Exploring Differential Equations in Astrophysics: A Project Idea

[SYoungblood]

Homework Statement

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Hello all, I am looking more for an idea than an actual problem to solve. (That comes later.)

Homework Equations

I am starting a Differential Equations class, and I am looking for an idea for a project. We are to complete an application assignment in which we have to set up a differential equation with a real-world application, solve it, and interpret the meaning of the solution.

The Attempt at a Solution

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I wanted to do something related to astrophysics and cosmology, but after looking through an intro to astronomy textbook, I really didn't get very far. Can someone offer a suggestion?

Thank you,

SY

 

[DEvens]

Maybe the text for your course is a good place to look for examples?

Are you required to get an exact solution? That is, an algebraic solution in closed form? Or is it supposed to be a numerical solution?

There are lots of differential equations that you could solve. But it depends on how much background you are prepared to learn, and what kind of differential equations you are prepared to solve. For example:
- The Oppenheimer-Snyder dust-cloud model of stellar collapse in general relativity has, very surprisingly, an exact solution. But it's kind of a leap. I'm guessing you are probably in second year undergrad.
- The Schrodinger equation for the Hydrogen atom in non-relativistic quantum mechanics has an exact solution. Maybe that's ok for a second year student.
- Maybe closer to home is the diffusion equation. In 1-dimension you can fairly easily find heat balance systems with exact solutions. If you do some jim-jam on the heat capacity or the thermal diffusivity you can get exact solutions in 2-d or 3-d as well.

That last is possibly an interesting idea. The method is called "solution generation." It is very valuable in the context of validating computer programs that numerically solve a differential equation. The problem is, you want the computer program to work on systems for which you don't have an exact solution. So one method is to create a system that is close to the real system, but that does have an exact solution. For example: Suppose you were doing a heat balance problem. You have some device that generates heat, say be electrical heating. And you want to know the temperature at each point in the device. So you have to solve the heat equation.

https://en.wikipedia.org/wiki/Diffusion_equation
https://en.wikipedia.org/wiki/Heat_equation

But for your system, the thermal diffusivity is a complicated function that makes solving the equation very difficult. What you do is, you pick a different function, one that is not drastically different from the real one. But that has properties that make it possible to produce an exact solution. So then you put this different diffusivity into your computer program, and compare the results it gives to the exact solution.

So maybe you can find a real-world system, and pick some idealized approximate properties that gives you an exact solution.

 

[pasmith]

Homework Statement

[/B]
Hello all, I am looking more for an idea than an actual problem to solve. (That comes later.)

Homework Equations

I am starting a Differential Equations class, and I am looking for an idea for a project. We are to complete an application assignment in which we have to set up a differential equation with a real-world application, solve it, and interpret the meaning of the solution.

Questions:

(1) When you say "Differential Equations", do you mean ordinary differential equations (one independent variable) or partial differential equations (two or more independent variables)?

(2) When you say "solve", do you mean "solve analytically" or "solve numerically"?

 

[SYoungblood]

Hello all,

Thank you for you help. I found some non-astrophysics examples in my text and another book in the library. It seems the vogue challenge is to use the rate of cooling to find the time of death of a person whose body temperature is measured at 88 F or a similar temperature, and to use that info to determine the time of death. I think I will run with something like that.

SY