Australian Mathematical Society (AustMS): E-Journals
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Fuzzy approaches against outliers and applications in wind energy
No full texthttp://dx.doi.org/10.1017/S000497271200033
Exact solutions of hyperbolic reaction-diffusion equations in two dimensions
No full textExact solutions are constructed for a class of nonlinear hyperbolic reaction-diffusion equations in two-space dimensions. Reduction of variables and subsequent solutions follow from a special nonclassical symmetry that uncovers a conditionally integrable system, equivalent to the linear Helmholtz equation. The hyperbolicity is commonly associated with a speed limit due to a delay, , between gradients and fluxes. With lethal boundary conditions on a circular domain wherein a species population exhibits logistic growth of Fisher–KPP type with equal time lag, the critical domain size for avoidance of extinction does not depend on . A diminishing exact solution within a circular domain is also constructed, when the reaction represents a weak Allee effect of Huxley type. For a combustion reaction of Arrhenius type, the only known exact solution that is finite but unbounded is extended to allow for a positive .
doi: 10.1017/S144618112300009
Contractive semigroups in topological vector spaces, on the 100th anniversary of Stefan Banach’s contraction principle
No full texthttp://dx.doi.org/10.1017/S000497272200162
High-order explicit second derivative methods with strong stability properties based on Taylor series conditions
No full textWhen faced with the task of solving hyperbolic partial differential equations (PDEs), high order, strong stability-preserving (SSP) time integration methods are often needed to ensure preservation of the nonlinear strong stability properties of spatial discretizations. Among such methods, SSP second derivative time-stepping schemes have been recently introduced and used for evolving hyperbolic PDEs. In previous works, coupling of forward Euler and a second derivative formulation led to sufficient conditions for a second derivative general linear method (SGLM), which preserve the strong stability properties of spatial discretizations. However, for such methods, the types of spatial discretizations that can be used are limited. In this paper, we use a formulation based on forward Euler and Taylor series conditions to extend the SSP SGLM framework. We investigate the construction of SSP second derivative diagonally implicit multistage integration methods (SDIMSIMs) as a subclass of SGLMs with order and stage order up to order eight, where r is the number of external stages and s is the number of internal stages of the method. Proposed methods are examined on some one-dimensional linear and nonlinear systems to verify their theoretical order, and show potential of these schemes in preserving some nonlinear stability properties such as positivity and total variation.
doi: 10.1017/S1446181122000128
A comparison of explicit Runge-Kutta methods
No full textRecent higher-order explicit Runge–Kutta methods are compared with the classic fourth-order (RK4) method in long-term integration of both energy-conserving and lossy systems. By comparing quantity of function evaluations against accuracy for systems with and without known solutions, optimal methods are proposed. For a conservative system, we consider positional accuracy for Newtonian systems of two or three bodies and total angular momentum for a simplified Solar System model, over moderate astronomical timescales (tens of millions of years). For a nonconservative system, we investigate a relativistic two-body problem with gravitational wave emission. We find that methods of tenth and twelfth order consistently outperform lower-order methods for the systems considered here.
doi: 10.1017/S144618112200014
Some criteria for solvability and nilpotency of finite groups by strongly monolithic characters
No full texthttp://dx.doi.org/10.1017/S000497271200033
A rigidity property of pluriharmonic maps from projective manifolds
No full texthttp://dx.doi.org/10.1017/S000497271200033