johan_ekh
Aug13-08, 11:22 AM
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
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