# Heat equation with source and Neumann B.C.

1. Aug 13, 2008

### johan_ekh

Hi all,
I'm trying to analytically solve the heat equation with a heat
source and Neumann B.C. The source term is creating some problems
for me as I cannot determine the coefficients in the series that
builds up the solution. If someone could could help me or at
least point me in the right direction I would be very thankful.

The geometry is a cylindrical disk with radius R. Thus, the
problem is defined as

$$\frac{\partial u}{\partial t} = a \left[ \frac{\partial^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} +q \right]$$

$$\frac{\partial u}{\partial n} +h(u-u_c)=0$$

$$u(r,\varphi,t=0)=u_0$$

where $$q$$ is a constant heat source.

I have developed the temperature and the source according to

$$u(r,t)=\sum_{n=1}^{\infty} \tilde{R_n}(t) R_n(r)$$

$$q=\sum_{n=1}^{\infty} \tilde{q_n}(t) R_n(r)$$

where $$\tilde{R_n}(t)$$ and $$\tilde{q_n}(t)$$ are time dependent
coefficients and $${R_n}(r)$$ are orthogonal functions.

Substituting these series into the equation separates the
variables $$r$$ and $$t$$ and results in two differential equations, one
involving $$r$$ and one involving $$t$$ . The equation involving $$r$$ is
Bessel's equation which in our case have the solution

$$R_n(r)=\tilde{c_n} J_0(\lambda_n r)$$

where $$\tilde{c_n}$$ are coefficients to be determined by the
I.C., $$J_0$$ is a Bessel function of the first type and order 0
and $$\lambda_n$$ are the eigenvalues to be determined by the B.C.

The equation involving $$t$$ looks like

$$\tilde{R_n}^{'}(t) - \frac{a}{\lambda^*} \tilde{q_n}(t) + \lambda_n^2 a \tilde{R_n}(t) = 0$$

where $$a$$ and $$\lambda^*$$ are constants.

The constant source $$q$$ is known which means that the
coefficients $$\tilde{q_n}(t)$$ can be determined by using the
properties of the orthogonal Bessel functions. I will denote the
known coefficients $$\tilde{q_n}^*$$ from now on.

Thus, the equation is now

$$\tilde{R_n}^{'}(t) + \lambda_n^2 a \tilde{R_n}(t) = \frac{a}{\lambda^*} \tilde{q_n}^*$$

and a solution can be obtained by the method of integrating
factor and looks like

$$\tilde{R_n}(t) = \frac{\tilde{q_n}^*}{\lambda^* \lambda_n^2} + b_n e^{-a \lambda_n^2 t}$$

where $$b_n$$ are coefficients yet to be determined.

The solution is thus on the form

$$u(r,t) = \sum_{n=1}^{\infty} \left[ \frac{\tilde{q_n}^*}{\lambda^* \lambda_n^2} + b_n e^{-a \lambda_n^2 t} \right] \tilde{c_n} J_0(\lambda_n r)$$

where $$\lambda_n$$ are determined from the B.C. (not shown
here). Since the coefficients $$b_n$$ and $$\tilde{c_n}$$ are not yet
determined we can write the solution on the form

$$u(r,t) = \sum_{n=1}^{\infty} \left[ \frac{\tilde{q_n}^*}{\lambda^* \lambda_n^2} + b_n e^{-a \lambda_n^2 t} \right] \tilde{c_n} J_0(\lambda_n r) + u_c$$

where $$u_c$$ is the ambient temperature. This helps the
calculation of $$\lambda_n$$ since it cancels the $$u_c$$ from the
B.C.

Thus, what remains is to determine the coefficients $$b_n$$ and
$$\tilde{c_n}$$ , and this is where I have problem. When no source
is present, there exist no $$b_n$$ and $$\tilde{c_n}$$ can be
determined by substituting the solution into the I.C. and
utilizing the orthogonality of the Bessel functions.

But how can I determine both $$b_n$$ and $$\tilde{c_n}$$ ? It seems
that I need an extra equation since I have two unknowns instead
of one. What am I missing?

Or is there some fundamental problem with my approach?

Any help is appreciated.

Best regards,
Johan