# Solution of the Hyperbolic Partial Differential Equation on Graphs and Digital Spaces: a Klein Bottle a Projective Plane and a 4D Sphere

## TL;DR

This paper explores computational solutions to hyperbolic PDEs on digital models of continuous manifolds like the Klein bottle and 4D sphere, emphasizing topologically correct digital domains for accurate results.

## Contribution

It introduces conditions for the existence of solutions and demonstrates numerical solutions on digital manifolds, bridging continuous PDEs and digital topology.

## Key findings

- Conditions for solution existence established
- Numerical solutions demonstrated on digital manifolds
- Highlights importance of topologically correct digital domains

## Abstract

In many cases, analytic solutions of partial differential equations may not be possible. For practical problems, it is more reasonable to carry out computational solutions. However, the standard grid in the finite difference approximation is not a correct model of the continuous domain in terms of digital topology. In order to avoid serious problems in computational solutions it is necessary to use topologically correct digital domains. This paper studies the structure of the hyperbolic partial differential equation on graphs and digital n-dimensional manifolds, which are digital models of continuous n-manifolds. Conditions for the existence of solutions are determined and investigated. Numerical solutions of the equation on graphs and digital n-manifolds are presented.

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Source: https://tomesphere.com/paper/1705.02878