General¶

compute_rad_robin_eigenfrequencies(param, l, n_roots=10, show_plot=False)[source]¶

Return the first n_roots eigenfrequencies $\omega$ (and eigenvalues $\lambda$ )

$\omega = \sqrt{-\frac{a_1^2}{4a_2^2}+\frac{a_0-\lambda}{a_2}}$

to the eigenvalue problem

$a_2\varphi''(z) + a_1&\varphi'(z) + a_0\varphi(z) = \lambda\varphi(z) \\ \varphi'(0) &= \alpha\varphi(0) \\ \varphi'(l) &= -\beta\varphi(l).$

Parameters:

param (array_like) – $\Big( a_2, a_1, a_0, \alpha, \beta \Big)^T$
l (numbers.Number) – Right boundary value of the domain $[0,l]\ni z$ .
n_roots (int) – Amount of eigenfrequencies to be compute.
show_plot (bool) – A plot window of the characteristic equation appears if it is True.

Returns:

$\Big(\big[\omega_1,...,\omega_\text{n\_roots}\Big], \Big[\lambda_1,...,\lambda_\text{n\_roots}\big]\Big)$

Return type:

tuple –> two numpy.ndarrays of length nroots

eliminate_advection_term(param, domain_end)[source]¶

This method performs a transformation

$\tilde x(z,t)=x(z,t) e^{\int_0^z \frac{a_1(\bar z)}{2 a_2}\,d\bar z} ,$

on the system, which eliminates the advection term $a_1 x(z,t)$ from a reaction-advection-diffusion equation of the type:

$\dot x(z,t) = a_2 x''(z,t) + a_1(z) x'(z,t) + a_0(z) x(z,t) .$

The boundary can be given by robin

$x'(0,t) = \alpha x(0,t), \quad x'(l,t) = -\beta x(l,t) ,$

dirichlet

$x(0,t) = 0, \quad x(l,t) = 0$

or mixed boundary conditions.

Parameters:	param (array_like) – $\Big( a_2, a_1, a_0, \alpha, \beta \Big)^T$ domain_end (float) – upper bound of the spatial domain
Raises:	`TypeError` – If $a_1(z)$ is callable but no derivative handle is defined for it.
Returns:	Parameters $\big(a_2, \tilde a_1=0, \tilde a_0(z), \tilde \alpha, \tilde \beta \big) for$ the transformed system $\dot{\tilde{x}}(z,t) = a_2 \tilde x''(z,t) + \tilde a_0(z) \tilde x(z,t)$ and the corresponding boundary conditions ( $\alpha$ and/or $\beta$ set to None by dirichlet boundary condition).
Return type:	SecondOrderOperator or tuple

get_parabolic_dirichlet_weak_form(init_func_label, test_func_label, input_handle, param, spatial_domain)[source]¶

Return the weak formulation of a parabolic 2nd order system, using an inhomogeneous dirichlet boundary at both sides.

Parameters:	init_func_label (str) – Label of shape base to use. test_func_label (str) – Label of test base to use. input_handle (`SimulationInput`) – Input. param (tuple) – Parameters of the spatial operator. spatial_domain (#) – Spatial domain of the problem. spatial_domain – Spatial domain of the problem. (#) –
Returns:	Weak form of the system.
Return type:	`WeakFormulation`

get_parabolic_robin_weak_form(shape_base_label, test_base_label, input_handle, param, spatial_domain, actuation_type_point=None)[source]¶

Parameters:	shape_base_label – test_base_label – input_handle – param – spatial_domain – actuation_type_point –
Returns:

get_in_domain_transformation_matrix(k1, k2, mode='n_plus_1')[source]¶

Returns the transformation matrix M. M is one part of a transformation

where x is the field variable of an interior point controlled parabolic system and y is the field variable of an boundary controlled parabolic system. T is a (Fredholm-) integral transformation (which can be approximated with M).

Parameters:	k1 – k2 – mode – Available modes: ’n_plus_1’: M.shape = (n+1,n+1), w = (w(0),…,w(n))^T, w in {x,y} ‘2n’: M.shape = (2n,2n), w = (w(0),…,w(n),…,w(1))^T, w in {x,y}
Returns:	Transformation matrix M.
Return type:	numpy.array