Classifying second-order Partial differential equations

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Discussion Overview

The discussion revolves around the classification of second-order partial differential equations (PDEs), exploring the definitions and characteristics of different types such as elliptic, parabolic, and hyperbolic equations. Participants seek to understand the classification process and its implications, particularly in relation to constant versus variable coefficients.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant inquires about the meaning and process of classifying second-order PDEs, indicating a need for foundational understanding.
  • Another participant explains that linear second-order PDEs can be classified into three standard forms—elliptic, parabolic, and hyperbolic—through a change of coordinates, referencing introductory PDE textbooks for further details.
  • It is noted that while PDEs with constant coefficients can be classified into these categories, those with variable coefficients may shift between classifications based on the values of the independent variables.
  • A later reply reiterates the point about variable coefficients affecting the classification of PDEs, acknowledging the previous contribution.

Areas of Agreement / Disagreement

Participants appear to agree on the classification framework for second-order PDEs, but there is an acknowledgment of complexity regarding variable coefficients that may lead to different classifications, indicating an area of ongoing discussion.

Contextual Notes

The discussion does not resolve the implications of variable coefficients on classification, nor does it clarify the conditions under which the classifications hold. There is also a lack of consensus on the specific methods for classification.

Who May Find This Useful

This discussion may be useful for students or individuals seeking to understand the classification of second-order partial differential equations, particularly in the context of mathematical physics or applied mathematics.

hellomrrobot
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What does it mean when it says to classify the second-order partial differential? (See attached)
How would I get started?
 

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Linear second order PDE's can, by a change of coordinates, be written in one of three (technically four, but one is degenerate) standard forms, called elliptic, parabolic, and hyperbolic equations. Any introductory PDE textbook should contain a description of these three types of equations. See e.g. http://www.math.psu.edu/tseng/class/Math251/Notes-PDE%20pt1.pdf
 
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Partial differential equations with constant coefficients can be so classified. If there are variable coefficients, the equation may change from "parabolic" to "hyperbolic" to "elliptic" depending on the value of the independent variables.
 
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HallsofIvy said:
Partial differential equations with constant coefficients can be so classified. If there are variable coefficients, the equation may change from "parabolic" to "hyperbolic" to "elliptic" depending on he value of the independent variables.

Good catch. Thanks.
 

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