Browsing by Author "Gwaxa, Bongumusa"

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    A symmetry perspective of third-order polynomial evolution equations
    (University of the Witwatersrand, Johannesburg, 2024) Gwaxa, Bongumusa; Jamal, Sameerah
    In this thesis, we analyse the full class of ten Fujimoto-Watanabe equations. In particular, these are highly nonlinear third-order and two fifth-order equations. With the aid of computer algebra software such as Mathematica, we calculate symmetries for these equations and we construct their commutator tables. The one dimensional system of optimal subalgebras is obtained via adjoint operators. Finally, we reduce these higher-order partial differential equations into ordinary differential equations, derive their solutions via a power series solution method and show how convergence may be tested. Lastly, we determine some conservation laws
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    The complex-valued class of Korteweg-de Vries equations
    (2022) Gwaxa, Bongumusa
    In this dissertation, we apply Lie theory to determine the one-parameter point transformations which leave complex-valued Korteweg-de Vries equations in- variant. The conserved vectors of the systems are constructed. We provide travelling wave reductions that lead to third-order ordinary differential equa- tions. These equations are highly nonlinear to solve directly, and therefore we first establish their first integrals. The latter is of second-order and facilitates the solution of the complex-valued system.