fronts.Solution

class fronts.Solution(sol, ob, oi, D)

Solution to a problem.

Represents a continuously differentiable function \(\theta\) of r and t such that:

\[\dfrac{\partial\theta}{\partial t} = \nabla\cdot\left[D(\theta) \dfrac{\partial \theta}{\partial r}\mathbf{\hat{r}}\right]\]

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 an ODE obtained with ode. For any float or one-dimensional NumPy array o with values in the closed interval [ob, oi], sol(o)[0] are the values of \(\theta\) at o, and sol(o)[1] are the values of the derivative \(d\theta/do\) at o`. sol will only be evaluated in this interval.

  • ob (float) – Parameter \(o_b\), which determines the behavior of the boundary in the problem.

  • oi (float) – Value of the Boltzmann variable at which the solution can be considered to be equal to the initial condition. Cannot be less than ob.

  • D (callable) – Function to evaluate \(D\) at arbitrary values of the solution. Must be callable with a float or NumPy array as its argument.

__init__(sol, ob, oi, D)

Methods

__init__(sol, ob, oi, D)

d_do([r, t, o])

Boltzmann-variable derivative of the solution.

d_dr(r, t)

Spatial derivative of the solution.

d_drb(t)

Spatial derivative of the solution at the boundary.

d_dt(r, t)

Time derivative of the solution.

d_dtb(t)

Time derivative of the solution at the boundary.

flux(r, t)

Diffusive flux.

fluxb(t)

Boundary flux.

rb(t)

Boundary location.

sorptivity(*[, o])

Sorptivity.

Attributes

b

Boundary value of the solution.

d_dob

Derivative of the solution with respect to the Boltzmann variable at the boundary.

i

Initial value of the solution.

ob

Parameter \(o_b\).

oi

Value of the Boltzmann variable at which the solution can be considered to be equal to the initial condition.
__call__(r=None, t=None, o=None)

Evaluate the solution.

Evaluates and returns \(\theta\). May be called either with arguments r and t, or with just o.

Parameters:

  • r (None or float or numpy.ndarray, shape (n,), optional) – Location(s). If this parameter is used, t must also be given.

  • t (None or float or numpy.ndarray, optional) – Time(s). Values must be positive.

  • o (None or float or numpy.ndarray, shape (n,) optional) – Value(s) of the Boltzmann variable. If this parameter is used, neither r nor t can be given.

Return type:

float or numpy.ndarray, shape (n,)

property b

Boundary value of the solution.

Type:

float

d_do(r=None, t=None, o=None)

Boltzmann-variable derivative of the solution.

Evaluates and returns \(d\theta/do\), the derivative of \(\theta\) with respect to the Boltzmann variable. May be called either with arguments r and t, or with just o.

Parameters:

  • r (None or float or numpy.ndarray, shape (n,), optional) – Location(s). If this parameter is used, t must also be given.

  • t (None or float or numpy.ndarray, shape (n,), optional) – Time(s). Values must be positive.

  • o (None or float or numpy.ndarray, shape (n,), optional) – Value(s) of the Boltzmann variable. If this parameter is used, neither r nor t can be given.

Return type:

float or numpy.ndarray, shape (n,)

property d_dob

Derivative of the solution with respect to the Boltzmann variable at the boundary.

Type:

float

d_dr(r, t)

Spatial derivative of the solution.

Evaluates and returns \(\partial\theta/\partial r\).

Parameters:
Return type:

float or numpy.ndarray, shape (n,)

d_drb(t)

Spatial derivative of the solution at the boundary.

Evaluates and returns \(\partial\theta/\partial r|_b\). Equivalent to self.d_dr(self.rb(t), t).

Parameters:

t (float or numpy.ndarray, shape (n,)) – Time(s). Values must be positive.

Return type:

float or numpy.ndarray, shape (n,)

d_dt(r, t)

Time derivative of the solution.

Evaluates and returns \(\partial\theta/\partial t\).

Parameters:
Return type:

float or numpy.ndarray, shape (n,)

d_dtb(t)

Time derivative of the solution at the boundary.

Evaluates and returns \(\partial\theta/\partial t|_b\). Equivalent to self.d_dt(self.rb(t), t).

Parameters:

t (float or numpy.ndarray, shape (n,)) – Time(s). Values must be positive.

Return type:

float or numpy.ndarray, shape (n,)

flux(r, t)

Diffusive flux.

Returns the diffusive flux of \(\theta\) in the direction \(\mathbf{\hat{r}}\), equal to \(-D(\theta)\partial\theta/\partial r\).

Parameters:
Return type:

float or numpy.ndarray, shape (n,)

fluxb(t)

Boundary flux.

Returns the diffusive flux of \(\theta\) at the boundary, in the direction \(\mathbf{\hat{r}}\). Equivalent to self.flux(self.rb(t), t).

Parameters:

t (float or numpy.ndarray, shape (n,)) – Time(s). Values must be positive.

Return type:

float or numpy.ndarray, shape (n,)

property i

Initial value of the solution.

Type:

float

property ob

Parameter \(o_b\).

Type:

float

property oi

Value of the Boltzmann variable at which the solution can be considered to be equal to the initial condition.

Type:

float

rb(t)

Boundary location.

Returns \(r_b\), the location of the boundary.

Depending on \(o_b\), the boundary may be fixed at \(r=0\) or it may move with time.

Parameters:

t (float or numpy.ndarray) – Time(s). Values must not be negative.

Returns:

rb – The return is of the same type and shape as t.

Return type:

float or numpy.ndarray

sorptivity(*, o=None)

Sorptivity.

Returns the sorptivity \(S\) of \(\theta\), equal to \(-2D(\theta)\partial\theta/\partial o\).

Parameters:

o (float or numpy.ndarray, shape (n,), optional) – Value(s) of the Boltzmann variable. If not given, the method will return the sorptivity at the boundary.

Return type:

float or numpy.ndarray, shape (n,)

References

[1] PHILIP, J. R. The theory of infiltration: 4. Sorptivity and algebraic infiltration equations. Soil Science, 1957, vol. 84, no. 3, pp. 257-264.