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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    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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    Comparative Analysis of Neural Networks and Numerical Methods for Solving Differential Equations
    (Library Information Services COMSATS University Lahore Campus, 2024-03-18) Samra Sarwar; SP23-RMT-035; Dr. Adeel Faqoor; LHR TP 9569
    In order to solve Time-Dependent Kohn-Sham (TDKS) equations in quantum systems, this thesis primarily uses analytical solutions, the Runge-Kutta 4th Order (RK4) method, and Physics-Informed Neural Networks (PINNs). Although analytical solutions provide exact results, they are applicable only to simplified or highly structured systems; this is because real-life problems tend to get more complicated [1, 8]. For small time steps in smooth systems, RK4 is a highly accurate method, though its performance degrades substantially for large, stiff, or non-linear equations that can become computationally intensive to iterate through recursively [7, 2]. PINNs offer a paradigm change by incorporating the governing equations intrinsically within the neural network architecture itself [3, 18]. That avoids iterative time-stepping, allowing for fast in-time simulations of general, high-dimensional, and nonlinear quantum circuits [9, 12]. Our results reveal that, while RK4 is effective for small, low-dimensional problems, when tackling more complex ones, PINNs triumph in terms of the scalability and robustness of the process [4, 11]. Moreover, PINNs not only demonstrate a huge potential to transform computational strategy in quantum mechanics but also pave the way to tackle the long-time dynamics and nonlinearity problems in quan tum many-body systems [10, 13