Department of Mathematics

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Author: search.filters.author.Dr. Yousaf HabibSubject: search.filters.subject.Department of Mathematics

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    Advance Numerical Methods for Special Differential Equations with Structure
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) Fozia Yasin; CIIT/FA19-RMT-017/LHR; Dr. Yousaf Habib; LHR TP 7422
    The current work is concerned with structure-preserving numerical meth- ods for particular differential equations representing dynamically in classical mechanics from the Hamiltonian perspective. These differential equations involve quantities that should remain constant throughout the investiga- tion. These methods includes the symplectic Runge-Kutta methods and the G-symplectic general linear methods. They faithfully maintain the un- derline characteristic properties of a Hamiltonian system during numerical discritization of the problem, whereas conventional numerical methods do not account for the preservation of these invariants. The conservation of total energy and the symplecticity of flow are two physically significant invariant properties of a Hamiltonian system which we to preserve numeri- cally. This thesis contains four chapters. In chapter 1, the fundamental theory of ordinary differential equations is presented. The exposition combines definitions with the development and application of numerical methods for special differential equations, and it serves as the foundation for all subsequent investigations in this work. In chapter 2, numerical methods for the Hamiltonian system are thoroughly discussed. In chapter 3, the crucial concept of projection is introduced. The basic concept of the standard projection technique is generalised to create the heterogeneous class of numerical methods known as general lin- ear methods. In chapter 4, the symmetric general linear methods are used with the Runge-Kutta methods as starting methods.
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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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    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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    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.
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    Growth of Matter Index and Cosmic Acceleration
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2020) Ahmad Hassan; FA18-RMT-005; LHR TP 6452; Dr. Yousaf Habib
    The aim of this research is to construct G-symplectic general linear methods for solution of ordinary differential equation in which parasitism can be controlled. Due to multi-value nature of general linear methods, there exist a parasitic component which disturb the actual solution and destroy the overall accuracy. We want to construct parasitic free G-symplectic general linear methods.