Solving the neutron diffusion equation.

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$\newcommand{\dd}{\mathrm{d}}$

Recall from the first part of this post a diffusion equation was derived to approximate the movement of neutrons in a nuclear reactor, which in two dimensions was,

\[\frac{\dd^2 \phi}{\dd r^2} + \frac{1}{r} \frac{\dd \phi}{\dd r} + B^{2}\phi = 0.\]

The main reason I was interested in solving this was due to the presence of non-constant coefficients, which was not a general form I had come across. Solving second order ODE’s of this form can be done using the Frobenius method, which produces an infinite series solution. We begin by defining an expression in the form which we want the solution to be:

\[\begin{equation*} \begin{aligned} \phi = \sum_{n=0}^{\infty}a_nr^{n+k} \end{aligned} \end{equation*}\]

The first and second derivatives are, therefore,

\[\begin{equation*} \begin{aligned} \frac{\text{d} \phi}{\text{d} r} = \sum_{n=0}^{\infty}(n + k)a_n r^{n+k-1} \end{aligned} \end{equation*}\] \[\begin{equation*} \begin{aligned} \frac{\text{d}^2 \phi}{\text{d} r^2} = \sum_{n=0}^{\infty}(n+k-1)(n + k)a_n r^{n+k-2} \end{aligned} \end{equation*}\]

Substituting the derivatives back into the ordinal expression gives

\[\begin{equation*} \begin{aligned} \sum_{n=0}^{\infty}(n+k-1)(n+k)a_nr^{n+k-2} + \frac{1}{r}\sum_{n=0}^{\infty}(n + k)a_n r^{n+k-1} + B^2\sum_{n=0}^{\infty}a_nr^{n+k} = 0 \\ \end{aligned} \end{equation*}\]

Ideally we would want each $r$ term to be raised to same power. Expanding out the first two terms of the $r^{n+k}$ expression, followed by some manipulation, gives:

\[\begin{equation*} \begin{aligned} \sum_{n=0}^{\infty}(n+k-1)(n+k)a_nr^{n+k-2} + \frac{1}{r}\sum_{n=0}^{\infty}(n + k)a_n r^{n+k-1} + B^2\sum_{n=2}^{\infty}a_nr^{n+k-2} = 0 \\ \end{aligned} \end{equation*}\] \[\begin{equation*} \begin{aligned} \sum_{n=2}^{\infty} (n+k-1)(n+k)a_{n}r^{n+k-2} + \frac{1}{r}\sum_{n=2}^{\infty} (n+k)a_{n}r^{n+k-1} + B^{2}\sum_{n=2}^{\infty} a_{n-2}r^{n+k-2} \\ + k(k-1)a_{0}r^{k-2} + k(k+1)a_{1}r^{k-1} + \frac{1}{r}\left( ka_{0}r^{k-1}+(k+1)a_{1}r^{k} \right) = 0 \nonumber \end{aligned} \end{equation*}\] \[\begin{equation*} \begin{aligned} \sum_{n=2}^{\infty} (n+k-1)(n+k)a_{n}r^{n+k-2} + \frac{1}{r}\sum_{n=2}^{\infty} (n+k)a_{n}r^{n+k-1} + B^{2}\sum_{n=2}^{\infty} a_{n-2}r^{n+k-2} \\ + k(k-1)a_{0}r^{k-2} + k(k-1)a_{1}r^{k-1} + ka_{0}r^{k-2} + (k+1)a_{1}r^{k-1} = 0 \end{aligned} \end{equation*}\]

We can equate the coefficients on the right side of the equation to zero. For the $r^{k-2}$ terms we get

\[\begin{equation*} \begin{aligned} k(k - 1)a_{0} + k a_{0} = 0 \\ a_{0}\left[ k(k - 1) + k \right] = 0 \\ \implies k = 0 \ \therefore \ a_{0} = 0 \end{aligned} \end{equation*}\]

Doing the same with the $r^{k-1}$ terms:

\[\begin{equation*} \begin{aligned} k(k + 1)a_{1} + (k + 1)a_{1} = 0 \\ a_{1}\left[ k(k + 1) + (k + 1) \right] = 0 \\ a_{1}(k + 1)(k + 1) = 0 \\ a_{1} = 0 \\ \end{aligned} \end{equation*}\]

The previous expression, therefore, becomes

\[\sum_{n=2}^{\infty} (n + k)(n + k - 1)a_{n}r^{n+i-2} + \sum_{n=2}^{\infty} (n + k)a_{n}r^{n+k-2} + B^{2}\sum_{n=2}^{\infty} a_{n-2}r^{n+k-2} = 0\] \[\sum_{n=2}^{\infty} \left\{ \left[ (n + k)(n + k - 1) + (n + k) \right] a_{n} + B^{2}a_{n-2}\right\} r^{n+k-2} = 0\]

We can see from this that the coefficients of $r^{n+k-2}$ must equate to 0, and so,

\[\begin{equation*} \begin{aligned} \left[ (n + k)(n + k - 1) + (n + k) \right] a_{n} + B^{2}a_{n-2} = 0 \end{aligned} \end{equation*}\] \[\begin{equation*} \begin{aligned} a_{n}(k) &= -\frac{B^{2}a_{n-2}}{(n + k)(n + k - 1) + (n + k)} \\ &= - \frac{B^{2}a_{n-2}}{(n + k)(n + k - 1 + 1)} \\ &= - \frac{B^{2}a_{n-2}}{(n + k)^{2}} \\ \end{aligned} \end{equation*}\]

If we now let $k = 0$,

\[a_{n}(0) = - \frac{B^{2}a_{n-2}}{n^{2}}\]

We know that $a_{1} = 0$, so we can say that

\[a_{1} = a_{3} = a_{5} = a_{2n+1} = 0\]

For $k = 0$, let $n = 2m$:

\[a_{2m}(0 = - \frac{B^{2}a_{2m-2}}{(2m^{2}}\]

We can see that, for even values of $m$,

\[a_{2} = - \frac{B^{2}a_{0}}{2^{2}}\] \[a_{4} = \frac{-B^{2}a_{2}}{2^{2} \cdot 2^{2}} = \frac{-B^{2}a_{0}}{2^{2}} \cdot \frac{-B^{2}}{2^{2}\cdot 2^{2}} = \frac{B^{4}a_{0}}{2^{4}\cdot 2^{2}} = \frac{B^{4}a_{0}}{2^{4}(2^{2} \cdot 1^{2})}\] \[a_{6} = \frac{-B^{2}a_{4}}{2^{2}3^{2}} = \frac{-B^{2}}{2^{2}3^{2}} \cdot \frac{B^{4}a_{0}}{2^{4}(2 \cdot 1)^{2}} = \frac{-B^{6}a_{0}}{2^{6}(3 \cdot 2 \cdot 1)^{2}}\] \[a_{8} = \frac{-B^{2}a_{6}}{2^{2}4^{2}} = \frac{-B^{2}}{2^{2}4^{2}} \cdot \frac{-B^{6}a_{0}}{2^{6}(3 \cdot 2 \cdot 1)^{2}} = \frac{B^{8}a_{0}}{2^{8}(4 \cdot 3 \cdot 2 \cdot 1)^{2}}\]

And so, in general we can say that

\[a_{2m} = \frac{(-1)^{m}B^{2m}a_{0}}{2^{2m}(m!)^{2}}\]

We can put this back into out original expression for $\phi$ to get a solution

\[\phi_{1} = a_{0} \sum_{n=0}^{\infty} \frac{(-1)^{m}B^{2m}}{2^{2m}(m!)^{2}}r^{2m}\]

This is known as the Bessel function of the first kind (i.e. for $k = 0$ ).

\[\phi_{1} = \mathcal{J}_{0} (Br)\]