What are the applications of non-linear PDEs in physics and engineering?

In summary, a non-linear partial differential equation (PDE) is a mathematical equation that describes a relationship between a function and its derivatives, where the function itself is non-linear. These equations have various applications in physics and engineering, such as modeling turbulent systems and designing aircraft wings. Solving non-linear PDEs is a challenging task and requires advanced mathematical techniques. Real-world examples of non-linear PDEs include weather forecasting and aerodynamics. However, these equations also come with challenges, such as difficulty in obtaining accurate solutions and determining initial and boundary conditions.
  • #1
DevacDave
15
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Hello Everybody!

I'm an applied mathematics researcher who found out about this website through the courtesy of one of my awesome students. My main degree is in physics, but afterwards I have been leaning more and more in the direction of mathematics and programming. Most of my work orbits around non-linear PDE that pop-up in physics and engineering. As far as hobbies go I think that I have too many of them to list, but I should mention that bashing on people who think that Mathematica will replace the need for mathematicians and mathematics education is among my cherished pleasures. I do like and use Mathematica, it's just the above mindset that bugs me. Other ones (interests and hobbies) would be condensed matter physics and DIY electronics projects.

English is not my primary language, but I think that I am far from being incomprehensible ;).
 
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  • #3
DevacDave said:
I should mention that bashing on people who think that Mathematica will replace the need for mathematicians and mathematics education is among my cherished pleasures. I do like and use Mathematica, it's just the above mindset that bugs me.
I like it when people cherish their annoyances, especially when they are right. Welcome here!

EDIT: Maybe you can briefly tell a bit more about your research? Is there a particular kind of PDE (a particular class of problems) that you are interested in?
 
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  • #4
Krylov said:
I like it when people cherish their annoyances, especially when they are right. Welcome here!

EDIT: Maybe you can briefly tell a bit more about your research? Is there a particular kind of PDE (a particular class of problems) that you are interested in?
Polar bear! you have a good like!
 
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  • #5
Krylov said:
I like it when people cherish their annoyances, especially when they are right. Welcome here!

EDIT: Maybe you can briefly tell a bit more about your research? Is there a particular kind of PDE (a particular class of problems) that you are interested in?

I'm definitely guilty on spending too much of my free time on giving a shot with Navier-Stokes Equation ;). But my fairly recent research in shallow water waves for modelling movement of sediments and shore erosion made me look closer and closer in that direction.
 
  • #6
DevacDave said:
I'm definitely guilty on spending too much of my free time on giving a shot with Navier-Stokes Equation ;).
This must be a beautiful topic, mathematically profound while simultaneously very close to direct applications. It is certainly on my long-time list of subjects that deserve much closer studying.
 

What is a non-linear PDE?

A non-linear partial differential equation (PDE) is a mathematical equation that describes a relationship between a function and its derivatives, where the function itself is non-linear. This means that the terms in the equation involving the function are not simply multiples of the function, but may involve powers, products, or other non-linear operations.

What are the applications of non-linear PDEs in physics and engineering?

Non-linear PDEs have a wide range of applications in physics and engineering, including fluid dynamics, quantum mechanics, electromagnetism, and general relativity. They are used to model complex systems that exhibit non-linear behavior, such as turbulence, shock waves, and chaotic systems.

How are non-linear PDEs solved?

Solving non-linear PDEs is a challenging task and often requires advanced mathematical techniques. Some common methods include numerical methods, perturbation methods, and variational methods. In some cases, non-linear PDEs can also be solved analytically using special techniques such as separation of variables or similarity transformations.

What are some real-world examples of non-linear PDEs?

One example of a real-world application of non-linear PDEs is in weather forecasting. The Navier-Stokes equations, which describe the motion of fluids, are non-linear PDEs and are used to model weather patterns and predict future weather conditions. Another example is in the design of aircraft wings, where non-linear PDEs are used to calculate the aerodynamic forces and optimize the wing shape for maximum efficiency.

What are the challenges in using non-linear PDEs?

Non-linear PDEs are notoriously difficult to solve and often require complex mathematical techniques. They can also exhibit chaotic behavior, making it challenging to predict their solutions accurately. Additionally, the accuracy of the solutions obtained from non-linear PDEs depends heavily on the initial and boundary conditions, which can be difficult to determine in real-world applications.

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