Dirichlet problem for a non-autonomous Bratu equation

Solution by M.C. Nucci (Dipartimento di Matematica e Informatica, Università di Perugia, 06123 Perugia, Italy) Lie’s monumental work on transformation groups, [12], [13] and [14], and in particular contact transformations [15], has provided systematic techniques for obtaining exact solutions of differential equations [16]. Many books have been dedicated to this subject and its generalizations ([1], [3], [19], [18], [4], [20], [21], [7], [9], [10], [11], [8], [2]). Lie group analysis is indeed the most powerful tool to find the general solution of ordinary differential equations. Any known integration technique can be shown to be a particular case of a general integration method based on the derivation of the continuous group of symmetries admitted by the differential equation, i.e. the Lie symmetry algebra, which can be easily derived by a straightforward although lengthy procedure. As computer algebra software becomes widely used, the integration of systems of ordinary differential equations by means of Lie group analysis is becoming easier to perform (see [6] for a list of such programs done with different computer algebra softwares). Hence let us apply Lie group analysis to equation (1) by using an ad-hoc interactive REDUCE program [17]. We easily find that equation (1) admits a two-dimensional non-abelian and transitive Lie symmetry algebra generated by the following operators: