fronts.Solution¶
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class
fronts.Solution(sol, ob, oi, D)¶ Continuous solution to a problem.
A subclass of BaseSolution, its methods describe a continuous solution to a problem of finding a function S of r and t such that:
\[\dfrac{\partial S}{\partial t} = \nabla\cdot\left[D\left(S\right) \dfrac{\partial S}{\partial r}\mathbf{\hat{r}}\right]\]with r bounded at \(r_b(t)=o_b\sqrt t\) on the left and unbounded to the right. For \(r<r_b(t)\), the methods will evaluate to NaNs.
- Parameters
sol (callable) – Solution to the corresponding ODE obtained with ode. For any o in the closed interval [ob, oi],
sol(o)[0]is the value of S at o, andsol(o)[1]is the value of the derivative \(dS/do\) at o. sol will only be evaluated in this interval.ob (float) – \(o_b\), which determines the behavior of the boundary.
oi (float) – Value of the Boltzmann variable at which the solution can be considered to be equal to the initial condition. Must be \(\geq o_b\).
D (callable) – D used to obtain sol. Must be the same function that was passed to ode.
See also
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__init__(sol, ob, oi, D)¶ Initialize self. See help(type(self)) for accurate signature.
Methods
S([r, t, o])S, the unknown function.
__init__(sol, ob, oi, D)Initialize self.
dS_do([r, t, o])\(dS/do\), derivative of S with respect to the Boltzmann variable.
dS_dr(r, t)\(\partial S/\partial r\), spatial derivative of S.
dS_dt(r, t)\(\partial S/\partial t\), time derivative of S.
flux(r, t)Diffusive flux of S.
rb(t)\(r_b\), the location of the boundary.
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S(r=None, t=None, o=None)¶ S, the unknown function.
May be called either with parameters r and t, or with just o.
- Parameters
r (float or numpy.ndarray, optional) – Location(s). If a numpy.ndarray, it must have a shape broadcastable with t. If this parameter is used, you must also pass t and cannot pass o.
t (float or numpy.ndarray, optional) – Time(s). If a numpy.ndarray, it must have a shape broadcastable ith r. Values must be positive. If this parameter is used, you must also pass r and cannot pass o.
o (float or numpy.ndarray, optional) – Value(s) of the Boltzmann variable. If this parameter is used, you cannot pass r or t.
- Returns
S – If o is passed, the return is of the same type and shape as o. Otherwise, return is a float if both r and t are floats, or a numpy.ndarray of the shape that results from broadcasting r and t.
- Return type
float or numpy.ndarray
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dS_do(r=None, t=None, o=None)¶ \(dS/do\), derivative of S with respect to the Boltzmann variable.
May be called either with parameters r and t, or with just o.
- Parameters
r (float or numpy.ndarray, optional) – Location(s). If a numpy.ndarray, it must have a shape broadcastable with t. If this parameter is used, you must also pass t and cannot pass o.
t (float or numpy.ndarray, optional) – Time(s). If a numpy.ndarray, it must have a shape broadcastable with r. Values must be positive. If this parameter is used, you must also pass r and cannot pass o.
o (float or numpy.ndarray, optional) – Value(s) of the Boltzmann variable. If this parameter is used, you cannot pass r or t.
- Returns
dS_do – If o is passed, the return is of the same type and shape as o. Otherwise, the return is a float if both r and t are floats, or a numpy.ndarray of the shape that results from broadcasting r and t.
- Return type
float or numpy.ndarray
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dS_dr(r, t)¶ \(\partial S/\partial r\), spatial derivative of S.
- Parameters
r (float or numpy.ndarray) – Location(s) along the coordinate. If a numpy.ndarray, it must have a shape broadcastable with t.
t (float or numpy.ndarray) – Time(s). If a numpy.ndarray, it must have a shape broadcastable with r. Values must be positive.
- Returns
dS_dr – The return is a float if both r and t are floats. Otherwise it is a numpy.ndarray of the shape that results from broadcasting r and t.
- Return type
float or numpy.ndarray
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dS_dt(r, t)¶ \(\partial S/\partial t\), time derivative of S.
- Parameters
r (float or numpy.ndarray) – Location(s). If a numpy.ndarray, it must have a shape broadcastable with t.
t (float or numpy.ndarray) – Time(s). If a numpy.ndarray, it must have a shape broadcastable with r. Values must be positive.
- Returns
dS_dt – The return is a float if both r and t are floats. Otherwise it is a numpy.ndarray of the shape that results from broadcasting r and t.
- Return type
float or numpy.ndarray
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flux(r, t)¶ Diffusive flux of S.
Returns the diffusive flux of S in the direction \(\mathbf{\hat{r}}\), equal to \(-D(S)\partial S/\partial r\).
- Parameters
r (float or numpy.ndarray) – Location(s). If a numpy.ndarray, it must have a shape broadcastable with t.
t (float or numpy.ndarray) – Time(s). If a numpy.ndarray, it must have a shape broadcastable with r. Values must be positive.
- Returns
flux – The return is a float if both r and t are floats. Otherwise it is a numpy.ndarray of the shape that results from broadcasting r and t.
- Return type
float or numpy.ndarray
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rb(t)¶ \(r_b\), the location of the boundary.
This is the point where the boundary condition of the problem is imposed.
- Parameters
t (float or numpy.ndarray) – Time(s). Values must be positive.
- Returns
rb – The return is of the same type and shape as t.
- Return type
float or numpy.ndarray
Notes
Depending on \(o_b\), the boundary may be fixed at \(r=0\) or it may move with time.