Eigenfunctions¶

This modules provides eigenfunctions for a certain set of parabolic problems. Therefore functions for the computation of the corresponding eigenvalues are included. The functions which compute the eigenvalues are deliberately separated from the predefined eigenfunctions in order to handle transformations and reduce effort by the controller implementation.

class AddMulFunction(function)[source]¶

Bases: object

(Temporary) Function class wich can multiplied with scalars and added with functions. Only needed to compute the matrix (of scalars) vector (of functions) product in FiniteTransformFunction. Will be no longer needed when pyinduct.core.Function is overloaded with __add__ and __mul__ operator.

Parameters:	function (callable) –

class FiniteTransformFunction(function, M, l, scale_func=None, nested_lambda=False)[source]¶

Bases: pyinduct.core.Function

Provide a transformed pyinduct.core.Function $\bar x(z)$ through the transformation $\bar{\boldsymbol{\xi}} = T * \boldsymbol \xi$ , with the function vector $\boldsymbol \xi\in\mathbb R^{2n}$ and with a given matrix $T\in\mathbb R^{2n\times 2n}$ . The operator $*$ denotes the matrix (of scalars) vector (of functions) product. The interim result $\bar{\boldsymbol{\xi}}$ is a vector

$\bar{\boldsymbol{\xi}} = (\bar\xi_{1,0},...,\bar\xi_{1,n-1},\bar\xi_{2,0},...,\bar\xi_{2,n-1})^T.$

of functions

$&\bar\xi_{1,j} = \bar x(jl_0 + z),\qquad j=0,...,n-1, \quad l_0=l/n, \quad z\in[0,l_0] \\ &\bar\xi_{2,j} = \bar x(l - jl_0 + z).$

Finally, the provided function $\bar x(z)$ is given through $\bar\xi_{1,0},...,\bar\xi_{1,n-1}$ .

Note

For a more extensive documentation see section 4.2 in:

Wang, S. und F. Woittennek: Backstepping-Methode für parabolische Systeme mit punktförmigem inneren Eingriff. Automatisierungstechnik, 2015.

http://dx.doi.org/10.1515/auto-2015-0023

Parameters:

function (callable) –
Function $x(z)$ which will subdivided in $2n$ Functions

$&\bar\xi_{1,j} = x(jl_0 + z),\qquad j=0,...,n-1, \quad l_0=l/n, \quad z\in[0,l_0] \\ &\bar\xi_{2,j} = x(l - jl_0 + z).$

The vector of functions $\boldsymbol\xi$ consist of these functions:

$\boldsymbol\xi = (\xi_{1,0},...,\xi_{1,n-1},\xi_{2,0},...,\xi_{2,n-1})^T) .$
M (numpy.ndarray) – Matrix $T\in\mathbb R^{2n\times 2n}$ of scalars.
l (numbers.Number) – Length of the domain ( $z\in[0,l]$ ).

class SecondOrderDirichletEigenfunction(omega, param, spatial_domain, norm_fac=1.0)[source]¶

Bases: pyinduct.core.Function

Provide the eigenfunction $\varphi(z)$ to an eigenvalue problem of the form

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

The eigenfrequency

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

must be provided.

Parameters:	om (numbers.Number) – eigenfrequency $\omega$ param (array_like) – $\Big( a_2, a_1, a_0, None, None \Big)^T$ spatial_domain (tuple) – Start point $z_0$ and end point $z_1$ of the spatial domain $[z_0,z_1]\ni z$ . norm_fac (numbers.Number) – Factor to scale the eigenfunctions.

class SecondOrderRobinEigenfunction(om, param, spatial_domain, phi_0=1)[source]¶

Bases: pyinduct.core.Function

Provide the eigenfunction $\varphi(z)$ to an eigenvalue problem of the form

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

The eigenfrequency

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

must be provided (with compute_rad_robin_eigenfrequencies).

Parameters:	om (numbers.Number) – eigenfrequency $\omega$ param (array_like) – $\Big( a_2, a_1, a_0, \alpha, \beta \Big)^T$ spatial_domain (tuple) – Start point $z_0$ and end point $z_1$ of the spatial domain $[z_0,z_1]\ni z$ . phi_0 (numbers.Number) – Factor to scale the eigenfunctions (correspond $\varphi(0)=\text{phi\_0}$ ).

class TransformedSecondOrderEigenfunction(target_eigenvalue, init_state_vect, dgl_coefficients, domain)[source]¶

Bases: pyinduct.core.Function

Provide the eigenfunction $\varphi(z)$ to an eigenvalue problem of the form

$a_2(z)\varphi''(z) + a_1(z)\varphi'(z) + a_0(z)\varphi(z) = \lambda\varphi(z)$

where $\lambda$ is a predefined (potentially complex) eigenvalue and $[z_0,z_1]\ni z$ is the domain.

Parameters:	target_eigenvalue (numbers.Number) – $\lambda$ init_state_vect (array_like) – $\Big(\text{Re}\{\varphi(0)\}, \text{Re}\{\varphi'(0)\}, \text{Im}\{\varphi(0)\}, \text{Im}\{\varphi'(0)\}\Big)^T$ dgl_coefficients (array_like) – $\Big( a2(z), a1(z), a0(z) \Big)^T$ domain (array_like) – $\Big( z_0, ..... , z_1 \Big)$

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 –> booth tuple elements are numpy.ndarrays of length `nroots`

get_adjoint_rad_evp_param(param)[source]¶

Return to the eigen value problem of the reaction-advection-diffusion equation with robin and/or dirichlet boundary conditions

$a_2\varphi''(z) + a_1&\varphi'(z) + a_0\varphi(z) = \lambda\varphi(z) \\ \varphi(0) = 0 \quad &\text{or} \quad \varphi'(0) = \alpha\varphi(0) \\ \varphi`(l) = 0 \quad &\text{or} \quad \varphi'(l) = -\beta\varphi(l)$

the parameters for the adjoint problem (with the same structure).

Parameters:	param (array_like) – $\Big( a_2, a_1, a_0, \alpha, \beta \Big)^T$
Returns:	Parameters $\big(a_2, \tilde a_1=-a_1, a_0, \tilde \alpha, \tilde \beta \big)$ for the adjoint problem $a_2\psi''(z) + a_1&\psi'(z) + a_0\psi(z) = \lambda\psi(z) \\ \psi(0) = 0 \quad &\text{or} \quad \psi'(0) = \tilde\alpha\psi(0) \\ \psi`(l) = 0 \quad &\text{or} \quad \psi'(l) = -\tilde\beta\psi(l).$
Return type:	tuple

return_real_part(to_return)[source]¶

Check if the imaginary part of to_return vanishes and return the real part.

Parameters:	to_return (numbers.Number or array_like) – Variable to check.
Raises:	`ValueError` – If (all) imaginary part(s) not vanishes.
Returns:	Real part of `to_return`.
Return type:	numbers.Number or array_like

transform2intermediate(param, d_end=None)[source]¶

Transformation $\tilde x(z,t)=x(z,t)e^{\int_0^z \frac{a_1(\bar z)}{2 a_2}\,d\bar z}$ which eliminate the advection term $a_1 x(z,t)$ from the reaction-advection-diffusion equation

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

with robin

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

or dirichlet

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

or mixed boundary condition.

Parameters:	param (array_like) – $\Big( a_2, a_1, a_0, \alpha, \beta \Big)^T$
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:	tuple