Speaker
Description
We consider a class of differential problems set in a Banach space, with integral boundary conditions:
\begin{equation}
\frac{dv}{dt} = Av, \qquad 0<t<T,\qquad \frac{1}{T}\int_0^T v(t) dt = f,
\end{equation}
where $A$ is a linear, closed, possibly unbounded operator (e.g., second derivative in space). Note that the finite-dimensional version of this problem, where $A$ is a matrix, is closely related to the task of computing matrix functions $\psi_{\ell}(A)$, where $\psi_{\ell}$ denotes reciprocals of the $\varphi_{\ell}$-functions used in exponential integrators.
We prove the existence and uniqueness of the solution $v(t)$ and characterize it via a family of mixed polynomial-rational expansions w.r.t. the operator $A$. From this result we design a general numerical procedure for computing an approximation of $v(t)$ up to a given tolerance. An interesting feature of this approach is the fact that successive rational terms can be computed independently: this allows us to fine-tune the accuracy of the approximation by adding further terms as needed, without having to recompute the whole approximation.
Numerical tests focus on a model problem involving a parabolic equation and highlight the effectiveness of this approach.
