Classifying a 2nd Order PDE: Understanding the Significance of the Discriminant

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In summary, when classifying a 2nd order PDE, the discriminant is typically used to determine if it is Hyperbolic, Parabolic, or Elliptic. However, if the discriminant is dependent on independent or dependent variables, the equation may change classifications at different points in the domain. There are specialized equations that combine classifications, such as Hyperbolic-Elliptic and Parabolic-Elliptic equations.
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Diophantus
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A quick question:

When classifying a 2nd order PDE as either Hyperbolic, Parabolic or Elliptic we look at whether the discriminant is either positive, zero or negative respectively. Right. What do we do if the discriminant depends on independent variables (or the dependent variable for that matter) such that its sign can vary? Eg D = x. Do we classify it for the different values of x?

Regards.
 
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Diophantus said:
A quick question:

When classifying a 2nd order PDE as either Hyperbolic, Parabolic or Elliptic we look at whether the discriminant is either positive, zero or negative respectively. Right. What do we do if the discriminant depends on independent variables (or the dependent variable for that matter) such that its sign can vary? Eg D = x. Do we classify it for the different values of x?

Regards.

Yep, yep. Here's a quote:

"If the coefficients A, B, C are functions of x, y, and/or u (dep. variable), the equation may change from one classification to another at various points in the domain".
 
  • #3
And, in fact, there are entire books written on "Hyperbolic-Elliptic" equations, "Parabolic-Elliptic" equations, etc.
 

What is the discriminant of a PDE?

The discriminant of a PDE (partial differential equation) is a mathematical term used to determine the nature of the solutions to a PDE. It is calculated based on the coefficients of the highest order derivatives in the equation.

How is the discriminant used to classify PDEs?

The discriminant is used to classify PDEs into three types: elliptic, parabolic, and hyperbolic. Elliptic PDEs have a negative discriminant, parabolic PDEs have a discriminant of zero, and hyperbolic PDEs have a positive discriminant.

What is the significance of the discriminant in solving PDEs?

The discriminant helps determine the type of solution (e.g. single-valued, multi-valued, etc.) that a PDE will have. This information is crucial in selecting an appropriate method for solving the PDE.

Can the discriminant be used to determine the stability of a solution to a PDE?

Yes, the discriminant can be used to determine the stability of a solution. For example, elliptic PDEs have stable solutions, while hyperbolic PDEs have solutions that can become unstable under certain conditions.

Are there any limitations to using the discriminant in solving PDEs?

While the discriminant is a useful tool in classifying and solving PDEs, it does have limitations. For example, it only applies to linear PDEs and may not be applicable to more complex nonlinear PDEs.

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