M.Phil / MS

Permanent URI for this collectionhttps://repository.cuilahore.edu.pk/handle/123456789/52

This collection archives the complete set of theses produced by students of the COMSATS University Islamabad, Lahore Campus.

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Author: search.filters.author.Dr. Yousaf HabibEnd date: 2019

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Now showing 1 - 5 of 5
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    The Group of Runge-Kutta Methods
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2018) Asad Ullah; CIIT/FA13-BSM-018/LHR; Dr. Yousaf Habib; LHR TP 5251
    In this report we gave a review of Runge Kutta methods and partitioned Runge Kutta meth- ods.We also studied the group properties of these methods.The order conditions of Runge Kutta methods are related to rooted tree ,where as the order condition of partitioned Runge Kutta methods are related to bi-color rooted trees.The compositions of Runge Kutta meth- ods and partitioned Runge Kutta methods are studied as well.
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    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.
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    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.
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    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
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    On the Construction of Higher Order Conservative Methods
    (Library Information Services COMSATS University Islamabad Lahore Campus, 2012) Zareen Akhtar; FA19-RMT-026; LHR TP 7417; Dr. Yousaf Habib
    For the numerical solution of conservative differential equations such as Hamiltonian system or differential equations that arise in different physical phenomena with some structure, we use structure preserving numerical methods. In particular, symplectic Runge-Kutta methods, G-symplectic General Linear methods and Variational Integrators have been explored. In this thesis Variational Integrators for degenerate Lagrangian and non- degenerate Lagrangian have been discussed. For degenerate Lagrangian, we obtain multi-step numerical methods which suffer from parasitic corruption. In order to control the parasitism we write Variational Integrators as General Linear methods and then employ the projection technique to project the numerical solution onto the desired manifold
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