How Does Diffusion Affect the Decay Chain of Uranium?

[Avarus]

Ok guys, I've got a problem which I'd like to solve in an elegant way, but I don't how to solve it (if it can be solved analytically).

I'm considering the decay chain of Uranium. So Uranium decays to Thorium, which decays to Protactinium, and so on. I know how to solve such a linear system, since the only variable is time, with the decay constants being constant (duh). Example:

$\frac{dU}{dt} = -{k}_{1}U$

$\frac{dTh}{dt} = -{k}_{2}Th + {λ}_{1}U$

$\frac{dPa}{dt} = -{k}_{3}Pa + {λ}_{2}Th$

in which the k's are decay constants. Or in vector notation:

${N}^{*} = AN$

in which ${N}^{*}$ is the time derivative of species ${N}_{i}$, A is a matrix consisting of k's. This solves to:

$N(t) = S\Lambda{S}^{-1}N(0)$

in which S is the matrix of eigenvectors and $\Lambda$ is a diagonal matrix with ${e}^{{λ}_{i}t}$ as the diagonal entries.

But now I want to solve it when diffusion into or out of the system is allowed. So my initial problem becomes something like this:

$\frac{\partial U}{\partial t} = -{k}_{1}U - \nabla*(-D \nabla U)$

$\frac{\partial Th}{\partial t} = -{k}_{2}Th + {k}_{1}U - \nabla*(-D\nabla Th)$

and so on...

with D being a tensor consisting of diffusion constants for that species and * denoting the dot product (couldn't find a nicer dot).

Now the concentration of species depends on both t and x, which are independent of each other. I was wondering if it even is possible to solve this analytically and if so, how? I imagine the solution would have a position and a time dependent component, something like:

$N(t) = PQ{P}^{-1}x + ST{S}^{-1}N(0)$

in which P is the eigenvector matrix of some position dependent matrix, Q the solution matrix in terms of position and x the initial position, and S is the eigenvector matrix of the time dependent matrix and T the solution matrix in terms of time and N(0) the initial concentrations.

Please note that the diffusion flux depends on the concentration gradient in the system, which in turn depends on time, since species decay.

Could anyone please help me with the above?

 

[anathema]

Thank you for sharing your problem with us. It is certainly an interesting and challenging one. The good news is that it is indeed possible to solve this system analytically, although it may require some advanced mathematical techniques. Let me walk you through the steps that you can take to solve this problem.

Firstly, we need to rewrite the system of equations in a more general form. Instead of using specific species such as Uranium, Thorium, and Protactinium, let's use a generic notation, such as {N}_{i}, for the concentrations of all the species in the decay chain. Similarly, let's use {k}_{i} for the decay constants and D_{ij} for the diffusion constants.

\frac{\partial {N}_{i}}{\partial t} = -{k}_{i}{N}_{i} - \nabla*(-D_{ij} \nabla {N}_{i})

Next, we can rewrite this system in vector form, similar to the way you have done it in your first example. Let's call this vector {N} and the matrix of decay and diffusion constants A and B, respectively.

\frac{\partial {N}}{\partial t} = AN - \nabla*(-BN)

Now, to solve this system, we need to find the eigenvalues and eigenvectors of the matrix A. Let's call these eigenvalues {\lambda}_{i} and eigenvectors {v}_{i}. We can then write the solution in terms of these eigenvalues and eigenvectors as follows:

{N}(t) = \sum_{i} {c}_{i}{e}^{{\lambda}_{i}t}{v}_{i}

where {c}_{i} are constants that can be determined from the initial conditions.

Now, to incorporate the diffusion term into the solution, we need to use a technique called separation of variables. This involves separating the time and position dependent components of the solution. Let's assume that the solution can be written as a product of a time-dependent term T(t) and a position-dependent term X(x).

{N}(t,x) = T(t)X(x)

Substituting this into our original equation, we get:

T'(t)X(x) = AT(t)X(x) - \nabla*(-BT(t)X(x))

Now, we can divide both sides by T(t)X(x) and