Physics nonlinear ODE example for numerical methods project?

In summary, the conversation is about a research project on numerical methods for solving ODEs. The first half of the project involves learning about the basics of numerical methods, while the second half focuses on a specific method (Linear multistep) and a particular example. The person has completed the first part and is now looking for an interesting non-linear example to work on. They have a background in ODEs and physics, and are interested in plasma physics but have been advised to use a simpler example. Some suggestions given include the Lorenz equations, Rössler attractor, and Chua's circuit.
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fletch-j
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I am doing a little research project into numerical methods of solving ODEs where I do 1 half of learning about the basics of numerical methods and then look at a particular method (Linear multistep) and then the second half is looking at a particular example, applying what I've learned and writing a report about it.
I've just about completed the first part.

I was wondering if anyone could give an interesting non-linear example which I might be able to work on, or instruct me how I could find such an example and the required equation/s.
I have done a semester course on ODEs and semester of physics, and I am interested in plasma physics in the future (but I have been advised that using a specific example from plasma physics might be too unwieldy with my limited experience).
 
Physics news on Phys.org

1. What is a nonlinear ODE in physics?

A nonlinear ODE, or ordinary differential equation, is a mathematical equation that describes the relationship between a variable and its derivatives, where the variable is not directly proportional to its derivatives. In physics, nonlinear ODEs are used to model complex phenomena that cannot be described by simple linear equations.

2. Why is it important to use numerical methods for solving nonlinear ODEs in physics?

Numerical methods are necessary for solving nonlinear ODEs in physics because most of these equations do not have analytical solutions. Numerical methods use algorithms and computer simulations to approximate the solutions of these equations, making it possible to study and predict behavior of complex physical systems.

3. What are some common numerical methods used for solving nonlinear ODEs in physics?

Some common numerical methods used for solving nonlinear ODEs in physics include Euler's method, Runge-Kutta methods, and finite difference methods. These methods involve breaking down the ODE into smaller, simpler equations and using iterative calculations to approximate the solution.

4. Can you provide an example of a physics problem that can be solved using a nonlinear ODE and numerical methods?

One example could be the motion of a pendulum, which can be described by a nonlinear ODE. Using numerical methods, we can solve for the position and velocity of the pendulum at any given time, taking into account factors such as gravity, mass, and length of the pendulum.

5. Are there any limitations or challenges associated with using numerical methods for solving nonlinear ODEs in physics?

One limitation is that the accuracy of the solution depends on the step size used in the numerical method. Smaller step sizes may provide more accurate results, but require more computational power. Additionally, numerical methods may struggle with stiff systems, where the solution changes rapidly over a small interval. In these cases, specialized techniques may be needed for accurate solutions.

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