Heat Equation with moving source

In summary, the conversation is about solving a heat PDE in a 1D domain for a moving source. The solution has already been found, but it is stationary in a reference frame moving with the source. The person is interested in a source starting to move at time zero and has attached their approach to the problem. There is a discussion about using LaTeX to write equations and the conversation ends with gratitude and a hope for future help.
  • #1
muzialis
166
1
Hello there,

I want to solve the heat PDE in a 1D domain for a source moving at constant speed. The problem has been solved already, the solution being stationary in a reference frame moving with the source.

This is highly un-intuitive, and I suppose the result originate from the fact the source is assumed to have been moving (exisiting) for an infinite amount of time.

Instead, I am interested in a source starting to move at time zero.

I attacked the problem as reported in the attachment, I wonder if the solution is correct.

Many thanks as usual for your help. I apologize for not writing the equation directly here, but I have not figured out yet how to use Latex in the post, I am getting there though.
 

Attachments

  • Green solution for the Heat equation over an infinite 1D domain and initial condition G.pdf
    41.7 KB · Views: 257
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  • #2
Looks fine.

You can use LaTeX with the [itex] (inline) and [tex] (new line) commands (or shorter: ##equation## and $$equation$$)
 
  • #3
Many thanks for looking at it and the hint.
 
  • #4
Hi muzialis !

Sorry, I have not the time to dactylography the whole solution.
If there is no mistake, the key points are in attachment :
 

Attachments

  • Integral.JPG
    Integral.JPG
    36.3 KB · Views: 561
  • #5
JJaquelin,

what to say, you reda my mind! Many thanks for this, hope to be able to help back some time

Cheers
 

1. What is the "Heat Equation with moving source"?

The "Heat Equation with moving source" is a mathematical model that describes the flow of heat in a system where there is a source of heat that is also moving. It is a partial differential equation that is commonly used to study heat transfer in various physical systems.

2. How is the "Heat Equation with moving source" different from the traditional heat equation?

The traditional heat equation only considers a stationary source of heat, while the "Heat Equation with moving source" takes into account the movement of the heat source. This allows for a more accurate representation of real-world systems where heat sources are not always stationary.

3. What types of problems can be solved using the "Heat Equation with moving source"?

The "Heat Equation with moving source" can be used to solve a wide range of heat transfer problems, such as predicting the temperature distribution in a moving object or analyzing the cooling of a heated material during transportation.

4. What are the key variables involved in the "Heat Equation with moving source"?

The key variables in the "Heat Equation with moving source" include the temperature distribution, the heat source function, the thermal diffusivity of the material, and the velocity of the heat source. These variables are used to solve the partial differential equation and determine the temperature at different points in the system.

5. What are some applications of the "Heat Equation with moving source" in real life?

The "Heat Equation with moving source" has many practical applications, such as predicting the temperature profile in a moving vehicle, analyzing the cooling of molten metal during casting, and studying the heat transfer in biological systems such as blood flow in the human body. It is also used in engineering and design to optimize heat management in various industrial processes.

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