Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Heat equation with source and Neumann B.C.

  1. Aug 13, 2008 #1
    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

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

    [tex]\frac{\partial u}{\partial n} +h(u-u_c)=0[/tex]


    where [tex]q[/tex] is a constant heat source.

    I have developed the temperature and the source according to

    [tex]u(r,t)=\sum_{n=1}^{\infty} \tilde{R_n}(t) R_n(r)[/tex]

    [tex]q=\sum_{n=1}^{\infty} \tilde{q_n}(t) R_n(r)[/tex]

    where [tex]\tilde{R_n}(t)[/tex] and [tex]\tilde{q_n}(t)[/tex] are time dependent
    coefficients and [tex]{R_n}(r)[/tex] are orthogonal functions.

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

    [tex]R_n(r)=\tilde{c_n} J_0(\lambda_n r)[/tex]

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

    The equation involving [tex]t[/tex] looks like

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

    where [tex]a[/tex] and [tex]\lambda^*[/tex] are constants.

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

    Thus, the equation is now

    [tex]\tilde{R_n}^{'}(t) + \lambda_n^2 a \tilde{R_n}(t) = \frac{a}{\lambda^*} \tilde{q_n}^*[/tex]

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

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

    where [tex]b_n[/tex] are coefficients yet to be determined.

    The solution is thus on the form

    [tex]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)[/tex]

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

    [tex]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[/tex]

    where [tex]u_c[/tex] is the ambient temperature. This helps the
    calculation of [tex]\lambda_n[/tex] since it cancels the [tex]u_c[/tex] from the

    Thus, what remains is to determine the coefficients [tex]b_n[/tex] and
    [tex]\tilde{c_n}[/tex] , and this is where I have problem. When no source
    is present, there exist no [tex]b_n[/tex] and [tex]\tilde{c_n}[/tex] 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 [tex]b_n[/tex] and [tex]\tilde{c_n}[/tex] ? 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,
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted