Hello, steady conduction is elliptic, and unsteady conduction is parabolic. Since D=b2-4ac in staedy conduction b=0, a=some value and c=o so it should be parabolic. as in unsteady conduction b=0 a= some value and c=0. thanks

Your example is a degenerate case - you're applying a result for second order pdes in two independent variables to an ode in one - the condition D = 0 tells you that the equation u_t = u_xx is parabolic. To understand the classification it's better to look at

u_t = grad^2 u = u_xx + u_yy (*)

which is a parabolic pde in three independent variables (see Courant and Hilbert vol II ch.3). Now in a steady state u_t = 0 and we are left with Laplaces equation in two independent variables which has D<0 and is thus elliptic. (It's the change in the number of independent variables that's important here!)

Its mean that if there is two independent variables, eq is elliptic and three independent variable eq is parabolic so what condition for hyperbolic. Also means that D=b2-4ac method is not applicable in PDEs, it is just for ODEs. If eq of three independent variables are parabolic then steady state heat conduction in three dimension should be parabolic but it is an elliptic equation, WHY??????? Thanks

(1) The classification of an ode as elliptic, parabolic or hyperbolic is meaningless.

(2) When you classify a second order pde in two independet variables you have the simple condition

D < 0 Elliptic => no real characteristics

D = 0 Parabolic => one real characteristic (usually time)

D > 0 Hyperbolic => two real characteristics

(for higher order or more independent variables the classification is more complex than this and does not reduce to a simple condition on D).

Now in applying the above to the heat equation with TWO independent variables ( t and x ) you obtain D=0 which means your pde is parabolic. If you set d/dt to zero for the steady state you CANNOT use the above classification because the number of independent variables has been reduced by 1.

This was the point of my example! If you take the heat equation in THREE independent variables x,y,t then the classification tells you it is parabolic. In the steady state there is no t dependence and you have Laplaces equation in TWO independent variables which is elliptic; i.e. for the unsteady problem you follow the classification procedure for THREE independent variables while for the steady problem you use the one for 2 - the usteady problem has one real characteristic and two complex (conjugate) ones and in going to a steady state you remove the real one leaving only the complex pair.

what is the classifiction of 1D heat conduction eq. why the heat equation in THREE independent variables x,y,t is parabolic, although for this equation a=some value b=0 and c= some value so D<0 therefor it should be elliptic not parabolic???????

your using the WRONG formula to classify the heat equation in three independent variables! - read my note carefully.

Think of it this way D is the disciminant of a quadratic equation implying at most two roots. In three independent variables you have a cubic which has three roots and the formula for soving a general quadratic tells you nothing about the solution of the general cubic.

As I've already pointed out the 1D heat equation (u_t= u_xx) is parabolic because D=0 (a=1,b=0,c=0). In a steady state (u_t=0) the classification is meaningless BUT two point boundary value problems for odes behave like elliptic equations. (This analogy should not be stretched too far though!)

I will study on sunday and then ask u

I have understand classificationof 2 independent variables, initialy I was mixind 2D flow(unsteady) in 2 independent variable case but now I have understand that both are different and the latter one is the case of 3 independent variables. Now I want to know what is the classification method for 3 and 4 independent variables i.e 3D steady flow, 2D unsteady flow and 3D unsteady flow Thanks

Technically the classification method for higher order problems follows exactly the derivation used for second order problems; i.e. look for lines along which the pde degenerates to an ode. You need to look at one of the books

Methods om mathematical physics (vol 2) by Courant and Hilbert,

Partial differential equations by Garabedian,

there's also a description in

Linear and nonlear waves by Whitham.

The Steady (2d or 3d) fluid equations are elliptic

The unsteady Euler equations are hyperbolic

The unsteady Navier-Stokes equations are parabolic