Sequences of related linear PDEs

AbstractWe introduce a transformation theory to generate chains of linear PDEs and related solution sequences. Our procedure depends on representing a given linear PDE in terms of a special equivalent system of two coupled linear PDEs where the auxiliary dependent variable satisfies the next PDE of a sequence. The solution of a PDE with variable coefficient depending on n + 1 constants α1, α2, …, αn + 1} is obtained from any solution of a PDE of the same type with variable coefficient depending on n constants {;α1, α2…, αn} by a simple Bäcklund transformation. Each sequence contains two inclusive chains since the PDE with n constants is a special case of the PDE with n + 2 constants. We generate solutions of wave equations with wave speeds C(x; α1, α2, …, αn), Fokker-Planck equations with drifts F(x; α1, α2, … αn), and diffusion equations with diffusivities K(x; α1 α2, …, αn), where {α1, α2, …, αn} are arbitrary constants, n = 1, 2, … New explicit general solutions are obtained for a class of wave equations with wave speeds depending on three parameters

Group classification of the Sachs equations for a radiating axisymmetric, non-rotating, vacuum space-time

We carry out a Lie group analysis of the Sachs equations for a time-dependent axisymmetric non-rotating space-time in which the Ricci tensor vanishes. These equations, which are the first two members of the set of Newman-Penrose equations, define the characteristic initial-value problem for the space-time. We find a particular form for the initial data such that these equations admit a Lie symmetry, and so defines a geometrically special class of such spacetimes. These should additionally be of particular physical interest because of this special geometric feature.Comment: 18 Pages. Submitted to Classical and Quantum Gravit

Construction of solutions to partial differential equations by the use of transformation groups

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. A systematic approach is given for finding similarity solutions to partial differential equations by the use of transformation groups. If a one-parameter group of transformations leaves invariant a partial differential equation and its accompanying boundary conditions, then the number of variables can be reduced by one. In order to find the group of a given partial differential equation, the "classical" and "non-classical" methods are discussed. Initially no special boundary conditions are imposed since the invariances of the equation are used to find the general class of invariant boundary conditions. New exact solutions to the heat equation are derived. In addition new exact solutions are found for the transition probability density function corresponding to a particular class of first order nonlinear stochastic differential equations. The equation of nonlinear heat conduction is considered from the classical point of view. The conformal group in n "space-like" and m "time-like" dimensions, C(n, m), which is the group leaving invariant [...], is shown to be locally isomorphic to S O (n+l, m+l) for n + m >= 3. Thus locally compact operators, besides pure rotations, leave invariant Laplace's equation in n >= 3 dimensions. These are used to find closed bounded geometries for which the number of variables in Laplace's equation can be reduced

ABSTRACT: NEW SYWETRIES FOR ORDINARY DIPPERKNTIAL EQUATIONS

In thls paper we present a theory for calculating new symmetries for ordinary differential equations. These new symmetries lead to a systematic reduction of the order of a differential equation. Our approach depends on computing the Lie symmetries of a differential equation related to a given differential equation by a Backlund transformation. Consequently we induce new symmetries of the given differential equation vhich are not in general of Lie, contact, or Lie-mcklund type. We obtain nev symmetries and corresponding new analytic results for a class of ordinary differential equations arising from nonlinear diffusion