  This article develops an approximate proximal approach for the generalized
method of lines. The present results are extensions and applications of
previous ones which have been published since 2011, in books and articles such
as [3,4,5,6]. We also recall that in the generalized method of lines, the
domain of the partial differential equation in question is discretized in lines
(or in curves) and the concerning solution is developed on these lines, as
functions of the boundary conditions and the domain boundary shape.
