Understanding the One-Dimensional Heat Equation

[shoogar]

why does the one-dimensional heat equation for temperature distribution contain a second derivative of the spatial variable?

 

[Integral]

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.

 

[Edgardo]

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"

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

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.

 

[chwala]

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