Should ρ, c, and k Be Included in the Heat Equation Solution?

In summary, the conversation revolved around solving an ODE problem and determining the accuracy of the solution. The solution was found to be u(x,t)= u_0 + \mbox{exp} \left( - \frac{m^2 \pi ^2 \kappa t}{L^2} \right) \cdot \sin(\frac{m \pi x}{L}), with the possibility of including the units ρ, c, and k.
  • #1
dirk_mec1
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  • #2
I thus only solved the ODE -problem.
 
  • #3
Show your working, and it will be easier to check whether you are correct or not.
 
  • #4
cristo said:
Show your working, and it will be easier to check whether you are correct or not.

OK, you got it:

[tex]\frac{ \partial{u} }{ \partial{t} } = \frac{df}{dt} \sin(\frac{m \pi x}{L}) [/tex]

[tex]\frac{ \partial{^2 u} }{ \partial{x^2} } = f(t) \left( \frac{-m^2 \pi ^2 }{L^2} \right) \sin(\frac{m \pi x}{L}) [/tex]
Therefore

[tex]
f(t) =\mbox{exp} \left( - \frac{m^2 \pi ^2 \kappa t}{L^2} \right)
[/tex]Conclusion:
[tex]
u(x,t)= u_0 + \mbox{exp} \left( - \frac{m^2 \pi ^2 \kappa t}{L^2} \right) \cdot \sin(\frac{m \pi x}{L})[/tex]
 
  • #5
Looks good, I'm just wondering one thing.

Should ρ, c, and k be in there somewhere, or do we assume units such that those are all =1?
 

Related to Should ρ, c, and k Be Included in the Heat Equation Solution?

1. What is the heat equation and why is it important in science?

The heat equation is a mathematical equation used to describe the flow of heat through a material. It is important in science because it helps us understand how heat is transferred and distributed in various systems, such as in engineering, meteorology, and physics.

2. What are the key concepts and variables involved in solving the heat equation?

The key concepts involved in solving the heat equation include temperature, thermal conductivity, and heat flux. The variables that are commonly used in the equation are time, distance, and temperature.

3. What methods are commonly used to solve the heat equation?

There are several methods that can be used to solve the heat equation, including the separation of variables method, the method of eigenfunction expansion, and the finite difference method. The choice of method depends on the specific problem and the available resources.

4. How do boundary conditions affect the solution of the heat equation?

The boundary conditions play a crucial role in determining the solution of the heat equation. They specify the values of temperature at the boundaries of the system and determine how heat is transferred in and out of the system. Different boundary conditions can result in different solutions.

5. Can the heat equation be applied in real-life situations?

Yes, the heat equation can be applied in various real-life situations, such as predicting the temperature distribution in a building, analyzing the cooling of a hot object, or understanding the thermal behavior of a material. It is a useful tool in many fields of science and engineering.

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