FracSym: automated symbolic computation of Lie symmetries of fractional differential equations

In this paper, we present an algorithm for the systematic calculation of Lie point symmetries for fractional order differential equations (FDEs) using the method as described by Buckwar & Luchko (1998) and Gazizov, Kasatkin & Lukashchuk (2007, 2009, 2011). The method has been generalised here to allow for the determination of symmetries for FDEs with n independent variables and for systems of partial FDEs. The algorithm has been implemented in the new MAPLE package FracSym (Jefferson and Carminati 2013) which uses routines from the MAPLE symmetry packages DESOLVII (Vu, Jefferson and Carminati, 2012) and ASP (Jefferson and Carminati, 2013). We introduce FracSym by investigating the symmetries of a number of FDEs; specific forms of any arbitrary functions, which may extend the symmetry algebras, are also determined. For each of the FDEs discussed, selected invariant solutions are then presented. © 2013 Elsevier B.V. All rights reserved.

The formation of ultrafine ferrite through static transformation in low carbon steels

A novel approach was used to produce an ultrafine grain structure in low carbon steels with a wide range of hardenability. This included warm deformation of supercooled austenite followed by reheating in the austenite region and cooling (RHA). The ultrafine ferrite structure was independent of steel composition. However, the mechanism of ferrite refinement changed with the steel quench hardenability. In a relatively low hardenable steel, the ultrafine structure was produced through dynamic strain induced transformation, whereas the ferrite refinement was formed by static transformation in steels with high quench hardenability. The use of a model Ni-30Fe austenitic alloy revealed that the deformation temperature has a strong effect on the nature of the intragranular defects. There was a transition temperature below which the cell dislocation structure changed to laminar microbands. It appears that the extreme refinement of ferrite is due to the formation of extensive high angle intragranular defects at these low deformation temperature that then act as sites for static transformation. © 2008 World Scientific Publishing Company.

A universal optimization strategy for ant colony optimization algorithms based on the Physarum-inspired mathematical model

Ant colony optimization (ACO) algorithms often fall into the local optimal solution and have lower search efficiency for solving the travelling salesman problem (TSP). According to these shortcomings, this paper proposes a universal optimization strategy for updating the pheromone matrix in the ACO algorithms. The new optimization strategy takes advantages of the unique feature of critical paths reserved in the process of evolving adaptive networks of the Physarum-inspired mathematical model (PMM). The optimized algorithms, denoted as PMACO algorithms, can enhance the amount of pheromone in the critical paths and promote the exploitation of the optimal solution. Experimental results in synthetic and real networks show that the PMACO algorithms are more efficient and robust than the traditional ACO algorithms, which are adaptable to solve the TSP with single or multiple objectives. Meanwhile, we further analyse the influence of parameters on the performance of the PMACO algorithms. Based on these analyses, the best values of these parameters are worked out for the TSP.