Ryad Ghanam

Symmetry analysis of wave equation on static spherically symmetric spacetimes with higher symmetries

Symmetry analysis of wave equation on all static spherically symmetric spacetimes admitting maximal isometry groups G10 or G7 or G6 is carried out. Symmetry algebras of the wave equation are found and their structural information-in the sense of Iwasawa decomposition-is obtained. Joint invariants of appropriate subalgebras are utilized to obtain many exact solutions of the wave equation on static spherically symmetric spacetimes.

On a generalized diffusion equation arising in petroleum engineering

In this work we investigate a model which describes diffusion in petroleum engineering. The original model, which already generalizes the standard one (usual diffusion equation) to a non-local model taking into account the memory effects, is here further extended to cope with many other different possible situations. Namely, we consider the Hilfer fractional derivative which by its nature interpolates the Riemann-Liouville fractional derivative and the Caputo fractional derivative (the one that has been studied previously). At the same time, this kind of derivative provides us with a whole range of other types of fractional derivatives. We treat both the Neumann boundary conditions case and the Dirichlet boundary conditions case and find explicit solutions. In addition to that, we also discuss the case of an infinite reservoir.

Bi-invariant and noninvariant metrics on Lie groups

A restricted version of the inverse problem of Lagrangian dynamics for the canonical linear connection on a Lie group is studied. Specifically for solvable Lie algebras of dimension up to and including six all algebras for which there is a compatible pseudo-Riemannian metric on the corresponding linear Lie group are found. Of the 19 such metrics four are bi-invariant. The Lie algebras are taken from tables compiled originally by Mubarakzyanov [Izv. Vyssh. Uchebn. Zaved., Mat. 4, 104–116 (1963)] and Morozov [Izv. Vyssh. Uchebn. Zaved., Mat. 4, 161–171 (1958)].

Embedding algorithms and applications to differential equations

Abstract Algorithms for embedding certain types of nilpotent subalgebras in maximal subalgebras of the same type are developed, using methods of real algebraic groups. These algorithms are applied to determine non-conjugate subalgebras of the symmetry algebra of the wave equation, which in turn are used to determine a large class of invariant solutions of the wave equation. The algorithms are also illustrated for the symmetry algebra of a classical system of differential equations considered by Cartan in the context of contact geometry.

Minimal representations of Lie algebras with non-trivial Levi decomposition

We obtain minimal dimension matrix representations for each of the Lie algebras of dimensions five, six, seven and eight obtained by Turkowski that have a non-trivial Levi decomposition. The key technique involves using the invariant subspaces associated to a particular representation of a semi-simple Lie algebra to help in the construction of the radical in the putative Levi decomposition.

Nonsolvable Subalgebras of

All the simple and then semisimple subalgebras of are found. Each such semisimple subalgebra acts by commutator on . In each case the invariant subspaces are found and the results are used to determine all possible subalgebras of that are not solvable.

New Examples of Einstein Metrics in Dimension Four

We construct new examples of four-dimensional Einstein metrics with neutral signature and two-dimensional holonomy Lie algebra.