3. Electronic Theses and Dissertations (ETDs) - All submissions
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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 resultItem Symmetry structures and conserved forms of di erential equations on curved manifolds(2019) Gadjagboui, Bourgeois Biova IreneeWe explore two methods in nding Noether symmetries and conservation laws of di erential equations on Riemannian manifolds. The rst one is based on the Noether's theorem while the second one is about the `multiplier approach'. Using the rst method, we try to nd the variational symmetries, here, denoted X. With geodesic equations, the second method consists of nding the Lagrangian multipliers. This yields the conserved quantities when one acts the multipliers on the geodesic equations. It turns out that the total number of conserved quantities is equal to the number of variational symmetries found. The Lie algebra of in nitesimal isometries of the Riemannian manifolds studied has the dimension not exceeding 1 2n(n + 1), where n = 4. In the rst case studied in Chapter 2, variational symmetries and conservation laws of a modi ed de Sitter metric are classi ed. We came up with the suggestion that where a nite dimensional group generated by conservation laws exists, the Noether symmetry group has at least one additional symmetry that is not given by the Killing Vectors. This is later con rmed in the rest of all the other cases studied.Item Symmetries and conservation laws of certain classes of complex partial differential equations(2018) Phidane, ThilivhaliLie symmetry analysis is an established method for generating symmetries of differential equations. We apply this method together with fundamental theorem of double reduction. In particular, Noether symmetries and some associated conservation laws are constructed in our investigation to find exact solutions of higher order partial differential equations and complex partial differential equations.