Answer to Question #205034 in Differential Equations for Laiba Shahbaz

Question

Solution of second dimensional heat equation in cylindrical cordinates with dririchlet boundary conditions?

Answer 1

The boundary condition T(r_b, z, t)=T_w  is not consistent with T(r,0,t) = T₀ --- unless T_w =  T₀, in which case the problem is trivial. Otherwise there will be a permanent diverging heat-flow in the vicinity of the point (r_b, 0). Although it is possible to analyse that to considerable extent, it would (without further, well-explained motivations) be the mathematical equivalent of self-flogging.

Let me add to the above remarks, that the inconsistency is not substantial. At least for the stationary solution. It can be compared to the following problem of seeking a harmonic function in a rectangle, say square [0,1]² with 

$T(x,1)=0\hspace{0.1cm} [for 0<x<1], T(1,y)=f(y) \hspace{0.1cm}[for 0<y<1]$

with the insulated other sides. Then the (stationary) solution is the series, whose general term is equal to 

an

$\sin\frac{(2n+1)\pi y} {2}\cosh\frac{(2n+1)\pi x}{2}, (x,y) in [0,1]^2$

By orthogonality of the sine functions, using the Fourier coefficients formula we have

an

$=2 \int_0^1 f(y) \sin\frac{(2n+1)\pi y}{ 2} dy \cosh\frac{(2n+1)\pi}{ 1 / 2}$

This will provide a solution satisfying the boundary conditions almost everywhere, for the family of piecewise monotone functions f , which contains the constant functions.

If we want to find the exact or approximate solution for a given time depending 1D physical problem whose mathematical model is a nonlinear second order PDE in terms of the solution function f(x,t) (t-is the time and x is the spatial variable) are required given

initial conditions (given function f(x, t=0) and

given boundary conditions ( f(x=a,t)=fa(t) and ( f(x=b,t)=fb(t) are given time functions on the interval boundaries x=a and x=b)

These given conditions must be attached to the PDE for obtain the unique solution of the considered physical problem.

Mathematically the PDE with the attached conditions is called Initial-Boundary-Value-Problem.

If initial conditions only are considered it's about Initial Value Problem (e.g. the problem which you refer in the cited paper). In this case on obtain exact or approximate solutions functions f(x,t) of the PDE for different initial conditions, defined through the given function f(x,t=0). This solution functions of the PDE don’t respect given boundary conditions. But, they naturally can be useful for solving given Initial- Boundary -Value- Problems.