Abstract

Invariant Equations for (1+1) Linear Parabolic Equations

F.M.Mahomed (Centre for Differential Equations, Continuum Mechanics and Applications, School of Computational and Applied Mathematics, University of the Witwatersrand, Johannesburg, South Africa)

e-mail: fmahomed@cam.wits.ac.za

Semi-invariants for linear parabolic equations in two independent variables and one dependent variable under linear changes of the dependent variable have been investigated in [1]. The joint invariant equation for (1+1) linear parabolic equations for reduction to the classical heat equation has also been deduced in [2]. We further study reduction for these type of equations under equivalence transformations. We prove necessary and sufficient conditions for (1+1) parabolic equations to be reducible via equivalence transformations to the heat equation and to another Lie canonical form. The results obtained provide practical criteria for reduction. Examples are given to illustrate the results. References [1] N H Ibragimov 2002 Laplace type invariants for parabolic equations, Nonlinear Dynamics, to appear. [2] I K Johnpillai and F M Mahomed 2001 Singular invariant equation for the (1+1) Fokker-Planck equation, J Phys A: Math. Gen., 34, 11033

Section : 12