As we can classify The 1D heat equation by using (b)^2- 4ac=0 . how to classify the same heat equation when in 3D? how to choose coefficient in 3D.
The steady state heat equation (also called the steady state heat conduction equation) is elliptic whether it is 2-D or 3-D. The unsteady heat conduction equation (in 1-D, 2-D or 3-D) is parabolic in time. This means that the heat conduction signal is felt immediately throughout the system- information travels at infinite speed. The mathematical classification of PDEs can be found in most upper division PDE textbooks. However, in the 1960's there was the discovery of ballistic phonon propagation in dielectric materials. To explain these effects it was suggested that the heat conduction equation be modified to include a term \tau\frac{\partial^2 T}{\partial t^2} - to yield the so-called telegraph equation, which makes the system hyperbolic, and became popular in irreversible thermodynamics. The telegraph equation has been used to study picosecond pulses in catalysis of fast reactions, explosions and collapse of supernovae.
To my knowledge, the heat equation is always parabolic, no matter if in 1D, 2D or 3D.
Edit: With 'heat equation' I meant of course the PDE, not the steady state ODE.
As Christoph said, the heat equation is always parabolic, and the dimension does not affect the type of PDE. There are several (equivalent) ways to determine the class of the PDE. One way is to seek characteristic curves, substitute them in the equation, and after some manipulation you obtain a form (B^2-4AC), which determines the class of the PDE and type of characteristics, just like in the 1D case. This is a general method and applies also to cases of variable coefficients, with just some more complex manipulation.
The steady state heat equation (also called the steady state heat conduction equation) is elliptic whether it is 2-D or 3-D. The unsteady heat conduction equation (in 1-D, 2-D or 3-D) is parabolic in time. This means that the heat conduction signal is felt immediately throughout the system- information travels at infinite speed. The mathematical classification of PDEs can be found in most upper division PDE textbooks. However, in the 1960's there was the discovery of ballistic phonon propagation in dielectric materials. To explain these effects it was suggested that the heat conduction equation be modified to include a term \tau\frac{\partial^2 T}{\partial t^2} - to yield the so-called telegraph equation, which makes the system hyperbolic, and became popular in irreversible thermodynamics. The telegraph equation has been used to study picosecond pulses in catalysis of fast reactions, explosions and collapse of supernovae.
Here is a link http://en.wikipedia.org/wiki/Partial_differential_equation
(It is after "If there are n independent variables ...)".
The classification is based on eigenvalues i.e., det(a-Lambda I)
For 2 independent variables, a11=A, a12=a21=B, and a22=C.
Then, characteristic eq.
det(a-lambda I)= (lambda)2- (A+C)lambda+AC-B2=0.
for e.g. AC-B2=0; parabolic, lambda=0
AC-B2>0, elliptic, lambda1, lambda2>0
(Steady state 3-D conduction eq. is elliptic with eigenvalues 1,1,2)
Thanks i got it Sumer Dribude,Brian G Higgins,Krsto Sbutega,Christoph Josef Backi
if conduction equation is steady state,it's Elliptic
else it's Parabolic
How we use the characteristic method to 1-D heat transfer equation also applied to 2-D and even 3-D. In fact, mathematically, this is how ppl implement the characteristic method to solve the "hyperbolic system" of the conservative equations in compressible flow problem.
Mathematically, characteristic method will tell you that the steady state heat transfer equation is "elliptic" and transient heat transfer equation is parabolic with the hyperbolic axis for time. That why we need to specify BC os temperature at all boundary, and IC at time zero. Unless you modify the PDE, this shall be true.
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