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A note on modelling with measures: two-features balance equations.

Michael Böhm1, Martin Höpker

  • 1Center for Industrial Mathematics, University of Bremen, Bibliothekstrasse 1, D-28359 Bremen, Germany. mbohm@math.uni-bremen.de.

Mathematical Biosciences and Engineering : MBE
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Summary

This study introduces balance equations using measures, not derivatives, for modeling systems like diffusion-reaction and mass conservation. It extends this to complex scenarios involving multiple features, such as aging and motion in populations.

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Area of Science:

  • Mathematical Modeling
  • Theoretical Physics
  • Population Dynamics

Background:

  • Traditional modeling often relies on derivatives and rates.
  • Existing models may struggle with complex, multi-feature systems.
  • A need exists for alternative modeling approaches in scientific domains.

Purpose of the Study:

  • To define and illustrate balance situations and balance equations using measures.
  • To propose a novel framework for modeling diffusion-reaction and mass-conservation scenarios.
  • To extend the concept to multi-feature balance situations with practical examples.

Main Methods:

  • Exemplification of balance situations and equations through a concrete example.
  • Development of balance equations based on measures instead of derivatives.
  • Application and extension of the concept to two-feature scenarios using three distinct examples.

Main Results:

  • A clear definition and understanding of balance situations and equations in terms of measures.
  • Demonstration of the applicability of measure-based modeling to diffusion-reaction and mass-conservation.
  • Successful extension to model complex systems combining traits like aging, motion, and polymer formation.

Conclusions:

  • Measure-based balance equations offer a viable alternative to derivative-based models.
  • The proposed framework effectively models single and multi-feature dynamic systems.
  • This approach provides a unified way to describe diverse phenomena in populations and material science.