3. Electronic Theses and Dissertations (ETDs) - All submissions
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Item Symmetry analysis of geometries in general relativity and mathematical models in quantitative finance(2024) Obaidullah, UsaamahThis thesis may be divided into two themes. The first, consists of a mathematical analysis of selected models in cosmology, while the second, is in the field of quantitative finance. The pp-wave spacetime, Bianchi I spacetime, and the Bianchi II spacetime are the three universes that we examine. The latter two are studied in the framework of f(R) theory of gravity, a plausible substitute for general relativity and a solution to the dark energy problem. We apply the Killing and homothetic vector fields to divide the spacetimes into classes or categories. Subsequently, potential functions are established using the geometry of the point symmetries of the space, while Noether’s theorem provides the first integrals connected to each isometry. We take advantage of the geometric fact that the homothetic algebra of spacetimes yields the Noether point symmetries of geodesic Lagrangians. For the Bianchi spacetimes, the Wheeler-DeWitt equations are derived by the quantisation of the spacetime Lagrangians, and from the Lie point symmetries it admits, we sequentially find invariant solutions to solve for the universe’s wave function. Finally, exact solutions to the field equations are also found. The research comprising of the second theme, investigates two nonlinear partial differential equations used in derivative pricing for financial markets. The point symmetries, invariant solutions, and conversation laws of these equations are found. In our analysis, we vary certain variables that change the nonlinearity of the models and thus give us unique symmetries and solutions. With various parameter settings, graphical solutions are investigated.Item An analysis of symmetries and conservation laws of nonlinear partial differential equations arising from Burgers’ hierarchy(2020) Obaidullah, UsaamahWe investigate the nonlinear evolutionary partial differential equations (PDEs) derived from Burgers’ hierarchy and give the exact solution of the complete hierarchy. The conservation laws of the hierarchy are studied and we proceed to establish the general nth conservation law. A transformation is used to render the hierarchy to a hierarchy of nonlinear ordinary differential equations (ODEs). These expressions are then linearised. Ultimately we give a novel exact solution of the entire Burgers’ hierarchy, that is, for all values of n. Several members of the hierarchy are solved, and the graphical counterparts of their solutions are provided to illustrate the applicability of our formula. Next we extend our study to the hierarchy of ODEs linked to this hierarchy. One-parameter Lie group of transformations that leave the ODEs invariant are constructed, from which it is established that these symmetries arise from the (n+ 1) complex roots of a certain polynomial. This gives us a formula to solve the ODE expressions, and finally we show how a more general exact solution of the complete hierarchy is obtained from this result