ODE

class probnum.diffeq.ODE(timespan, rhs, jac=None, hess=None, sol=None)

Bases: abc.ABC

Ordinary differential equations.

Extended by the types of ODEs, e.g. IVPs, BVPs. This class describes systems of first order ordinary differential equations (ODEs),

\[\dot y(t) = f(t, y(t)), \quad t \in [t_0, T].\]

It provides options for defining custom right-hand side (RHS) functions, their Jacobians and closed form solutions.

Parameters

• timespan ((float, float)) – Time span of IVP.

• rhs (callable, signature: (t, y, **kwargs)) – RHS function \(f : [t_0, T] \times \mathbb{R}^d \rightarrow \mathbb{R}^d\) of the ODE system. As such it takes a float and an np.ndarray of shape (d,) and returns a np.ndarray of shape (d,). As of now, no vectorization is supported (nor needed).

• jac (callable, signature: (t, y, **kwargs), optional) – Jacobian of RHS function \(J_f : [0, T] \times \mathbb{R}^d \rightarrow \mathbb{R}^d\) of the ODE system. As such it takes a float and an np.ndarray of shape (d,) and returns a np.ndarray of shape (d,). As of now, no vectorization is supported (nor needed).

• sol (callable, signature: (t, **kwargs), optional) – Solution of the ODE system. Only well-defined in subclasses like IVP or BVP.

See also

IVP

Extends ODE for initial value problems.

Attributes Summary

dimension	Abstract, in order to force subclassing.
t0
timespan	Returns \((t_0, T)\) as [self.t0, self.tmax].
tmax

Methods Summary

__call__(t, y, **kwargs)	Piggybacks on self.rhs(t, y).
hessian(t, y, **kwargs)	Hessian of model function f.
jacobian(t, y, **kwargs)	Jacobian of model function f.
rhs(t, y, **kwargs)	Evaluates model function f.
solution(t, **kwargs)	Solution of the IVP.

Attributes Documentation

dimension

Abstract, in order to force subclassing.

t0

timespan

Returns \((t_0, T)\) as [self.t0, self.tmax].

Mainly here to provide an interface to scipy.integrate. Both \(t_0\) and \(T\) can be accessed via self.t0 and self.tmax respectively.

tmax

Methods Documentation

__call__(t, y, **kwargs)

Piggybacks on self.rhs(t, y).

hessian(t, y, **kwargs)

Hessian of model function f.

For \(d=3\), the Hessian \(H_f(t, y) \in \mathbb{R}^{3 \times 3 \times 3}\) is expected be evaluated as

\[H_f(t, y) = \left[H_{f_1}(t, y), H_{f_2}(t, y), H_{f_3}(t, y) \right]^\top\]

since for any directions \(v_1, v_2\) the outcome of \(H_f(t_0, y_0) \cdot v_1 \cdot v_2\) is expected to contain the incline of \(f_i\) in direction \((v_1, v_2)\).

jacobian(t, y, **kwargs)

Jacobian of model function f.

rhs(t, y, **kwargs)

Evaluates model function f.

solution(t, **kwargs)

Solution of the IVP.