Practical application of partial,simple differential equation

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Discussion Overview

The discussion focuses on the practical applications of partial and ordinary differential equations, particularly in the context of mechanical engineering, including statics and dynamics. Participants explore how these equations model real-world phenomena and their relevance in various engineering scenarios.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Homework-related

Main Points Raised

  • One participant requests examples of practical applications of partial and ordinary differential equations in mechanical engineering, specifically in statics.
  • Another participant emphasizes that real-world mathematical physics often involves non-linear partial differential equations due to the complexity of real-world interactions, suggesting that linearization is a common approach to manage these complexities.
  • A participant notes that partial differential equations are relevant in elasticity theory, particularly for analyzing beam bending and stress distribution under loads, indicating the complexity of the equations involved.
  • Specific applications in statics and dynamics are suggested, including beam deflection, virtual work, finite element methods, catenary equations, oscillators, least action principles, Kepler's equations, and Maxwell's equations.

Areas of Agreement / Disagreement

Participants express a range of views on the applications of differential equations, with no consensus on specific examples or methods. The discussion remains open-ended, with various perspectives on the complexity and utility of these equations in engineering contexts.

Contextual Notes

Some discussions involve assumptions about the nature of real-world systems and the necessity of non-linear equations, which may not be universally applicable. The mention of specific applications suggests a dependence on particular definitions and contexts within mechanical engineering.

chandran
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can anybody tell with equation,the practical applications of partial
differential equation and ordinary differential equation. If it is in
mechanical engineering statics etc will be helpful
 
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chandran said:
can anybody tell with equation,the practical applications of partial
differential equation and ordinary differential equation. If it is in
mechanical engineering statics etc will be helpful

Most of the equations of Mathematical Physics are in terms of not only partials but non-linear ones to boot. I mean really, we do easy ones in school to just learn how to work them but in real-life, the equations include more variables (hence partials), since as you know "everything is connected to everything out there", and if our mathematical models are to have any chance of genuinely reflecting the real world out there, then they usually need to be non-linear since of course it's a non-linear world out there as well. However, non-linear ones are really tough to work with so just so that we can get a handle on things, we often "linearize" them just to cope.

As far as "practical", well we live in a "changing world". Derivatives represent change. Ergo: Differential equations are good at modeling the world.
 
As for statics, partial differential equations naturally occurs in elasticity theory.
If you want to find out how much a beam is going to bend, or find the stress distribution within it when subjected to some load, you're in the middle of some ugly non-linear partial diff.eq. problems.
 
Here are some statics and dynamics applications to get you started... google these

beam deflection "differential equation"
"virtual work" "differential equation"
"finite element method" "differential equation"
catenary "differential equation"


for dynamics:

oscillator "differential equation"
"least action" "differential equation"
kepler "differential equation"

maxwell "differential equation"
 

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