Finite Difference Schemes as Algebraic Correspondences between Layers

Mikhail Malykh,Leonid Sevastianov,Gh Adam,J Buša,M Hnatič,D Podgainy

doi:10.1051/epjconf/201817303016

Mikhail Malykh, Leonid Sevastianov + Show 4 more

Open Access

https://doi.org/10.1051/epjconf/201817303016

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Abstract

For some differential equations, especially for Riccati equation, new finite difference schemes are suggested. These schemes define protective correspondences between the layers. Calculation using these schemes can be extended to the area beyond movable singularities of exact solution without any error accumulation.

Highlights

Modern development of computer science revived the old investigations on the solvability of differential equations in the finite terms
There are remarkable differential equations which can be integrated in computer algebra systems (CAS)
If a singular point of the solution to the Cauchy problem dy = f (x, y), dx y|x=a = y0 depends on the initial data it is called movable singularity of ordinary differential equation (ODE)

Summary

Introduction

Modern development of computer science revived the old investigations on the solvability of differential equations in the finite terms. There are remarkable differential equations which can be integrated in computer algebra systems (CAS). In our work we want to talk about remarkable differential equations in another sense: for these equations there are finite difference schemes that correctly describe singularities of exact solutions

Finite difference method and singularities

Riccati equation

Finite difference method and algebraic geometry

Conclusions