Volume 18, Issue Supplement 1, September 2011

M. Bruschi, F. Calogero, F. Leyvraz, M. Sommacal

Pages: 1 - 31

Several applications of an explicitly invertible transformation are reported. This transformation is elementary and therefore all the results obtained via it might be considered trivial; yet the findings highlighted in this paper are generally far from appearing trivial until the way they are obtained...

J. C. Camacho, M. S. Bruzón, J. Ramírez, M. L. Gandarias

Pages: 33 - 49

In this paper we make a full analysis of the symmetry reductions of a beam equation by using the classical Lie method of infinitesimals and the nonclassical method. We consider travelling wave reductions depending on the form of an arbitrary function. We have found several new classes of solutions that...

P. G. Estévez, M. L. Gandarias, J. Lucas

Pages: 51 - 60

The classical Lie method is applied to a nonisospectral problem associated with a system of partial differential equations in 2 + 1 dimensions (Maccari A, J. Math. Phys. 12 (1998) 6547–6551.). Identification of the classical Lie symmetries provides a set of reductions that give rise to different nontrivial...

Marianna Euler, Norbert Euler

Pages: 61 - 75

We apply a list of criteria which leads to a class of fifth-order symmetry-integrable evolution equations. The recursion operators for this class are given explicitly. Multipotentialisations are then applied to the equations in this class in order to extend this class of integrable equations.

Norbert Euler, Marianna Euler

Pages: 77 - 105

We propose a method to identify and classify evolution equations and systems that can be multipotentialised in given target equations or target systems. We refer to this as the converse problem. Although we mainly study a method for (1 + 1)-dimensional equations/system, we do also propose an extension...

V. A. Galaktionov, S. R. Svirshchevskii

Pages: 107 - 121

Nonlinear n-dimensional second-order diffusion equations admitting maximal Lie algebras of point symmetries are considered. Examples of invariant solutions, as well as of solutions on invariant subspaces for some nonlinear operators, are constructed for arbitrary n. A complete description of all possible...

M. L. Gandarias, M. S. Bruzón

Pages: 123 - 133

In [19] we derive nonlocal symmetries for ordinary differential equations by embedding the given equation in an auxiliary system. Since the nonlocal symmetries of the ODE's are local symmetries of the associated auxiliary system this result provides an algorithmic method to derive this kind of nonlocal...

Keshlan S. Govinder, Barbara Abraham-Shrauner

Pages: 135 - 142

The inheritance of symmetries of partial differential equations occurs in a different manner from that of ordinary differential equations. In particular, the Lie algebra of the symmetries of a partial differential equation is not sufficient to predict the symmetries that will be inherited by a resulting...

N. H. Ibragimov

Pages: 143 - 162

The paper is devoted to the Lie group analysis of a nonlinear equation arising in metallurgical applications of Magnetohydrodynamics. Self-adjointness of the basic equations is investigated. The analysis reveals two exceptional values of the exponent playing a significant role in the model.

Vladimir F. Kovalev

Pages: 163 - 175

An application of approximate transformation groups to study dynamics of a system with distinct time scales is discussed. The utilization of the Krylov–Bogoliubov–Mitropolsky method of averaging to find solutions of the Lie equations is considered. Physical illustrations from the plasma kinetic theory...

Roman Kozlov

Pages: 177 - 187

Lie point symmetry group classification of a scalar stochastic differential equation (SDE) with one-dimensional Brownian motion is presented. First we prove that the admitted symmetry group is at most three-dimensional. Then the classification is carried out with the help of Lie algebra realizations...

M. Maldonado, J. Prada, M. J. Senosiain

Pages: 189 - 194

We study conditions for a sequence of Appell polynomials to be a basis on a sequence space.

Sergey V. Meleshko, Vasilii P. Shapeev

Pages: 195 - 212

The manuscript is devoted to nonisentropic solutions of simple wave type of the gas dynamics equations. For an isentropic flow these equations (in one-dimensional and steady two-dimensional cases) are reduced to the equations written in the Riemann invariants. The system written in the Riemann invariants...

Sibusiso Moyo, Sergey V. Meleshko

Pages: 213 - 236

In the literature, the generalized Sundman transformation has been used for obtaining necessary and sufficient conditions for a single second- and third-order ordinary differential equation to be equivalent to a linear equation in the Laguerre form. As far as we are aware, the generalized Sundman transformation...

C. Muriel, J. L. Romero

Pages: 237 - 250

We study the class of the ordinary differential equations of the form ẍ + a2(t, x)ẋ2 + a1(t, x)ẋ + a0(t, x) = 0, that admit v = ∂x as λ-symmetry for some λ = α(t, x)ẋ + β(t, x). This class coincides with the class of the second-order equations that have first integrals of the form C(t) + 1/(A(t, x)ẋ...

Andreas M. Rosteck, Martin Oberlack

Pages: 251 - 264

We briefly derive the infinite set of multi-point correlation equations based on the Navier–Stokes equations for an incompressible fluid. From this we reconsider the previously derived set of Lie symmetries, i.e. those directly induced by the ones from classical mechanics and also new symmetries. The...