The structure of lie algebras and the classification problem for partial differential equations
2001 (English)In: Acta Applicandae Mathematicae - An International Survey Journal on Applying Mathematics and Mathematical Applications, ISSN 0167-8019, Vol. 69, no 1, 43-94 p.Article in journal (Refereed) Published
The present paper solves completely the problem of the group classification of nonlinear heat-conductivity equations of the form u(t)=F(t,x,u,u(x))u(xx)+G(t,x,u,u(x)). We have proved, in particular, that the above class contains no nonlinear equations whose invariance algebra has dimension more than five. Furthermore, we have proved that there are two, thirty-four, thirty-five, and six inequivalent equations admitting one-, two-, three-, four- and five-dimensional Lie algebras, respectively. Since the procedure which we use relies heavily upon the theory of abstract Lie algebras of low dimension, we give a detailed account of the necessary facts. This material is dispersed in the literature and is not fully available in English. After this algebraic part we give a detailed description of the method and then we derive the forms of inequivalent invariant evolution equations, and compute the corresponding maximal symmetry algebras. The list of invariant equations obtained in this way contains (up to a local change of variables) all the previously-known invariant evolution equations belonging to the class of partial differential equations under study.
Place, publisher, year, edition, pages
2001. Vol. 69, no 1, 43-94 p.
group classification, equivalence group, symmetry analysis, Lie algebras
Engineering and Technology
IdentifiersURN: urn:nbn:se:liu:diva-49058OAI: oai:DiVA.org:liu-49058DiVA: diva2:269954