How Do We Classify Higher-Order PDEs with Multiple Independent Variables?

In summary, there is a classification system for PDEs with an arbitrary number of independent variables and of order n, based on the Monge cone. However, there are also equations that do not fit into this classification system, possibly due to the complexity of forming a general classification or because it serves little purpose. The classification system may vary depending on whether it is for second order PDEs in n variables or for PDEs with order n derivatives.
  • #1
SpaceWalrus
18
0
Suppose you have a PDE with an arbitrary number of independent variables (not necessarily two), and of order n. Is there a nice classification akin to the hyperbolic, parabolic, etc.

Thanks
 
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  • #2
There is, it's to do with the Monge cone (I think). I also am informed that there are equations which have no classification.
 
  • #3
I also am informed that there are equations which have no classification.

This surprises me. Is this because forming a general classification system its more complicated than I imagine, or just that doing so serves little to no purpose?
 
  • #4
It depends if you're saying is there a classification system for second order PDEs in n variables or if there is a classification system for PDEs with order n derivatives.
 
  • #5
Either really... Second order with n variables, or n order with 2 variables (or n order with m variables).
 
  • #6
For second order equations in n variables, then it's to do with the Monge cone, with the other case I am not too sure as I am not an expert in this topic.
 

FAQ: How Do We Classify Higher-Order PDEs with Multiple Independent Variables?

What is the difference between ordinary and partial differential equations?

Ordinary differential equations (ODEs) involve only one independent variable, while partial differential equations (PDEs) involve multiple independent variables. ODEs also have only one type of derivative (usually with respect to time), while PDEs have multiple types of derivatives (with respect to different independent variables).

What are the main types of PDEs?

The main types of PDEs are elliptic, parabolic, and hyperbolic. Elliptic PDEs have a solution that is smooth and continuous, while parabolic PDEs have a solution that evolves over time. Hyperbolic PDEs have a solution that exhibits both wave-like and diffusion behavior.

How are PDEs classified based on their order?

PDEs can be classified as first-order or second-order, depending on the highest derivative present in the equation. First-order PDEs involve only first derivatives, while second-order PDEs involve second derivatives.

What is the importance of boundary and initial conditions in solving PDEs?

Boundary conditions specify the behavior of the solution at the boundaries of the domain, while initial conditions specify the behavior of the solution at the starting time or initial position. These conditions are crucial in determining a unique solution to a PDE.

How are PDEs used in real-world applications?

PDEs are used in many fields of science and engineering, such as physics, chemistry, biology, and finance. They are used to model and analyze complex systems and phenomena, such as fluid flow, heat transfer, diffusion, and population dynamics.

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