Linearisation and potential symmetries of certain systems of diffusion equations

We consider systems of two pure one-dimensional diffusion equations that have considerable interest in Soil Science and Mathematical Biology. We construct non-local symmetries for these systems. These are determined by expressing the equations in a partially and wholly conserved form, and then by performing a potential symmetry analysis on those systems that can be linearised. We give several examples of such systems, and in a specific case we show how linearising and hodograph-type mappings can lead to new solutions of the diffusion system.     

A deductive approach to the solution of the problem of optimal pairs trading from the viewpoint of stochastic control with time‐dependent parameters

In a fairly recent paper (2008 American Control Conference, June 11‐13, 1035‐1039), the problem of dealing with trading in optimal pairs was treated from the viewpoint of stochastic control. The analysis of the subsequent nonlinear evolution partial differential equation was based upon a succession of Ansätze, which can lead to a solution of the terminal‐value problem. Through an application of the Lie Theory of Continuous Groups to this equation, we show that the Ansätze are based upon the underlying symmetries of the equation (their (14)). We solve the problem in a more general context by allowing the parameters to be explicitly time dependent. The extension means thatmore realistic problems are amenable to the samemode of solution. Copyright © 2014 JohnWiley & Sons, Ltd.     

Lie group analysis of two-dimensional variable-coefficient Burgers equation

The modern group analysis of differential equations is used to study a class of two-dimensional variable coefficient Burgers equations. The group classification of this class is performed. Equivalence transformations are also found that allow us to simplify the results of classification and to construct the basis of differential invariants and operators of invariant differentiation. Using equivalence transformations, reductions with respect to Lie symmetry operators and certain non-Lie ansätze, we construct exact analytical solutions for specific forms of the arbitrary elements. Finally, we classify the local conservation laws.     

Conservation laws and hierarchies of potential symmetries for certain diffusion equations

We show that the so-called hidden potential symmetries considered in a recent paper [M.L. Gandarias, New potential symmetries for some evolution equations, Physica A 387 (2008) 2234-2242] are ordinary potential symmetries that can be obtained using the method introduced by Bluman and collaborators [G.W. Bluman, S. Kumei, Symmetries and Differential Equations, Springer, New York, 1989; G.W. Bluman, G.J. Reid, S. Kumei, New classes of symmetries for partial differential equations, J. Math. Phys. 29 (1988) 806-811]. In fact, these are simplest potential symmetries associated with potential systems which are constructed with single conservation laws having no constant characteristics. Furthermore we classify the conservation laws for classes of porous medium equations, and then using the corresponding conserved (potential) systems we search for potential symmetries. This is the approach one needs to adopt in order to determine the complete list of potential symmetries. The provenance of potential symmetries is explained for the porous medium equations by using potential equivalence transformations. Point and potential equivalence transformations are also applied to deriving new results on potential symmetries and corresponding invariant solutions from known ones. In particular, in this way the potential systems, potential conservation laws and potential symmetries of linearizable equations from the classes of differential equations under consideration are exhaustively described. Infinite series of infinite-dimensional algebras of potential symmetries are constructed for such equations.     

On the invariants of two dimensional linear parabolic equations

We consider the most general two dimensional linear parabolic equations. Motivated by the recent work of Ibragimov et al. [1], [2], [3] we construct differential invariants, semi-invariants and invariant equations. These results are achieved with the employment of the equivalence group admitted by this class of parabolic equations. We derive those variable coefficient equations of this class of linear parabolic equations that can be mapped into constant coefficient equations. Further applications are presented.     

Spatial visualizers, object visualizers and verbalizers: their mathematical creative abilities

This paper investigates the relationship between the creative process in mathematical tasks and spatial, object and verbal cognitive styles. A group of 96 prospective primary school teachers completed the Object-Spatial Imagery and Verbal Questionnaire and took a mathematical creativity test. The results of a multiple regression analysis demonstrated that whereas visual cognitive styles (spatial and object imagery) were statistically significant predictors of participants’ creative abilities in mathematics, verbal cognitive style did not predict these abilities. Further analysis of the data indicated that spatial imagery cognitive style was related to mathematical fluency, flexibility and originality. On the other hand, object imagery cognitive style was negatively related to mathematical originality and verbal cognitive style was negatively related to mathematical flexibility. The study also revealed that individuals with a tendency towards different cognitive styles employed different strategies in the creative mathematical tasks.     

Classification of Noether Symmetries for Lagrangians with Three Degrees of Freedom

We classify all possible Noether symmetries of the Euler—Lagrange equations for a Hamiltonian system with three degrees of freedom. We also review the results for the cases of one and two degrees of freedom.     

Symmetry and singularity analyses of some equations of the fifth and sixth order in the spatial variable arising from the modelling of thin films

In the modelling of the flow of thin films higher-order derivatives in the spatial variable are introduced to model nonlinear effects. We examine nonlinear evolution equations of the fifth and sixth orders in the spatial variable from the viewpoint of Lie symmetry analysis. Values of the parameters which allow for a greater number of Lie point symmetries are identified. As the equations can be recast in potential form, we consider their potential symmetries. We also consider the singularity properties of the corresponding steady-state equations.     

Transformation properties of a variable-coefficient Burgers equation

In this paper we consider the variable coefficient equation u_t=b(t)uu_x+a(t)u_xx which among other applications has considerable interest in nonlinear acoustics. We present transformation properties of this generalised equation. In particular, we classify the Lie classical symmetries, the nonclassical symmetries, the potential symmetries, point and potential form preserving transformations. Finally, using these transformations we give examples of exact solutions.     

Enhanced group analysis and conservation laws of variable coefficient reaction–diffusion equations with power nonlinearities

A class of variable coefficient (1+1)-dimensional nonlinear reaction–diffusion equations of the general form f(x)u_t=(g(x)uⁿu_x)_x+h(x)u^m is investigated. Different kinds of equivalence groups are constructed including ones with transformations which are nonlocal with respect to arbitrary elements. For the class under consideration the complete group classification is performed with respect to convenient equivalence groups (generalized extended and conditional ones) and with respect to the set of all local transformations. Usage of different equivalences and coefficient gauges plays the major role for simple and clear formulation of the final results. The corresponding set of admissible transformations is described exhaustively. Then, using the most direct method, we classify local conservation laws. Some exact solutions are constructed by the classical Lie method.