Definition of order of a partial differential equation

In summary, the order of a partial differential equation is defined by the highest order derivative present in the equation, whether it is in terms of time or space. For example, an equation with a second order time derivative would be considered second order. Further information and resources on this topic can be found online.
  • #1
Kashmir
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How is the order of a partial differential equation defined?

This is said to be first order: ##\frac{d}{d t}\left(\frac{\partial L}{\partial s_{i}}\right)-\frac{\partial L}{\partial q_{i}}=0##

And this second order :##\frac{d}{d t}\left(\frac{\partial L}{\partial \dot{q_{i}}}\right)-\frac{\partial L}{\partial q_{i}}=0##

What's the proper definition?

Thank you
 
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  • #2
The dot over the q makes the second line second order.
 
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Likes Kashmir
  • #3
mathman said:
The dot over the q makes the second line second order.
Thank you, could you please tell me the definition of second order for partial D.E ?
 
  • #4
The order for partial D.E., like an ordinary D.E., refers to the highest order derivative in the D.E.

For example, a D.E. with ##\partial^2{y}/\partial{x}^2##, would be 2nd order if no higher derivatives were present, and similarly with d2y/dx2.

Unfortunately, I've gone blank about mixing time and position/space derivatives, and ordinary with partial.

There are many online tutorials concerning DEs, both ODE and PDE.
https://users.aber.ac.uk/ruw/teach/260/classification.php
https://tutorial.math.lamar.edu/classes/calciii/highorderpartialderivs.aspx

https://www.math.toronto.edu/jko/APM346_summary_1_2020.pdf
https://www.csc.kth.se/utbildning/kth/kurser/DN1213/numme06/utdelat/kap10.pdf
 
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What is the definition of order of a partial differential equation?

The order of a partial differential equation (PDE) is the highest derivative present in the equation. It indicates the complexity of the equation and determines the number of independent variables needed to solve it.

How is the order of a PDE different from the degree of a polynomial?

The order of a PDE refers to the highest derivative, while the degree of a polynomial refers to the highest power of the variable. In a PDE, the highest derivative may have a different power than the other derivatives present.

How does the order of a PDE affect its solution?

The order of a PDE determines the number of initial or boundary conditions needed to find a unique solution. A higher order PDE requires more conditions, making it more difficult to solve.

Can a PDE have a fractional or negative order?

Yes, a PDE can have a fractional or negative order. These types of PDEs are known as fractional or noninteger order PDEs and are often used to model complex systems with non-local behavior.

What is the significance of the order of a PDE in real-world applications?

The order of a PDE is crucial in understanding the behavior of physical systems. It helps determine the complexity of the system and the number of variables needed to accurately describe it. In many cases, higher order PDEs are needed to model real-world phenomena accurately.

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