Similarity variables and reduction of the heat equation on torus

The problem of symmetry classification for the heat equation on torus is studied by means of classical Lie group theory. The Lie point symmetries are constructed and Lie algebra is formed for equation under consideration. Then these algebras are used to classify its subalgebras up to conjugacy classes. In general the heat equation on torus admits one-, two-, three- and four-dimensional algebras. For one-dimensional algebra £1 and £2 the heat equation on torus is reduced in independent variables whereas in two-dimensional algebras £3 and £4 the considered heat equation is investigated by quadrature. While three- and four-dimensional algebras lead to a trivial solution.

► Heat equation on torus is discussed.
► Conjugacy classes for Lie symmetry generators are discussed.
► Reductions and solutions of the considered equation are constructed via conjugacy classes.
► It is proved that symmetries of torus are dominant in the construction of the Lie symmetries of the considered equation.