Group classification and symmetry reductions for a class of nonlinear Poisson equations on the line and certain surfaces
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Saudi Digital Library
Abstract
Lie symmetry method is a technique to find exact solutions of differential equations. One of the significant applications of Lie symmetry theory is to achieve a complete classification of Lie symmetries and symmetry reductions of differential equations. This project is concerned with carrying out a symmetry analysis of a class of nonlinear Poisson equations of the form [special characters omitted]
The Laplacian operator Δ will be considered in four cases: on the line (one dimension), and the two dimensional cases consisting of plane, sphere and helicoid. The function f(u) will be assumed to be nonlinear.
In all four cases, the aim is to • find the minimal symmetry algebra • find all forms of f(u) which give larger symmetry algebras and determine these symmetry algebras • find some symmetry reductions and exact solutions for each case of f(u).