Is the 1D Heat Equation Parabolic?

  • Thread starter Saladsamurai
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In summary, Anderson used a method called "systems of PDEs" in order to solve a 1st order parabolic PDE.
  • #1
Saladsamurai
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Homework Statement



I am learning about PDE classification from a text on CFD (by Anderson). This section is not complete enough to be able to extend his example problems into more general cases. I read that to classify a system of PDEs as being parabolic, elliptic, or hyperbolic, I need to do some crazy stuff with Cramer's rule. However, the examples that he has shown are 1st order PDEs and like I said, are *systems* of PDEs. Now I have been asked to show that the 1D heat equation is parabolic and I am not sure how to apply what I have learned since a) it is 2nd order and b) it is only 1 eqaution:

[tex]\frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial x^2} \qquad(1)[/tex]

Homework Equations



Cramers rule

The Attempt at a Solution



I thought that I could form a system by forming the total differential of T(x, t)

[tex] dT = \frac{\partial T}{\partial x} \,dx + \frac{\partial T}{\partial t} \,dt \qquad(2)[/tex]However I am not sure if this is helpful. Any hints?
 
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  • #2
Saladsamurai said:

Homework Statement



I am learning about PDE classification from a text on CFD (by Anderson). This section is not complete enough to be able to extend his example problems into more general cases. I read that to classify a system of PDEs as being parabolic, elliptic, or hyperbolic, I need to do some crazy stuff with Cramer's rule. However, the examples that he has shown are 1st order PDEs and like I said, are *systems* of PDEs. Now I have been asked to show that the 1D heat equation is parabolic and I am not sure how to apply what I have learned since a) it is 2nd order and b) it is only 1 eqaution:

[tex]\frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial x^2} \qquad(1)[/tex]

Use the definition at:
http://en.wikipedia.org/wiki/Parabolic_partial_differential_equation
 
  • #3
LCKurtz said:

Hi LCKurtz :smile: I actually found that at wolfram as well. The problem is that he does not use that in the text (explicitly). I think what he has done is kind of derived that equation, but in a less general sense. I am just trying to learn a little more about where that equation at the wiki comes from. Let me show you in broad terms what Anderson did on his example:

The starting point:
Screenshot2012-01-07at114911PM.png


Putting in matrix form:

Screenshot2012-01-07at114924PM.png


He then set's det(A) = 0 in order to get a quadratic algebraic equation in (dy/dx). If the discriminant of this quadratic meets certain criteria, we can classify the PDE as parabolic. I am just having trouble putting everything into matrix form since I don't have a system of PDEs (unless I was right by taking the total differentials in post #1).
 
  • #4
I don't know if I can help you or not with this. I haven't done that much with PDE's. But I have one thought that occurs to me that you might or might not find useful. I'm thinking about how in ordinary DE's you can take a second order equation and write it as a first order linear system. For example, given the DE ##ay''+ by' + cy = 0## you make the substitution ##y_1=y, y_2= y', y_2' = y'' = \frac 1 a (-cy-by')=\frac 1 a(-cy_1-by_2)##

This gives you a first order linear system$$
\pmatrix{y_1\\y_2}'=\pmatrix{0&1\\-\frac c a &-\frac b a}\pmatrix{y_1\\y_2}$$

Maybe you can try something like that beginning with$$
U = T, V = U_x, V_x = U_{xx} = \frac 1 \alpha T_t=\frac 1 \alpha U_t$$Or maybe not, who knows?
 

Related to Is the 1D Heat Equation Parabolic?

1. What is a PDE?

A PDE, or partial differential equation, is a type of mathematical equation that involves multiple independent variables and their partial derivatives. It is commonly used to model physical phenomena in fields such as physics, engineering, and finance.

2. How can you determine if a PDE is parabolic?

A PDE is considered parabolic if it satisfies the following condition: the highest order derivative in the equation is of second order, and there is exactly one positive and one negative eigenvalue in the coefficient matrix of the second-order derivatives.

3. What does it mean to show that a PDE is parabolic?

Showcasing that a PDE is parabolic involves proving that it satisfies the criteria for a parabolic equation, as mentioned in the previous question. This is typically done through mathematical analysis and manipulation of the equation.

4. Why is it important to determine if a PDE is parabolic?

The classification of a PDE as parabolic is important because it helps with understanding the behavior of the equation and finding appropriate solutions. Parabolic PDEs are commonly used to model diffusion processes, heat conduction, and other phenomena that involve gradual changes over time.

5. Can a PDE be both parabolic and elliptic?

No, a PDE cannot be both parabolic and elliptic. These two types of equations have distinct properties and cannot be classified as both at the same time. A PDE can only be classified as one of the three types: hyperbolic, parabolic, or elliptic.

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