Department of Mathematics
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Item 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 5674The 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.Item 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 5666Geometric 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