# Structure of a Parabolic Partial Differential Equation on Graphs and Digital spaces. Solution of PDE on Digital Spaces: a Klein Bottle, a Projective Plane, a 4D Sphere and a Moebius Band

## TL;DR

This paper explores the structure and solutions of parabolic PDEs on digital models of manifolds like Klein bottles and Moebius strips, providing conditions for solutions and numerical examples on complex digital spaces.

## Contribution

It introduces conditions for the existence of solutions of parabolic PDEs on digital manifolds and demonstrates numerical solutions on various complex digital surfaces.

## Key findings

- Conditions for solution existence are established.
- Numerical solutions are provided for complex digital manifolds.
- The study extends PDE analysis to digital models of continuous manifolds.

## Abstract

This paper studies the structure of a parabolic partial differential equation on graphs and digital n-dimensional manifolds, which are digital models of continuous n-manifolds. Conditions for the existence of solutions of equations are determined and investigated. Numerical solutions of the equation on a Klein bottle, a projective plane, a 4D sphere and a Moebius strip are presented.

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Source: https://tomesphere.com/paper/1701.05232