Resource for modelling with differential equations

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Discussion Overview

The discussion focuses on seeking resources for modeling with differential equations, particularly in the context of engineering applications. Participants share recommendations for books and tools that address both ordinary differential equations (ODEs) and partial differential equations (PDEs), while also highlighting the need for practical modeling applications in various engineering fields.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant requests recommendations for resources on modeling with differential equations, noting a lack of application focus in previous courses.
  • Another participant suggests "Mathematical Biology" by J.D. Murray as a valuable resource for modeling in biology.
  • A different participant expresses interest in engineering applications, particularly in mechanical and materials engineering, indicating a need for resources tailored to these fields.
  • It is mentioned that modeling mechanical systems often leads to differential algebraic equations (DAEs) or PDEs, with references to books by Hairer and Wanner for mechanical examples.
  • One participant humorously cites a paper on modeling a zombie outbreak as an engaging example of how modeling with differential equations can be presented.
  • A participant emphasizes the importance of modeling in various engineering disciplines, listing areas such as solid deformation mechanics, heat transfer, and fluid mechanics, and describes the role of ODEs and PDEs in capturing physical system behaviors.
  • Another participant shares their use of computer algebra systems, specifically Mathematica and Maple, as tools for modeling, noting their capabilities and costs.

Areas of Agreement / Disagreement

Participants express differing interests in the application of differential equations, with some focusing on biological modeling and others on engineering contexts. No consensus is reached on a single resource or approach, as multiple perspectives and recommendations are presented.

Contextual Notes

Participants highlight the need for resources that bridge the gap between theoretical knowledge of ODEs and PDEs and their practical applications in modeling, indicating a potential limitation in existing educational materials.

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I'm looking for a good resource about modelling for differential equations. I've completed two courses on ODE's and PDE's, but they both lacked in the applications/modelling department.

Can anyone recommend a good resource/book on modelling?
 
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A very nice but definitely not trivial book for modeling in biology is
J.D. Murray – Mathematical Biology
 
I've actually heard about that book before, but my interests have to do with engineering - particularly mechanical and materials engineering.
 
Often when modeling mechanical systems or materials one ends up with differential algebraic (DAE) or partial differential equations (PDE).

In case of DAEs the books of Hairer and Wanner contain some mechanical examples. You can find some of the examples at the hompage of Hairer
http://www.unige.ch/~hairer/software.html.

Unfortunately, I only model biological systems therefore I cannot really help you. (Nevertheless, I recommend the book from Murray because it is excellent written.)

For fun, one of the craziest papers about modeling a biological system is from the famous R. Smith:

WHEN ZOMBIES ATTACK!: MATHEMATICAL
MODELLING OF AN OUTBREAK OF ZOMBIE
INFECTION

http://mysite.science.uottawa.ca/rsmith43/Zombies.pdf

It is very well written and idependently(!) of the topic you get a very good idea how modeling with differential equations works.
 
abstracted6 said:
I'm looking for a good resource about modelling for differential equations. I've completed two courses on ODE's and PDE's, but they both lacked in the applications/modelling department.

Can anyone recommend a good resource/book on modelling?

Modeling using ODEs and PDEs is abundantly necessary in engineering.

Solid deformation mechanics
Heat Transfer
Fluid Mechanics
Diffusion
Chemical Reaction engineering
Thermodynamics
Transport Phenomena
Continuum Mechanics
Quantum mechanics

Problems in all these areas should be part of the basic engineering curriculum. Modeling comes in when you are applying physical principles to describe the behavior of a physical system. The ODEs and PDEs capture the description of the physical system in terms of the language of mathematics. You then have to solve the equations to understand and predict in advance the behavior of the physical system.
 
I use two so-called 'computer algebra systems' and they are quite outstanding. Student, and Home versions are available. One is 'Mathematica' which is a huge 4.5Gbyte system, and the other is Maple - much smaller but very powerful. The only drawback is that they are relatively expensive - but remember you usually get what you pay for - in systems like that.
 

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