Group-invariant Solutions And Group Classification Of Nonlinear Partial Differential Equations

In this dissertation, symmetry group analysis is applied to the geometric heat equation and the affine heat equation, which are widely used in a number of different research areas such as differential geometry, crystal growth and image processing etc. The similarity reductions and group-invariant solutions of the two geometric flows are obtained. Furthermore, symmetry classification of a class of the fourth-order parabolic equations in one spatial variable is also performed. A large list of nonequivalent equations together with their symmetry structures is provided. The main achievements contained here are as follows:1. Classical Lie group analysis of the geometric heat flowis performed. Its Lie symmetries are derived. The basic similarity reductions for the equation are performed. Reduced equations and exact solutions associated to the symmetries are obtained.2. Lie symmetry group method is applied to study the affine heat equation for surfaceIts symmetry groups are determined, and the equation is reduced to 2-dimensional PDEs in the case of F_i(u) = K_i, (i=1,2,…,8). The symmetry groups, optimal systems, and the corresponding group-invariant solutions of these 2-dimensional reduced equations are obtained and classified. The reduced equation of a special case k₁ = k₆ = k₈ = 0, k₃ = k₄ = k₇ = 1, k₂ = -1 and k₅ = 2/3 is further discussed.3. Symmetry classification of a class of the fourth-order parabolic equations in one spatial variableis discussed. The classification basically consists of three steps. Firstly, we con-struct the equivalence group, that is, the most general Lie transformation group preserving the form of PDEs under consideration. And we also find the most gen-eral symmetry group together with a classifying equation for F. Secondly, using the structures of abstract Lie algebra, Lie algebras are realized by vector fields of the form obtained before in the first step. Lastly, inserting the canonical forms of symmetry generators into the classifying equations and solving them, yield the functions F. And then the corresponding invariant equations are obtained. Using this method, all inequivalent equations admitting either semi-simple or solvable Lie groups of dimensions up to and including four are constructed together with their symmetry algebras. Some equations admitting five-dimensional solvable Lie algebras are also listed. Main symmetry properties of these equations can be sum-marized as follows,·three inequivalent equations admitting one-dimensional Lie algebra.·two equations which admit semi-simple Lie algebras isomorphic to (?)(2, R).·nine equations admitting two-dimensional Lie algebras, among them, four equations admitting Abelian algebras and five admitting non-Abelian algebras.·forty-four equations admitting three-dimensional solvable Lie algebras. ·sixty-one equations admitting four-dimensional solvable Lie algebras.·some equations admitting five-dimensional solvable Lie algebras.The presented invariant equations include not only many well-known equa-tions but also a broad family of new equations possessing rich nontrivial Lie sym-metries. These new equations may have potential significance to model some new nonlinear phenomena.