On the generalized method of lines and its proximal explicit and hyper-finite difference approaches

This article firstly develops a proximal explicit approach for the generalized method of lines. In such a method, the domain of the PDE in question is discretized in lines and the equation solution is written on these lines as functions of the boundary conditions and domain shape. The main objective of introducing a proximal formulation is to minimize the solution error as a typical parameter $\varepsilon>0$ is too small. In a second step we present another procedure to minimize this same error, namely, the hyper-finite differences approach. In this last method the domain is divided in sub-domains on which the solution is obtained through the generalized method of lines allowing the parameter $\varepsilon>0$ to be very small without increasing the solution error. The solutions for the sub-domains are connected through the boundary conditions and the solution of the partial differential equation in question on the node lines which separate the sub-domains. In the last sections of each text part we present the concerning softwares and perform numerical examples.