. In Chapter 3, the important notion of projection is also introduced here. The basic idea of the standard projection technique is generalized to develop the heterogeneous class of numerical methods collectively known as the general linear methods. Projection technique is applied to the general linear methods and the similarity of the symplectic Runge-Kutta methods to the G-symplectic general linear methods is discussed.
The Numerical Preservation of Multiple First Integrals of Dynamical Systems
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2019) Atia Rashid; FA17-RMT -045; Dr. Yousaf Habib; LHR TP 5666
Geometric numerical integrators are numerical methods for differential equa tions with invariants such that the numerical solutions preserve the geometric properties or invariants like energy, integrals, symplectic structure, phase-space volume and symmetries. An ordinary differential equation may contain linear, quadratic, cubic or higher order first integrals. For the preservation of quadratic f irst integrals, symplectic numerical method is an appropriate candidate. If we want to preserve cubic or higher order first integrals, we use discrete gradient method and projection method. Discrete gradient method lies in a particular area of geometric numerical integra tion known as integral preserving integrators. Integral preserving integrators are numerical integration algorithms that preserves one or more first integrals of a differential equation. The integrals include energy, momentum and angular mo mentum. The differential equations include Hamiltonian or non-Hamiltonian ones. We consider harmonic oscillator and simple pendulum from Hamiltonian class be cause these are having the differential equations with quadratic first integrals. On the other hand from non-Hamiltonian class, we consider Lorenz and SIRI models because they are having the differential equations with cubic or higher order first integrals. SIRI model is basically a general model of disease transmission with a constant population size N in which we analyze the effect of any disease on a special commu nity. The under observation population is divided into three sections: susceptible, infected, and recovered individuals, labeled as S, I, R. It was firstly formulated by Tudor. The first integrals of SIRI model was calculated by Rehana Naz et al in their research work [8]. They used the partial Hamiltonian approach to obtain the first integrals. In the present work, we firstly studied the discrete gradient method which is a ge ometric numerical integrator for the preservation of first integral of the differential equations. Secondly, we have applied the discrete gradient method on Hamilto ix nian system with the quadratic first integral and report good energy preservation. Moreover, we have applied the discrete gradient method to solve the Lorenz and SIRI model and report good preservation of polynomial first integral of degree greater than two
Variational Integrators As General Linear Methods
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2020) Monazzam Khan; FA18-RMT-056; LHR TP 6460; Dr. Yousaf Habib
icular class of topological indices that are based on degrees of vertices of the graph. An ??-graph of G is newly defined class of graph that can be constructed by the replication of G in a particular fashion. In this Thesis, we study topological indices of line graph of ??-graph where the ??-graphs are produced by very known graphs which is ladder graph its extension triangular ladder graph, fan graph, friendship graph, circle graph, path graph.

M.Phil / MS

Browse

Filters

Settings

Sort By

Results per page

Search Results