fronts.inverse

fronts.inverse(o, samples)

Extract D from samples of a solution.

Given a function \(\theta\) of r and t, and scalars \(\theta_i\), \(\theta_b\) and \(o_b\), finds a positive function D of the values of \(\theta\) such that:

\[\begin{split}\begin{cases} \dfrac{\partial\theta}{\partial t} = \dfrac{\partial}{\partial r}\left(D(\theta)\dfrac{\partial\theta} {\partial r}\right) & r>r_b(t),t>0\\ \theta(r, 0) = \theta_i & r>0 \\ \theta(r_b(t), t) = \theta_b & t>0 \\ r_b(t) = o_b\sqrt t \end{cases}\end{split}\]

\(\theta\) is taken as its values on a discrete set of points expressed in terms of the Boltzmann variable. Problems in radial coordinates are not supported.

Parameters:

o (numpy.array_like, shape (n,)) – Points where \(\theta\) is known, expressed in terms of the Boltzmann variable. Must be strictly increasing.
samples (numpy.array_like, shape (n,)) – Values of \(\theta\) at o. Must be monotonic (either non-increasing or non-decreasing) and samples[-1] must be the initial value \(\theta_i\).

Returns:

D –

Function to evaluate \(D\) and its derivatives:

D(theta) evaluates and returns \(D\) at theta

D(theta, 1) returns both the value of \(D\) and its first derivative at theta

D(theta, 2) returns the value of \(D\), its first derivative, and its second derivative at theta

In all cases, the argument theta may be a single float or a NumPy array.

\(D\) is guaranteed to be continuous; however, its derivatives are not.

Return type:

callable