Analytical solution for heat equation with simple boundary conditions

[uqjzhang]

I am trying to solve the following heat equation ODE:
d^2T/dr^2+1/r*dT/dr=0 (steady state) or
dT/dt=d^2T/dr^2+1/r*dT/dr (transient state)

The problem is simple: a ring with r1<r<r2, T(r1)=T1, T(r2)=T2.
I have searched the analytical solution for this kind of ODEs in polar coordinate systems, found a lot of solutions for more difficult problems

Can anybody help me with this one please?

 

[JJacquelin]

The stady state problem :
d^2T/dr^2+1/r*dT/dr=0 is a simple linear ODE. With boundary conditions T(r1)=T1, T(r2)=T2, did you find the solution ?
The transiant problem :
dT/dt=d^2T/dr^2+1/r*dT/dr is a linear PDE. Boundary conditions T(r1)=T1, T(r2)=T2 are not enough. A unique solution requires more conditions related with time (for exemple what is the state at time 0, and/or how the T(r1), T(r2) are estabished as a function of time, or...)
Various methods are possible, with more or less difficulties depending on the initial conditions.
If the initial conditions are not stated, one can find the general solutions on the form of a series of particular solutions T(r,t)=Sum(Ak*Fk(r)*Gk(t)) where Ak are constants, Fk are Bessel functions or r and Gk is exponential functions of t. Other representations are possible.
So, first you have to clarify the initial conditions.

 

[JJacquelin]

I found it difficult to solve the equation?
Anyone of you can help me?

Steady state solution in attachment :

 

[JJacquelin]

I found it difficult to solve the equation?
Anyone of you can help me?

Transient solutions in attachment :

 

[blue_raver22]

I can provide some insight into the analytical solution for this heat equation with simple boundary conditions. This type of equation is known as the radial heat equation and it describes the temperature distribution in a ring with a constant heat source at its center. The solution to this equation depends on whether the system is in a steady state or a transient state.

In the steady state, the temperature distribution does not change with time and is only a function of the radial distance from the center of the ring. This means that the term dT/dt in the heat equation becomes zero, and we are left with the following equation:

d^2T/dr^2 + 1/r * dT/dr = 0

To solve this equation, we can use the method of separation of variables. Let us assume that the solution takes the form of T(r) = R(r)/r. Substituting this into the equation, we get:

d^2R/dr^2 + R = 0

The solution to this ordinary differential equation is R(r) = A*sin(r) + B*cos(r), where A and B are constants. Therefore, the solution to the steady state heat equation is:

T(r) = (A*sin(r) + B*cos(r))/r

To determine the values of A and B, we can use the boundary conditions. At r = r1, T(r1) = T1, and at r = r2, T(r2) = T2. This gives us the following equations:

T1 = (A*sin(r1) + B*cos(r1))/r1
T2 = (A*sin(r2) + B*cos(r2))/r2

Solving for A and B, we get:

A = (T2*r1*sin(r1) - T1*r2*sin(r2))/(r1*sin(r1) - r2*sin(r2))
B = (T1*r2*cos(r2) - T2*r1*cos(r1))/(r1*sin(r1) - r2*sin(r2))

Therefore, the final solution for the steady state heat equation is:

T(r) = [(T2*r1*sin(r1) - T1*r2*sin(r2))*sin(r) + (T1*r2*cos(r2) - T2*r1*cos(r1))*cos(r)]/[(r1*sin(r1) - r2*sin(r2))*r]