Exponential quadrature rules for problems with time-dependent fractional source

In this manuscript, we propose newly-derived exponential quadrature rules for stiff linear differential equations with time-dependent fractional sources in the form ℎ(𝑡^𝑟 ), with 0 < 𝑟 < 1 and ℎ a sufficiently smooth function. To construct the methods, the source term is interpolated at 𝜈 collocation points by a suitable non-polynomial function, yielding to time marching schemes that we call Exponential Quadrature Rules for Fractional sources (EQRF𝜈). We write the integrators in terms of special instances of the Mittag–Leffler functions that we call fractional 𝜑 functions. We perform the error analysis of the schemes in the abstract framework of strongly continuous semigroups. Compared to classical exponential quadrature rules, which in our case of interest converge with order 1 + 𝑟 at most, we prove that the new methods may reach order 1 + 𝜈𝑟 for proper choices of the collocation points. Several numerical experiments demonstrate the theoretical findings and highlight the effectiveness of the approach.