Equivalence problem for scalar linear (1+1) hyperbolic equations

I.K.Johnpillai, F.M.Mahomed

School of Computational and Applied Mathematics

Centre for Differential Equations, Continuum Mechanics and Applications

University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa

fmahomed@cam.wits.ac.zaљљљљ ignatius@uow.edu.au

 

We deduce new semi-invariants for the scalar linear second-order hyperbolic partial differential equation (PDE) that remains invariant under transformations of the independent variables t and x of the PDE. These are the analogues of the Laplace invariants for the scalar linear (1+1) hyperbolic PDE which transforms invariantly under linear changes of the dependent variable of the PDE. We prove necessary and sufficient conditions for the equivalence of two hyperbolic equations using the basis of joint invariants J₁ , J₂ , J₃ , J₄ and J₅ obtained in a previous paper by the authors. We show that, in general, the joint differential invariants J₄ and J₃ for the four Lie canonical forms of the scalar linear hyperbolic equation in t and x, provided that one of the Laplace invariants in the pair (h, k) does not vanish, are zero. If one of the Laplace invariants (h, k) is zero for any of the four Lie canonical forms of scalar linear (1+1) hyperbolic PDEs, then the Lie canonical form is factorisable. It is shown, via the use of invariants, that any scalar linear second-order hyperbolic PDE in t and x either belongs to the Lie canonical forms or it admits only the trivial homogeneity symmetry and its trivial infinite solution symmetries. Finally, several examples are given to illustrate the results obtained.

 

 

 

Ignatius Kenneth Johnpillai: PhD student at the University of the Witwatersrand while this work was undertaken; obtained his PhD at the beginning of 2002; presently has a postdoctoral position in Australia.

Fazal Mahmood Mahomed: Graduate of the University of Durban-Westville and of the University of the Witwatersrand, Johannesburg, in South Africa; Professor in the School of Computational and Applied Mathematics, and Director of the Centre for Differential Equations, Continuum Mechanics and Applications at the University of the Witwatersrand, Johannesburg; active in symmetry methods for differential equations and their applications.