M. L. Gandarias

Weak self-adjoint differential equations

The concepts of self-adjoint and quasi self-adjoint equations were introduced by Ibragimov (2006 J. Math. Anal. Appl. 318 742?57; 2007 Arch. ALGA 4 55?60). In Ibragimov (2007 J. Math. Anal. Appl. 333 311?28), a general theorem on conservation laws was proved. In this paper, we generalize the concept of self-adjoint and quasi self-adjoint equations by introducing the definition of weak self-adjoint equations. We find a class of weak self-adjoint quasi-linear parabolic equations. The property of a differential equation to be weak self-adjoint is important for constructing conservation laws associated with symmetries of the differential equation.

Classical point symmetries of a porous medium equation

The Lie-group formalism is applied to deduce symmetries of the porous medium equation . We study those spatial forms that admit the classical symmetry group. The reduction obtained from the optimal system of subalgebras are derived. Some new exact solutions can be obtained.

New symmetries for a model of fast diffusion

Abstract In this Letter we prove that, for some partial differential equations that model diffusion, by using the nonclassical method we obtain several new solutions which are not invariant under any Lie group admitted by the equations and consequently which are not obtainable through the classical Lie method. For these partial differential equations that model fast diffusion new classes of symmetries are derived. These nonclassical potential symmetries allow us to increase the number of exact explicit solutions of these nonlinear diffusion equations. These solutions are neither nonclassical solutions of the diffusion equation nor solutions arising from classical potential symmetries. Some of these solutions exhibit an interesting behavior as a shrinking pulse formed out of the interaction of two kinks.

Symmetry classification and optimal systems of a non-linear wave equation

Abstract In this paper the complete Lie group classification of a non-linear wave equation is obtained. Optimal systems and reduced equations are achieved in the case of a hyperelastic homogeneous bar with variable cross section.

Classical and Nonclassical Symmetries of a Generalized Boussinesq Equation

We apply the Lie-group formalism and the nonclassical method due to Bluman and Cole to deduce symmetries of the generalized Boussinesq equation, which has the classical Boussinesq equation as an special case. We study the class of functions f (u) for which this equation admit either the classical or the nonclassical method. The reductions obtained are derived. Some new exact solutions can be derived.

POTENTIAL SYMMETRIES OF A POROUS MEDIUM EQUATION

Potential symmetries, which are not local symmetries, are carried out for the porous medium equation where when it can be written in a conserved form. These symmetries are realized as local symmetries of a related auxiliary system, and lead to the construction of corresponding invariant solutions, as well as to the linearization of the equation by non-invertible mappings.

New Symmetry Reductions for some Ordinary Differential Equations

Abstract In this work we derive potential symmetries for ordinary differential equations. By using these potential symmetries we find that the order of the ODE can be reduced even if this equation does not admit point symmetries. Moreover, in the case for which the ODE admits a group of point symmetries, we find that the potential symmetries allow us to perform further reductions than its point symmetries. Some diffusion equations admitting an infinite number of potential symmetries and the scaling group as a Lie symmetry are considered and some general results are obtained. For all the equations that we have studied, a set of potential symmetries admitted by the diffusion equation is “inherited” by the ODE that emerges as the reduced equation under the scaling group.

On some differential invariants for a family of diffusion equations

The equivalence transformation algebra LE and some of its differential invariants for the class of equations ut = (h(u)ux)x + f(x, u, ux) (h ? 0) are obtained. Using these invariants, we characterize subclasses which can be mapped by means of an equivalence transformation into the well-studied family of equations vt = (vkvx)x.

Nonclassical symmetry reductions of a porous medium equation with convection

In this paper new symmetry reductions and exact solutions are presented for the porous medium equation with convection Those spatial forms for which the equation can be reduced to an ordinary differential equation are studied. The symmetry reductions and exact solutions presented are derived by using the nonclassical method developed by Bluman and Cole and are unobtainable by Lie classical method. The asymptotic behaviour of some of the new solutions is analysed.

THE CALOGERO-BOGOYAVLENSKII-SCHIFF EQUATION IN 2+1 DIMENSIONS

We use the classical and nonclassical methods to obtain symmetry reductions and exact solutions of the (2+1)-dimensional integrable Calogero–Bogoyavlenskii–Schiff equation. Although this (2+1)-dimensional equation arises in a nonlocal form, it can be written as a system of differential equations and, in potential form, as a fourth-order partial differential equation. The classical and nonclassical methods yield some exact solutions of the (2+1)-dimensional equation that involve several arbitrary functions and hence exhibit a rich variety of qualitative behavior.