Understanding the One-Dimensional Heat Equation

In summary, the one-dimensional heat equation for temperature distribution contains a second derivative of the spatial variable because it is derived from the divergence of the gradient of temperature, which is equivalent to the Laplace operator in one dimension. To understand the article, it is helpful to have knowledge about the divergence theorem, specific heat capacity, and gradient. These concepts are also explained in various online resources such as lectures and videos. The form of the wave equation, specifically the second derivative of time and the second derivative of space, leads to the d'Alembert's equation.
  • #1
shoogar
1
0
why does the one-dimensional heat equation for temperature distribution contain a second derivative of the spatial variable?
 
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  • #2
Because it wouldn't be the heat equation if it didn't?

Perhaps you need to refer to a text on PDEs for a derivation.
 
  • #3
Have a look at these derviations ("banach.millersville.edu/~bob/math467/HeatEquation3D.pdf"[/URL]. The divergence of grad(u) is laplace(u), or in one dimension u_xx.
To understand the article it is helpful to know about the following topics:

1) Divergence theorem:
- Examples for the divergence theorem (also known as Gauss theorem) can be found here:
http://math.bard.edu/~mbelk/math601/GaussExamples.pdf"
[PLAIN]http://tutorial.math.lamar.edu/Classes/CalcIII/DivergenceTheorem.aspx" [Broken]

2) Specific heat capacity:
- http://www.taftan.com/thermodynamics/CP.HTM" [Broken]

3) Gradient:
- Lecture by Edward Frenkel (Math Berkeley)
At 3:56 he gives an intuitive explanation of the gradient.
http://www.youtube.com/watch?v=7cPcutRLLXE"
- Videos by Salman Khan:
http://www.youtube.com/watch?v=U7HQ_G_N6vo"
http://www.youtube.com/watch?v=OB8b8aDGLgE"
In the second video he shows the gradient of a scalar field T(x,y,z) defined in 3 dimensional space.
 
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  • #4
kindly refer to the pde notes online ....its like asking why does the wave equation have u{tt} - c^2 u{xx}=2t=f(x,t) have this form that leads to d alemberts equation
 
  • #5


The one-dimensional heat equation for temperature distribution contains a second derivative of the spatial variable because it takes into account the rate of change of temperature with respect to both time and space. This second derivative allows us to account for the curvature of the temperature distribution and how it changes over space. In other words, it takes into consideration the change in temperature at different points along the one-dimensional space, rather than just the overall change in temperature over time. This is important in accurately predicting the behavior of heat transfer in a one-dimensional system and is a fundamental aspect of the heat equation.
 

1. What is the one-dimensional heat equation?

The one-dimensional heat equation is a mathematical model used to describe the flow of heat in a one-dimensional system such as a rod or wire. It takes into account factors such as the initial temperature, thermal conductivity, and boundary conditions to determine how the temperature changes over time.

2. How is the one-dimensional heat equation solved?

The one-dimensional heat equation can be solved using various methods such as separation of variables, Fourier series, or finite difference methods. These methods involve breaking down the equation into simpler parts and solving them individually to find the solution for the entire system.

3. What are the applications of the one-dimensional heat equation?

The one-dimensional heat equation has numerous applications in fields such as engineering, physics, and meteorology. It is used to predict the temperature distribution in various systems, design heating and cooling systems, and understand heat transfer in different materials.

4. What are the limitations of the one-dimensional heat equation?

The one-dimensional heat equation is a simplification of the more complex three-dimensional heat equation and therefore has its limitations. It assumes that heat transfer only occurs in one direction and does not take into account factors such as convection and radiation. It is also limited to homogeneous materials with constant thermal properties.

5. How does the one-dimensional heat equation relate to the laws of thermodynamics?

The one-dimensional heat equation is based on the laws of thermodynamics, specifically the law of conservation of energy. It describes the transfer of thermal energy from higher to lower temperatures, which follows the second law of thermodynamics. It can also be used to calculate the change in entropy in a system, in accordance with the third law of thermodynamics.

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