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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
    \right][/tex]

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

    [tex]u(r,\varphi,t=0)=u_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
    B.C.

    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,
    Johan
     
  2. jcsd
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