School of Mathematics (Journal articles)

Permanent URI for this collectionhttps://hdl.handle.net/10539/38037

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    A Technique to Solve a Parabolic Equation by Point Symmetries that Incorporate Initial Data
    (Springer, 2025-03) Jamal, Sameerah; Maphanga, Rivoningo
    In this paper, we show how transformation techniques coupled with a convolution integral can be used to solve a generalised option-pricing model, including the Black–Scholes model. Such equations are parabolic and the special convolutions are extremely involved as they arise from an initial value problem. New symmetries are derived to obtain solutions through an application of the invariant surface condition. The main outcome is that the point symmetries are effective in producing exact solutions that satisfy a given initial condition, such as those represented by a call-option.
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    Forward scattering on the line with a transfer condition
    (SpringerOpen [Commercial Publisher], 2013-12) Currie, Sonja; Nowaczyk, Marlena; Watson, Bruce A.
    We consider scattering on the line with a transfer condition at the origin. Under suitable growth conditions on the potential, the spectrum consists of a finite number of eigenvalues which are negative real numbers, while the remainder is continuous spectrum which is comprised of the positive real axis. Asymptotics are provided for the Jost solutions. Conditions which characterize transfer conditions resulting in self-adjoint problems are found. Properties are given of the scattering coefficient linking it to the spectrum.
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    Self-adjoint higher order differential operators with eigenvalue parameter dependent boundary conditions.
    (SpringerOpen [Commercial Publisher], 2015-12) Möller, Manfred; Zinsou, Bertin
    Eigenvalue problems for even order regular quasi-differential equations with boundary conditions which depend linearly on the eigenvalue parameter λ can be represented by an operator polynomial (Formula presented.) where M is a self-adjoint operator. Necessary and sufficient conditions are given such that also K and A are self-adjoint.