English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

This article firstly develops a proximal explicit approach for thegeneralized method of lines. In such a method, the domain of the PDE inquestion is discretized in lines and the equation solution is written on theselines as functions of the boundary conditions and domain shape. The mainobjective of introducing a proximal formulation is to minimize the solutionerror as a typical parameter $\varepsilon>0$ is too small. In a second step wepresent another procedure to minimize this same error, namely, the hyper-finitedifferences approach. In this last method the domain is divided in sub-domainson which the solution is obtained through the generalized method of linesallowing the parameter $\varepsilon>0$ to be very small without increasing thesolution error. The solutions for the sub-domains are connected through theboundary conditions and the solution of the partial differential equation inquestion on the node lines which separate the sub-domains. In the last sectionsof each text part we present the concerning softwares and perform numericalexamples.

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