Volume : V, Issue : II, February - 2016

A STUDY ON SIMILARITY SOLUTIONS OF A NONLINEAR DIFFUSION EQUATION

L. Sivakami

Abstract :

 Nonlinear diffusion equations, an important class of parabolic appear widely in nature. They are suggested as mathematical models of physical problems in many fields, such as filtration, phase transition, bio chemistry and dynamics of biological groups. In many cases, the equations possess degeneracy or singularity. The appearance of degeneracy or singularity makes the study more involved and challenging. Many new ideas and methods have been developed to overcome the special difficulties caused by the degeneracy and singularity, which enrich the theory of partial differential equations. The motivation for the present study had its origin in my attempt to carry over these techniques, either singly or collectively as the case may be, for obtaining the nonlinear diffusion equation and its symmetry reductions, namely, the second–order nonlinear ordinary differential equations via the isovector approach and further group invariance techniques respectively. The fundamental basis of the techniques is that , when a differential equation in invariant under a Lie group of transformations, a reduction transformation exists. The machinery of the Lie group theory provides a systematic method to search for these special group invariant solutions, a single group reduction transforms the partial differential equation into ordinary differential equations which are generally easier to solve than the original partial differential equation. Local symmetries admitted by a nonlinear partial differential equation are useful to discover whether or not the equation can be linearized by an invertible mapping and construct an explicit linearization when one exists. Nonlinear diffusion equations, an important class of parabolic appear widely in nature. They are suggested as mathematical models of physical problems in many fields, such as filtration, phase transition, bio chemistry and dynamics of biological groups. In many cases, the equations possess degeneracy or singularity. The appearance of degeneracy or singularity makes the study more involved and challenging. Many new ideas and methods have been developed to overcome the special difficulties caused by the degeneracy and singularity, which enrich the theory of partial differential equations. The motivation for the present study had its origin in my attempt to carry over these techniques, either singly or collectively as the case may be, for obtaining the nonlinear diffusion equation and its symmetry reductions, namely, the second–order nonlinear ordinary differential equations via the isovector approach and further group invariance techniques respectively. The fundamental basis of the techniques is that , when a differential equation in invariant under a Lie group of transformations, a reduction transformation exists. The machinery of the Lie group theory provides a systematic method to search for these special group invariant solutions, a single group reduction transforms the partial differential equation into ordinary differential equations which are generally easier to solve than the original partial differential equation. Local symmetries admitted by a nonlinear partial differential equation are useful to discover whether or not the equation can be linearized by an invertible mapping and construct an explicit linearization when one exists.

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Article: Download PDF   DOI : 10.36106/ijsr  

Cite This Article:

L.Sivakami A Study on Similarity Solutions of a Nonlinear Diffusion Equation International Journal of Scientific Research, Vol : 5, Issue : 2 February 2016

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