976 research outputs found
A precise definition of reduction of partial differential equations
We give a comprehensive analysis of interrelations between the basic concepts
of the modern theory of symmetry (classical and non-classical) reductions of
partial differential equations. Using the introduced definition of reduction of
differential equations we establish equivalence of the non-classical
(conditional symmetry) and direct (Ansatz) approaches to reduction of partial
differential equations. As an illustration we give an example of non-classical
reduction of the nonlinear wave equation in (1+3) dimensions. The conditional
symmetry approach when applied to the equation in question yields a number of
non-Lie reductions which are far-reaching generalization of the well-known
symmetry reductions of the nonlinear wave equations.Comment: LaTeX, 21 page
Algorithmic Thomas Decomposition of Algebraic and Differential Systems
In this paper, we consider systems of algebraic and non-linear partial
differential equations and inequations. We decompose these systems into
so-called simple subsystems and thereby partition the set of solutions. For
algebraic systems, simplicity means triangularity, square-freeness and
non-vanishing initials. Differential simplicity extends algebraic simplicity
with involutivity. We build upon the constructive ideas of J. M. Thomas and
develop them into a new algorithm for disjoint decomposition. The given paper
is a revised version of a previous paper and includes the proofs of correctness
and termination of our decomposition algorithm. In addition, we illustrate the
algorithm with further instructive examples and describe its Maple
implementation together with an experimental comparison to some other
triangular decomposition algorithms.Comment: arXiv admin note: substantial text overlap with arXiv:1008.376
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